text: DUALITY THEOREMS 2 (cid:0) (cid:1) Let θ be a non-constant inner function in the open unit disk D. For each complex number α of unit modulus the function Re α−+θ is positive and harmonic in D. α θ Hence there exists the unique positive Borel measure σ supported on the unit circle T such that (cid:90) α − | | Re α +− θ(z) = |1 − z |2 dσ (ξ), z ∈ D. (3) α θ(z) T 1 ξ¯z 2 α { } The measures σ | | are usually referred to as Clark measures of the inner α α =1 function θ due to seminal work [11] of D. N. Clark where their close connection to rank-one perturbations of singular unitary operators was discovered. For a modern exposition of this topic and subsequent results see survey [21]. Each Clark measure σ of an inner function θ is singular with respect to the Lebesgue measure on theαunit circle T. Conversely if µ is a ﬁnite positive Borel singular measure supported on T and |α| = 1, then there exists the unique inner function θ satisfying (3) with σ = µ. Thus, there is one-to-one correspondence be- tween inner functions in the unαit disk D and singular measures on the unit circle T. It was unknown which singular measures on T correspond to the Clark measures of one-component inner functions. We ﬁll this gap in Theorem 1 below. For every Borel measure µ on the unit circle T denote by a(µ) the set of isolated \ atoms of µ. Then the set ρ(µ) = supp µ a(µ) consists of accumulating points in ∈ the support supp µ of µ. We will say that an atom ξ a(µ) has two neighbours in a(µ) if there is an open arc (ξ−, ξ ) of the unit circle T with endpoints ξ± ∈ a(µ) + ∩ such that ξ is the only point in (ξ−, ξ ) supp µ. By m we will denote the Lebesgue measure on T normalized so that m(+T) = 1. | | Theorem 1. Let α = 1. The following conditions are necessary and suﬃcient for a Borel measure µ to be the Clark measure σ of a one-component inner function: α (a) µ is a discrete measure on T with isolated atoms, m(supp µ) = 0, every ∈ atom ξ a(µ) has two neighbours ξ± in a(µ), and every connected compo- nent of T \ ρ(µ) contains atoms of µ; (cid:82) | − | (cid:54) { } (cid:54) | − | ∈ ∞ (b) A ξ ξ± µ ξ B ξ ξ± for all ξ a(µ) and some A > 0, B < ; µ µ µ µ (c) the discre|te Hilbert| (cid:54)transform (Hµ1∈)(z) = T\{z} 1d−µ(ξ¯ξz) is bounded on a(µ): we have (H 1)(z) C for all z a(µ). µ µ The necessity of conditions (a) and (b) in Theorem 1 is well-known. I would like to thank A. D. Baranov who tell me the fact that condition (c) is necessary as well. The proof of suﬃciency part in Theorem 1 relies on a characterization of one-component inner functions in terms of their derivatives which is due to A. B. Aleksandrov [3]. Having a description of the Clark measures of one-component inner functions, − we now turn back (cid:26)to formula (2). For a measure µ(cid:90)with properties (a) ((cid:27)c) deﬁne the space BMO(µ) by ∈ (cid:107) (cid:107) 1 | − (cid:104) (cid:105) | ∞ BMO(µ) = b L1(µ) : b ∗ = sup b b dµ < , µ µ(∆) ∆,µ ∆ ∆ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

text: THEOREMS

text: Let

text: θ

text: be

text: a

text: non-constant

text: inner

text: function

text: in

text: the

text: open

text: unit

text: disk

text: D.

text: For

text: each

text: complex

text: number

text: α

text: of

text: unit

text: modulus

text: the

text: function

text: Re

text: α−+θ

text: is

text: positive

text: and

text: harmonic

text: in

text: D.

text: α

text: θ

text: Hence

text: there

text: exists

text: the

text: unique

text: positive

text: Borel

text: measure

text: σ

text: supported

text: on

text: the

text: unit

text: circle

text: T

text: such

text: that

label: equation

text: α

text: Re

text: α

text: θ(z)

text: z

text: dσ

text: (ξ),

text: z

text: D.

text: α

text: θ(z)

text: T

text: ξ¯z

text: α

text: The

text: measures

text: σ

text: are

text: usually

text: referred

text: to

text: as

text: Clark

text: measures

text: of

text: the

text: inner

text: α

text: α

text: function

text: θ

text: due

text: to

text: seminal

text: work

text: of

text: D.

text: N.

text: Clark

text: where

text: their

text: close

text: connection

text: to

text: rank-one

text: perturbations

text: of

text: singular

text: unitary

text: operators

text: was

text: discovered.

text: For

text: a

text: modern

text: exposition

text: of

text: this

text: topic

text: and

text: subsequent

text: results

text: see

text: survey

text: Each

text: Clark

text: measure

text: σ

text: of

text: an

text: inner

text: function

text: θ

text: is

text: singular

text: with

text: respect

text: to

text: the

text: Lebesgue

text: measure

text: on

text: theαunit

text: circle

text: T.

text: Conversely

text: if

text: µ

text: is

text: a

text: ﬁnite

text: positive

text: Borel

text: singular

text: measure

text: supported

text: on

text: T

text: and

text: |α|

text: then

text: there

text: exists

text: the

text: unique

text: inner

text: function

text: θ

text: satisfying

text: with

text: σ

text: µ.

text: Thus,

text: there

text: is

text: one-to-one

text: correspondence

text: be-

text: tween

text: inner

text: functions

text: in

text: the

text: unαit

text: disk

text: D

text: and

text: singular

text: measures

text: on

text: the

text: unit

text: circle

text: T.

text: It

text: was

text: unknown

text: which

text: singular

text: measures

text: on

text: T

text: correspond

text: to

text: the

text: Clark

text: measures

text: of

text: one-component

text: inner

text: functions.

text: We

text: ﬁll

text: this

text: gap

text: in

text: Theorem

text: below.

text: For

text: every

text: Borel

text: measure

text: µ

text: on

text: the

text: unit

text: circle

text: T

text: denote

text: by

text: a(µ)

text: the

text: set

text: of

text: isolated

text: atoms

text: of

text: µ.

text: Then

text: the

text: set

text: ρ(µ)

text: supp

text: µ

text: a(µ)

text: consists

text: of

text: accumulating

text: points

text: in

text: the

text: support

text: supp

text: µ

text: of

text: µ.

text: We

text: will

text: say

text: that

text: an

text: atom

text: ξ

text: a(µ)

text: has

text: two

text: neighbours

text: in

text: a(µ)

text: if

text: there

text: is

text: an

text: open

text: arc

text: (ξ−,

text: ξ

text: of

text: the

text: unit

text: circle

text: T

text: with

text: endpoints

text: ξ±

text: a(µ)

text: such

text: that

text: ξ

text: is

text: the

text: only

text: point

text: in

text: (ξ−,

text: ξ

text: supp

text: µ.

text: By

text: m

text: we

text: will

text: denote

text: the

text: Lebesgue

text: measure

text: on

text: T

text: normalized

text: so

text: that

text: m(+T)

text: Theorem

text: Let

text: α

text: The

text: following

text: conditions

text: are

text: necessary

text: and

text: suﬃcient

text: for

text: a

text: Borel

text: measure

text: µ

text: to

text: be

text: the

text: Clark

text: measure

text: σ

text: of

text: a

text: one-component

text: inner

text: function:

text: α

text: (a)

text: µ

text: is

text: a

text: discrete

text: measure

text: on

text: T

text: with

text: isolated

text: atoms,

text: m(supp

text: µ)

text: every

text: atom

text: ξ

text: a(µ)

text: has

text: two

text: neighbours

text: ξ±

text: in

text: a(µ),

text: and

text: every

text: connected

text: compo-

text: nent

text: of

text: T

text: ρ(µ)

text: contains

text: atoms

text: of

text: µ;

text: (b)

text: A

text: ξ

text: ξ±

text: µ

text: ξ

text: B

text: ξ

text: ξ±

text: for

text: all

text: ξ

text: a(µ)

text: and

text: some

text: A

text: B

text: µ

text: µ

text: µ

text: µ

text: (c)

text: the

text: discre|te

text: Hilbert|

text: T\{z}

text: 1d−µ(ξ¯ξz)

text: is

text: bounded

text: on

text: a(µ):

text: we

text: have

text: (H

text: C

text: for

text: all

text: z

text: a(µ).

text: µ

text: µ

text: The

text: necessity

text: of

text: conditions

text: (a)

text: and

text: (b)

text: in

text: Theorem

text: is

text: well-known.

text: I

text: would

text: like

text: to

text: thank

text: A.

text: D.

text: Baranov

text: who

text: tell

text: me

text: the

text: fact

text: that

text: condition

text: (c)

text: is

text: necessary

text: as

text: well.

text: The

text: proof

text: of

text: suﬃciency

text: part

text: in

text: Theorem

text: relies

text: on

text: a

text: characterization

text: of

text: one-component

text: inner

text: functions

text: in

text: terms

text: of

text: their

text: derivatives

text: which

text: is

text: due

text: to

text: A.

text: B.

text: Aleksandrov

text: Having

text: a

text: description

text: of

text: the

text: Clark

text: measures

text: of

text: one-component

text: inner

text: functions,

text: we

text: now

text: turn

text: back

text: formula

text: For

text: a

text: measure

text: properties

text: (a)

text: deﬁne

text: the

text: space

text: BMO(µ)

text: by

text: BMO(µ)

text: b

text: b

text: sup

text: b

text: b

text: dµ

text: µ

text: µ(∆)

text: ∆,µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DUALITY THEOREMS 12 Proof. It is suﬃcient to prove the statement in the case where ∆ contains only ﬁnite number of atoms of σ . Enumerate the atoms clockwise: ξ , . . . , ξ . Find the α \ 1 n neighbours of ξ , ξ in a(σ ) ∆ and denote them by ξ and ξ , correspondingly. 1 n (cid:88) α (cid:88) 0 n+1 We have (cid:54) n (cid:54) 2 n { } 2 m(∆) m[ξ , ξ ] σ ξ = σ (∆). k k+1 A˜ α k A˜ α k=0 σα k=1 σα (cid:62) In the case where n 2 we have (cid:88) n { } (cid:54) (cid:54) \ σ (∆) = σ ξ 2B˜ m(∆) 2B˜ m(∆ D (κ)) + B˜ κσ (∆). α α k σα σα σα σα α k=1 (cid:54) − (cid:54) \ Now use the assumption κ (2B˜ ) 1 and get σ (∆) 4B˜ m(∆ D (κ)). σα α σα σα (cid:3) Lemma 3.2. There exists ε > 0 such that |α − θ(z)| (cid:62) ε for all z ∈ D \ D (κ). σα Proof. Let δ ∈ (0, 1) be a number such that the set Ω = {z ∈ D : |θ(z)| < 1} is δ connected. The set { ∈ D ∪ | | } Ω = z G : δ < θ(z) < 1/δ δ,1 θ δ O is at most countable union of the open connected components, . It was proved k O by B. Cohn [12] that the restriction of the inner function θ to each of the sets is a covering map from O to the ring R = {z ∈ C : δ < |z| < 1/δ}. Take a positkive k− δ number ε < min(κ/2, 1 δ). We claim that every connected component E of the set L = {z ∈ D ∪ G : |α − θ(z)| < ε} contains an atom of σ . Indeed, we have ⊂ Oε θ ⊂ α O E for some index k because L Ω . Since θ is a covering map from to k ε δ,1 k R , there exists a number ε (which can beδ taken to be less than ε) such that the δ { | − | 1} O preimage of ζ : α ζ < ε under θ on is at most countable union of the open O ⊂ O 1 k O { | − | } disjoint sets and θ is a homeomorphism from to ζ : ζ α < ε km k | − km| 1 for every m. By the minimum principle, inf ∈ θ(z) α = 0. It follows that ∩ O (cid:54) ∅ z E | − | E = for some index m. Since E is connected and θ α < ε < ε on O km O ⊂ O 1 , we have E. But every set contains the unique point ξ with km km ∈ km ∩ (cid:54) ∅ θ(ξ) = α. By Lemma 2.1, ξ a(σ ) and thus E a(σ ) = . To prove the lemma ⊂ α ∈α ∩ it is suﬃcient to show that E D (κ). For every z E D (κ) we get from (14) ξ ξ the estimate | − | (cid:54) | − | { } (cid:54) { } ξ z 2 α θ(z) σ ξ 2εσ ξ . α α Hence E does not intersect the circle {z ∈ C : |z−ξ| = rσ {ξ}} for every r ∈ (2ε, κ). α ⊂ (cid:3) Since the set E is connected this yields the desired inclusion E D (κ). ξ Lemma 3.3. Let θ be an inner function with ρ(θ) (cid:54)= T. Then every function in K1 admits the analytic continuation from the unit disk D to the domain D ∪ G . θ θ Consequently, if σ (ρ(θ)) = 0 for a Clark measure σ of θ, then every function in K1 has a trace oαn the set a(σ ) ⊂ D ∪ G of full meαasure σ . θ α θ α ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

label: paragraph

text: THEOREMS

text: Proof.

text: It

text: is

text: suﬃcient

text: to

text: prove

text: the

text: statement

text: in

text: the

text: case

text: where

text: contains

text: only

text: ﬁnite

text: number

text: of

text: atoms

text: of

text: σ

text: Enumerate

text: the

text: atoms

text: clockwise:

text: ξ

text: ξ

text: Find

text: the

text: α

text: n

text: neighbours

text: of

text: ξ

text: ξ

text: in

text: a(σ

text: and

text: denote

text: them

text: by

text: ξ

text: and

text: ξ

text: correspondingly.

text: n

text: α

text: We

text: have

text: n

text: n

text: m(∆)

text: m[ξ

text: ξ

text: σ

text: ξ

text: σ

text: k

text: A˜

text: α

text: k

text: A˜

text: α

text: σα

text: σα

text: In

text: the

text: case

text: where

text: n

text: we

text: have

text: n

text: σ

text: σ

text: ξ

text: m(∆)

text: m(∆

text: D

text: (κ))

text: B˜

text: κσ

text: α

text: α

text: k

text: σα

text: σα

text: σα

text: σα

text: α

text: Now

text: use

text: the

text: assumption

text: κ

text: and

text: get

text: σ

text: m(∆

text: D

text: (κ)).

text: σα

text: α

text: σα

text: σα

text: Lemma

text: There

text: exists

text: ε

text: such

text: that

text: |α

text: θ(z)|

text: ε

text: for

text: all

text: z

text: D

text: D

text: (κ).

text: σα

text: Proof.

text: Let

text: δ

text: be

text: a

text: number

text: such

text: that

text: the

text: set

text: Ω

text: {z

text: D

text: |θ(z)|

text: is

text: δ

text: connected.

text: The

text: set

text: D

text: Ω

text: z

text: G

text: δ

text: θ(z)

text: θ

text: δ

text: O

text: is

text: at

text: most

text: countable

text: union

text: of

text: the

text: open

text: connected

text: components,

text: It

text: was

text: proved

text: k

text: O

text: by

text: B.

text: Cohn

text: that

text: the

text: restriction

text: of

text: the

text: inner

text: function

text: θ

text: to

text: each

text: of

text: the

text: sets

text: is

text: a

text: covering

text: map

text: from

text: O

text: to

text: the

text: ring

text: R

text: {z

text: C

text: δ

text: |z|

text: Take

text: a

text: positkive

text: k−

text: δ

text: number

text: ε

text: δ).

text: We

text: claim

text: that

text: every

text: connected

text: component

text: E

text: of

text: the

text: set

text: L

text: {z

text: D

text: G

text: |α

text: θ(z)|

text: ε}

text: contains

text: an

text: atom

text: of

text: σ

text: Indeed,

text: we

text: have

text: Oε

text: θ

text: α

text: O

text: E

text: for

text: some

text: index

text: k

text: because

text: L

text: Ω

text: Since

text: θ

text: is

text: a

text: covering

text: map

text: from

text: to

text: k

text: ε

text: k

text: R

text: there

text: exists

text: a

text: number

text: ε

text: (which

text: can

text: beδ

text: taken

text: to

text: be

text: less

text: than

text: ε)

text: such

text: that

text: the

text: δ

text: O

text: preimage

text: of

text: ζ

text: α

text: ζ

text: ε

text: under

text: θ

text: on

text: is

text: at

text: most

text: countable

text: union

text: of

text: the

text: open

text: O

text: O

text: k

text: O

text: disjoint

text: sets

text: and

text: θ

text: is

text: a

text: homeomorphism

text: from

text: to

text: ζ

text: ζ

text: α

text: ε

text: km

text: k

text: km|

text: for

text: every

text: m.

text: By

text: the

text: minimum

text: principle,

text: inf

text: θ(z)

text: α

text: It

text: follows

text: that

text: O

text: z

text: E

text: E

text: for

text: some

text: index

text: m.

text: Since

text: E

text: is

text: connected

text: and

text: θ

text: α

text: ε

text: ε

text: on

text: O

text: km

text: O

text: O

text: we

text: have

text: E.

text: But

text: every

text: set

text: contains

text: the

text: unique

text: point

text: ξ

text: with

text: km

text: km

text: km

text: θ(ξ)

text: α.

text: By

text: Lemma

text: ξ

text: a(σ

text: and

text: thus

text: E

text: a(σ

text: To

text: prove

text: the

text: lemma

text: α

text: ∈α

text: it

text: is

text: suﬃcient

text: to

text: show

text: that

text: E

text: D

text: (κ).

text: For

text: every

text: z

text: E

text: D

text: (κ)

text: we

text: get

text: from

text: ξ

text: ξ

text: the

text: estimate

text: ξ

text: z

text: α

text: θ(z)

text: σ

text: ξ

text: 2εσ

text: ξ

text: α

text: α

text: Hence

text: E

text: does

text: not

text: intersect

text: the

text: circle

text: {z

text: C

text: |z−ξ|

text: rσ

text: {ξ}}

text: for

text: every

text: r

text: κ).

text: α

text: Since

text: the

text: set

text: E

text: is

text: connected

text: this

text: yields

text: the

text: desired

text: inclusion

text: E

text: D

text: (κ).

text: ξ

text: Lemma

text: Let

text: θ

text: be

text: an

text: inner

text: function

text: with

text: ρ(θ)

text: T.

text: Then

text: every

text: function

text: in

text: admits

text: the

text: analytic

text: continuation

text: from

text: the

text: unit

text: disk

text: D

text: to

text: the

text: domain

text: D

text: G

text: θ

text: θ

text: Consequently,

text: if

text: σ

text: (ρ(θ))

text: for

text: a

text: Clark

text: measure

text: σ

text: of

text: θ,

text: then

text: every

text: function

text: in

text: has

text: a

text: trace

text: oαn

text: the

text: set

text: a(σ

text: D

text: G

text: of

text: full

text: meαasure

text: σ

text: θ

text: α

text: θ

text: α

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DUALITY THEOREMS 13 Proof. For every function(cid:90) F ∈ K1 we have θ¯F ∈ zH1 on T. Hence, θ − F(z) = F(ξ)1 θ−(z)θ(ξ) dm(ξ), z ∈ D. (27) T 1 zξ¯ Extend the inner function θ to the domain D ∪ G by formula (10). The right θ hand side of (27) then determines the analytic continuation of the function F to D ∪ G . By Lemma 2.1 we have ρ(σ ) = ρ(θ) which completes the proof. (cid:3) θ α ∈ ∩ Lemma 3.4. Let θ be an inner function and let G K1 zH1. Then there exist functions G , G ∈ K1 ∩zH1 such that G = G +iG andθ G = θG on T\ρ(θ). 1 2 (cid:107) θ (cid:107) (cid:54) (cid:107) (cid:107) 1 2 1,2 1,2 Moreover, we have G T G T . 1,2 L1( ) L1( ) Proof. Consider the function G˜ = θG on the unit circle T. We have ∈ ∩ ∩ ∩ G˜ θ(zH1 zθ¯H1) = z¯θH1 zH1 = K1 zH1. θ This shows that G can be continued to the open unit disk D as a function from ∩ − the space K1 zH1. Now put G = (G + G˜)/2, G = (G G˜ )/2i and obtain the θ 1 2 1 (cid:3) desired representation. Let µ be a measure on the unit circle T with properties (a) − (c) and let |α| = 1. Consider the one-component inner function θ with the Clark measure σ = µ. Take a function F ∈ K1 ∩ zH1. By Lemma 3.3, F is analytic on the domαain D ∪ G . Denote by f its tθrace on the set a(σ ) ⊂ D ∪ G of full measure σ . Our aimθ ∈ (cid:107) (cid:107)α (cid:46) (cid:107)θ (cid:107) α is to prove that f H1 (σ ) and f F T . At ﬁrst, assume that F ∈ K2 ∩ zH2 and F =atθFαon T \ ρ(θH).a1tW(σαe) will neeLd1(th)e following modiﬁcation θ of the Lusin-Privalov construction (see Section III.D in [16] for the standard one). Consider the non-tangential maximal function of F, ∗ | | ∈ T F (ξ) = sup F(z) , ξ , ∈ z Λξ √ where Λ denotes the convex hull of the set {ξ} ∪ {z ∈ D : |z| (cid:54) 1/ 2}. Put ξ S (λ) = D \ {z ∈ D : z ∈ Λ for some ξ ∈ T with F∗(ξ) < λ}. F ξ Let D (κ) be the set deﬁned at the beginning of Section . By Lemma 3.2, we have |ασα− θ| (cid:62) ε on D \ D (κ). Denote by R (λ) the union of those connected components of the set S (σλα) ∪ D (κ) for whFich we have E ∩ S (λ) (cid:54)= ∅ and E ∩ D (κ) (cid:54)= ∅. The setsFR (λ) aσreα closed and have the following Fproperties: σα F ⊂ (1) If λ < λ , then R (λ ) R (λ ); (2) |F(z1)| (cid:54) 2λ for σ -aFlmo2st all pFoin1ts z ∈ T \ R (λ); (3) |F(z)| (cid:54) λ and |αα − θ(z)| (cid:62) ε for z ∈ ∂R (λ)F∩ D. F More special properties of the sets R (λ) are collected in the following lemma. F Lemma 3.5. Let E be a connected component of the set R (λ). Put γ = ∂E ∩ D and ∆ = ∂E ∩ T. There exist constants c , c , c dependingFonly on θ such that 4 | |5(cid:54) 6 (4) γ is a rectiﬁable curve with length γ c σ (∆); (cid:54) ∩ 4 α (5) σ (∆) c m(∆ S (λ)) if E contains at least two atoms of σ ; α 5 F α ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

text: THEOREMS

text: Proof.

text: For

text: every

text: F

text: we

text: have

text: θ¯F

text: on

text: T.

text: Hence,

text: θ

text: F(z)

text: θ−(z)θ(ξ)

text: dm(ξ),

text: z

text: D.

text: T

text: zξ¯

text: Extend

text: the

text: inner

text: function

text: θ

text: to

text: the

text: domain

text: D

text: G

text: by

text: formula

text: The

text: right

text: θ

text: hand

text: side

text: of

text: then

text: determines

text: the

text: analytic

text: continuation

text: of

text: the

text: function

text: F

text: to

text: D

text: G

text: By

text: Lemma

text: we

text: have

text: ρ(σ

text: ρ(θ)

text: which

text: completes

text: the

text: proof.

text: θ

text: α

text: Lemma

text: Let

text: θ

text: be

text: an

text: inner

text: function

text: and

text: let

text: G

text: Then

text: there

text: exist

text: functions

text: G

text: G

text: such

text: that

text: G

text: G

text: +iG

text: andθ

text: G

text: θG

text: on

text: T\ρ(θ).

text: θ

text: Moreover,

text: we

text: have

text: G

text: T

text: G

text: T

text: Proof.

text: Consider

text: the

text: function

text: G˜

text: θG

text: on

text: the

text: unit

text: circle

text: T.

text: We

text: have

text: G˜

text: θ

text: This

text: shows

text: that

text: G

text: can

text: be

text: continued

text: to

text: the

text: open

text: unit

text: disk

text: D

text: as

text: a

text: function

text: from

text: the

text: space

text: Now

text: put

text: G

text: (G

text: G

text: (G

text: G˜

text: and

text: obtain

text: the

text: θ

text: desired

text: representation.

text: Let

text: µ

text: be

text: a

text: measure

text: on

text: the

text: unit

text: circle

text: T

text: with

text: properties

text: (a)

text: (c)

text: and

text: let

text: |α|

text: Consider

text: the

text: one-component

text: inner

text: function

text: θ

text: with

text: the

text: Clark

text: measure

text: σ

text: µ.

text: Take

text: a

text: function

text: F

text: By

text: Lemma

text: F

text: is

text: analytic

text: on

text: the

text: domαain

text: D

text: G

text: Denote

text: by

text: f

text: its

text: tθrace

text: on

text: the

text: set

text: a(σ

text: D

text: G

text: of

text: full

text: measure

text: σ

text: Our

text: aimθ

text: α

text: is

text: to

text: prove

text: that

text: f

text: (σ

text: and

text: f

text: F

text: T

text: At

text: ﬁrst,

text: assume

text: that

text: F

text: and

text: F

text: =atθFαon

text: T

text: will

text: following

text: modiﬁcation

text: θ

text: of

text: the

text: Lusin-Privalov

text: construction

text: (see

text: Section

text: III.D

text: in

text: for

text: the

text: standard

text: one).

text: Consider

text: the

text: non-tangential

text: maximal

text: function

text: of

text: F,

text: T

text: F

text: (ξ)

text: sup

text: F(z)

text: ξ

text: z

text: Λξ

text: where

text: Λ

text: denotes

text: the

text: convex

text: hull

text: of

text: the

text: set

text: {ξ}

text: {z

text: D

text: |z|

text: Put

text: ξ

text: S

text: (λ)

text: D

text: {z

text: D

text: z

text: Λ

text: for

text: some

text: ξ

text: T

text: with

text: F∗(ξ)

text: λ}.

text: F

text: ξ

text: Let

text: D

text: (κ)

text: be

text: the

text: set

text: deﬁned

text: at

text: the

text: beginning

text: of

text: Section

text: By

text: Lemma

text: we

text: have

text: |ασα−

text: θ|

text: ε

text: on

text: D

text: D

text: (κ).

text: Denote

text: by

text: R

text: (λ)

text: the

text: union

text: of

text: those

text: connected

text: components

text: of

text: the

text: set

text: S

text: (σλα)

text: D

text: (κ)

text: for

text: whFich

text: we

text: have

text: E

text: S

text: (λ)

text: and

text: E

text: D

text: (κ)

text: The

text: setsFR

text: (λ)

text: aσreα

text: closed

text: and

text: have

text: the

text: following

text: Fproperties:

text: σα

text: F

text: If

text: λ

text: λ

text: then

text: R

text: (λ

text: R

text: (λ

text: 2λ

text: for

text: σ

text: all

text: z

text: T

text: R

text: (λ);

text: |F(z)|

text: λ

text: and

text: |αα

text: θ(z)|

text: ε

text: for

text: z

text: ∂R

text: (λ)F∩

text: D.

text: F

text: More

text: special

text: properties

text: of

text: the

text: sets

text: R

text: (λ)

text: are

text: collected

text: in

text: the

text: following

text: lemma.

text: F

text: Lemma

text: Let

text: E

text: be

text: a

text: connected

text: component

text: of

text: the

text: set

text: R

text: (λ).

text: Put

text: γ

text: ∂E

text: D

text: and

text: ∂E

text: T.

text: There

text: exist

text: constants

text: c

text: c

text: c

text: dependingFonly

text: on

text: θ

text: such

text: that

text: γ

text: is

text: a

text: rectiﬁable

text: curve

text: with

text: length

text: γ

text: c

text: σ

text: α

text: σ

text: c

text: m(∆

text: S

text: (λ))

text: if

text: E

text: contains

text: at

text: least

text: two

text: atoms

text: of

text: σ

text: α

text: F

text: α

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DUALITY THEOREMS 15 (cid:48) Lemma 3.5 is the key argument in the proof of Theorem 2 . The rest of the proof is a standard Caldero´n-Zigmund decomposition. We will follow the exposition in Section VII.E of [16]. For each λ > 0 the set ∆ (λ) = R (λ) ∩ T is a union of closed disjo(cid:40)int arcs ∆k (λ), ∆ (λ) = ∪ ∈ ∆k (cid:40)(λ).F ConsiderFthe functions F F k Iλ F ∈ T \ ∈ T \ f, ξ ∆ (λ), 0, ξ ∆ (λ), F F G = (cid:104) (cid:105) ∈ B = − (cid:104) (cid:105) ∈ λ f , ξ ∆k (λ), λ f f , ξ ∆k (λ). ∆kF(λ),σα F ∆kF(λ),σα F By Lemma 3.5 we have |G | (cid:54) c λ σ -almost everywhere on T. The function B has zero σ -mean on each arcλ∆k (λ6), kα∈ I . For every integer n ∈ Z set g = G λ and b = Bα . Fix a number N F∈ Z such tλhat n 2n n 2n 0 | | (cid:54) 2N0 < inf√ F(z) 2N0+1. | |(cid:54) (cid:90) z 1/ 2 Note that ∆F (2N0) = T. By formula (21), 1 F(0) gN0 = σ (T) T f dσα = σ (T)(1 − α¯θ(0)) = 0. α α ∈ Since f is ﬁnite at each point ξ a(σ ) we have f(ξ) = g (ξ) for every suﬃciently α N big number N. Hence (cid:88) (cid:0) (cid:1) ∞ − ∈ f(ξ) = g (ξ) g (ξ) , ξ a(σ ), (31) n+1 n α n=N0 where the sum converges pointwise (in fact, only ﬁnite number of summands in ∈ − (31) are non-zero for every ξ a(σ )). Note that f = b + g and g g = − (cid:62) (cid:48) α n∈ n n+1 n b b for all n N . Let I be the set of indexes k I such that the set n n+1 0 2n 2n − ∆k (2n) contains at least two atoms of the measure σ . The function g g F α ∈ \ (cid:48) n+1 n vanishes σ -almost everywhere on each of the sets ∆k (2n), k I I . Indeed, for α F 2n 2n such index k we have by the construction. Hence g (ξ) = g (ξ) = f(ξ) because n ∈ n+1 the σ -mean of f on any arc containing the only point ξ a(σ ) equals f(ξ). Deﬁne α α − (cid:62) ∈ (cid:48) a˜ = χ (b b ), n N , k I , n,k ∆k (2n) n n+1 0 2n F where χ is the indicator of the set ∆k (2n). The functions a˜ have zero σα-me(cid:90)an on∆kFT(2.n)Inde(cid:90)ed, let I ⊂denote the set oFf i(cid:88)ndex(cid:90)es m such that n∆,kmF (2n+1) ⊂ ∆k (2n) (note that ∆ (2n+1) ∆ (2n) by property (1) of the sets R (λ)). Then F F F F − − a˜ dσ = (b b ) dσ = b dσ = 0. T n,k α ∆kF(2n) n n+1 α m∈I ∆mF (2n+1) n+1 α Also, we have |a˜ | (cid:54) |g | + |g | (cid:54) 3c · 2n on T for every n (cid:62) N and k ∈ I(cid:48) . n,k n n+1 6 0 2n Now put a˜ (cid:62) ∈ (cid:48) a = · · n,k , n N , k I , n,k 3c 2n σ (∆k (2n)) 0 2n 6 α F and observe that a are atoms with respect to the measure σ . It follows from n,k (cid:88) (cid:88) α formula (31) that ∈ f(ξ) = λ a (ξ), ξ a(σ ), (32) n,k n,k α (cid:62) ∈ (cid:48) n N0 k I2n ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

text: THEOREMS

text: Lemma

text: is

text: the

text: key

text: argument

text: in

text: the

text: proof

text: of

text: Theorem

text: The

text: rest

text: of

text: the

text: proof

text: is

text: a

text: standard

text: Caldero´n-Zigmund

text: decomposition.

text: We

text: will

text: follow

text: the

text: exposition

text: in

text: Section

text: VII.E

text: of

text: For

text: each

text: λ

text: the

text: set

text: (λ)

text: R

text: (λ)

text: T

text: is

text: a

text: union

text: of

text: closed

text: arcs

text: ∆k

text: (λ),

text: (λ)

text: ∆k

text: ConsiderFthe

text: functions

text: F

text: F

text: k

text: Iλ

text: F

text: T

text: T

text: f,

text: ξ

text: (λ),

text: ξ

text: (λ),

text: F

text: F

text: G

text: B

text: λ

text: f

text: ξ

text: ∆k

text: (λ),

text: λ

text: f

text: f

text: ξ

text: ∆k

text: (λ).

text: ∆kF(λ),σα

text: F

text: ∆kF(λ),σα

text: F

text: By

text: Lemma

text: we

text: have

text: |G

text: c

text: λ

text: σ

text: -almost

text: everywhere

text: on

text: T.

text: The

text: function

text: B

text: has

text: zero

text: σ

text: -mean

text: on

text: each

text: arcλ∆k

text: kα∈

text: I

text: For

text: every

text: integer

text: n

text: Z

text: set

text: g

text: G

text: λ

text: and

text: b

text: Bα

text: Fix

text: a

text: number

text: N

text: F∈

text: Z

text: such

text: tλhat

text: n

text: n

text: inf√

text: F(z)

text: z

text: Note

text: that

text: ∆F

text: T.

text: By

text: formula

text: σ

text: (T)

text: T

text: f

text: dσα

text: σ

text: α

text: α

text: Since

text: f

text: is

text: ﬁnite

text: at

text: each

text: point

text: ξ

text: a(σ

text: we

text: have

text: f(ξ)

text: g

text: (ξ)

text: for

text: every

text: suﬃciently

text: α

text: N

text: big

text: number

text: N.

text: Hence

text: f(ξ)

text: g

text: (ξ)

text: g

text: (ξ)

text: ξ

text: a(σ

text: n

text: α

text: where

text: the

text: sum

text: converges

text: pointwise

text: (in

text: fact,

text: only

text: ﬁnite

text: number

text: of

text: summands

text: in

text: are

text: non-zero

text: for

text: every

text: ξ

text: a(σ

text: Note

text: that

text: f

text: b

text: g

text: and

text: g

text: g

text: α

text: n∈

text: n

text: n

text: b

text: b

text: for

text: all

text: n

text: N

text: Let

text: I

text: be

text: the

text: set

text: of

text: indexes

text: k

text: I

text: such

text: that

text: the

text: set

text: n

text: ∆k

text: contains

text: at

text: least

text: two

text: atoms

text: of

text: the

text: measure

text: σ

text: The

text: function

text: g

text: g

text: F

text: α

text: n

text: vanishes

text: σ

text: -almost

text: everywhere

text: on

text: each

text: of

text: the

text: sets

text: ∆k

text: k

text: I

text: I

text: Indeed,

text: for

text: α

text: F

text: such

text: index

text: k

text: we

text: have

text: by

text: the

text: construction.

text: Hence

text: g

text: (ξ)

text: g

text: (ξ)

text: f(ξ)

text: because

text: n

text: the

text: σ

text: -mean

text: of

text: f

text: on

text: any

text: arc

text: containing

text: the

text: only

text: point

text: ξ

text: a(σ

text: equals

text: f(ξ).

text: Deﬁne

text: α

text: α

text: a˜

text: χ

text: (b

text: b

text: n

text: N

text: k

text: I

text: n,k

text: ∆k

text: n

text: F

text: where

text: χ

text: is

text: the

text: indicator

text: of

text: the

text: set

text: ∆k

text: The

text: functions

text: a˜

text: have

text: zero

text: let

text: I

text: ⊂denote

text: the

text: set

text: oFf

text: m

text: such

text: that

text: n∆,kmF

text: ∆k

text: (note

text: that

text: by

text: property

text: of

text: the

text: sets

text: R

text: (λ)).

text: Then

text: F

text: F

text: F

text: F

text: a˜

text: dσ

text: (b

text: b

text: dσ

text: b

text: dσ

text: T

text: n,k

text: α

text: n

text: α

text: m∈I

text: ∆mF

text: α

text: Also,

text: we

text: have

text: |a˜

text: |g

text: |g

text: on

text: T

text: for

text: every

text: n

text: N

text: and

text: k

text: n,k

text: n

text: Now

text: put

text: a˜

text: a

text: n,k

text: n

text: N

text: k

text: I

text: n,k

text: σ

text: (∆k

text: α

text: F

text: and

text: observe

text: that

text: a

text: are

text: atoms

text: with

text: respect

text: to

text: the

text: measure

text: σ

text: It

text: follows

text: from

text: n,k

text: α

text: formula

text: that

text: f(ξ)

text: λ

text: a

text: (ξ),

text: ξ

text: a(σ

text: n,k

text: n,k

text: α

text: n

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DUALITY THEOREMS 5 Theorem 3. Let θ be a one-component inner function, and let ν be the Clark α → measure of the inner function θ2. The truncated Hankel operator Γ : K2 zK2 ∈ ϕ θ θ with standard symbol ϕ is bounded if and only if ϕ BMO(ν ). Moreover, we have α (cid:107) (cid:107) (cid:54) (cid:107) (cid:107) (cid:54) (cid:107) (cid:107) c ϕ ∗ Γ c ϕ ∗, (8) 1 ν ϕ 2 ν α α for some constants c , c depending only on the inner function θ. 1 2 Similarly, one can describe compact truncated Hankel operators in terms of their ∈ ∈ standard symbols: we have Γ S∞ if and only if ϕ VMO(ν ), see Section . ϕ α Theorem 3 for the inner function θ = zn yields the following interesting corollary for ﬁnite Hankel matrices. × Corollary 1. Let Γ = (γ ) (cid:54) (cid:54) − be a Hankel matrix of size n n; consider j+k 0 k,j n 1 − its standard symbol ϕ = γ z¯ + γ z¯2 + . . . γ − z¯2n 1. We have 0 1 2n 2 c (cid:107)ϕ(cid:107) ∗ (cid:54) (cid:107)Γ(cid:107) (cid:54) c (cid:107)ϕ(cid:107) ∗ , (cid:80) (9) 1 µ 2 µ 2n 2n where the constants c , c do not depend on n and µ = 1 δ √ is the Haar measure on the group {ξ1∈2C : ξ2n = 1}. 2n 2n 2n 1 Corollary 1 implies the boundedness criterium for the standard Hankel operators on H2. Recall that the Hankel operator H : H2 → zH2 with symbol ϕ ∈ L2(T) is ϕ densely deﬁned by (cid:55)→ ∈ ∞ H : f P−(ϕf), f H , ϕ where P− denotes the orthogonal projection in L2(T) to zH2. It follows from the classical Feﬀerman duality theorem that H is bounded if and only if its antianalytic symbol P−ϕ lies in BMO(T). Moreover, thϕe operator norm of H is comparable to (cid:107)P−ϕ(cid:107)∗, the norm of P−ϕ in BMO(T). Taking the limit in (9) aϕs n → ∞ one can (cid:107) (cid:107) (cid:54) (cid:107) (cid:107) (cid:54) (cid:107) (cid:107) prove the estimate c ϕ ∗ H c ϕ ∗ for every antianalytic polynomial ϕ. 1 ϕ 2 This is already suﬃcient to obtain the general version of the boundedness criterium for Hankel operators on H2, see details in Section . 2. Proof of Theorem 1 Given an inner function θ, denote by ρ(θ) its boundary spectrum, that is, the set of points ζ ∈ T such that lim inf → ∈D |θ(z)| = 0. In this paper we always assume that ρ(θ) (cid:54)= T, because this iszso ζf,ozr one-component inner functions and for (cid:0) (cid:1) functions satisfying condition (a) in Theorem 1 (see Lemma 2.1 below). As is well- known, the function θ admits the analytic continuation from the open unit disk D to the open domain D ∪ G , where G = T \ ρ(θ) ∪ {z : |z| > 1, θ(1/z¯) (cid:54)= 0}. θ θ The analytic continuation is given by 1 ∈ θ(z) = , z G . (10) θ θ(1/z¯) Moreover, D∪G is the maximal domain to which θ can be extended analytically. θ We need the following known lemma. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

text: THEOREMS

text: Theorem

text: Let

text: θ

text: be

text: a

text: one-component

text: inner

text: function,

text: and

text: let

text: ν

text: be

text: the

text: Clark

text: α

text: measure

text: of

text: the

text: inner

text: function

text: The

text: truncated

text: Hankel

text: operator

text: Γ

text: ϕ

text: θ

text: θ

text: with

text: standard

text: symbol

text: ϕ

text: is

text: bounded

text: if

text: and

text: only

text: if

text: ϕ

text: BMO(ν

text: Moreover,

text: we

text: have

text: α

text: c

text: ϕ

text: Γ

text: c

text: ϕ

text: ν

text: ϕ

text: ν

text: α

text: α

text: for

text: some

text: constants

text: c

text: c

text: depending

text: only

text: on

text: the

text: inner

text: function

text: θ.

text: Similarly,

text: one

text: can

text: describe

text: compact

text: truncated

text: Hankel

text: operators

text: in

text: terms

text: of

text: their

text: standard

text: symbols:

text: we

text: have

text: Γ

text: S∞

text: if

text: and

text: only

text: if

text: ϕ

text: VMO(ν

text: see

text: Section

text: ϕ

text: α

text: Theorem

text: for

text: the

text: inner

text: function

text: θ

text: zn

text: yields

text: the

text: following

text: interesting

text: corollary

text: for

text: ﬁnite

text: Hankel

text: matrices.

text: Corollary

text: Let

text: Γ

text: (γ

text: be

text: a

text: Hankel

text: matrix

text: of

text: size

text: n

text: n;

text: consider

text: j+k

text: k,j

text: n

text: its

text: standard

text: symbol

text: ϕ

text: γ

text: z¯

text: γ

text: γ

text: We

text: have

text: c

text: c

text: µ

text: µ

text: where

text: the

text: constants

text: c

text: c

text: do

text: not

text: depend

text: on

text: n

text: and

text: µ

text: δ

text: is

text: the

text: Haar

text: measure

text: on

text: the

text: group

text: Corollary

text: implies

text: the

text: boundedness

text: criterium

text: for

text: the

text: standard

text: Hankel

text: operators

text: on

text: Recall

text: that

text: the

text: Hankel

text: operator

text: H

text: with

text: symbol

text: ϕ

text: is

text: ϕ

text: densely

text: deﬁned

text: by

text: H

text: f

text: P−(ϕf),

text: f

text: H

text: ϕ

text: where

text: P−

text: denotes

text: the

text: orthogonal

text: projection

text: in

text: to

text: It

text: follows

text: from

text: the

text: classical

text: Feﬀerman

text: duality

text: theorem

text: that

text: H

text: is

text: bounded

text: if

text: and

text: only

text: if

text: its

text: antianalytic

text: symbol

text: P−ϕ

text: lies

text: in

text: BMO(T).

text: Moreover,

text: thϕe

text: operator

text: norm

text: of

text: H

text: is

text: comparable

text: to

text: the

text: norm

text: of

text: P−ϕ

text: in

text: BMO(T).

text: Taking

text: the

text: limit

text: in

text: aϕs

text: n

text: one

text: can

text: prove

text: the

text: estimate

text: c

text: ϕ

text: H

text: c

text: ϕ

text: for

text: every

text: antianalytic

text: polynomial

text: ϕ.

text: ϕ

text: This

text: is

text: already

text: suﬃcient

text: to

text: obtain

text: the

text: general

text: version

text: of

text: the

text: boundedness

text: criterium

text: for

text: Hankel

text: operators

text: on

text: see

text: details

text: in

text: Section

text: Proof

text: of

text: Theorem

text: Given

text: an

text: inner

text: function

text: θ,

text: denote

text: by

text: ρ(θ)

text: its

text: boundary

text: spectrum,

text: that

text: is,

text: the

text: set

text: of

text: points

text: ζ

text: T

text: such

text: that

text: lim

text: inf

text: ∈D

text: |θ(z)|

text: In

text: this

text: paper

text: we

text: always

text: assume

text: that

text: ρ(θ)

text: T,

text: because

text: this

text: iszso

text: ζf,ozr

text: one-component

text: inner

text: functions

text: and

text: for

text: functions

text: satisfying

text: condition

text: (a)

text: in

text: Theorem

text: (see

text: Lemma

text: below).

text: As

text: is

text: well-

text: known,

text: the

text: function

text: θ

text: admits

text: the

text: analytic

text: continuation

text: from

text: the

text: open

text: unit

text: disk

text: D

text: to

text: the

text: open

text: domain

text: D

text: G

text: where

text: G

text: T

text: ρ(θ)

text: {z

text: |z|

text: θ

text: θ

text: The

text: analytic

text: continuation

text: is

text: given

text: by

text: θ(z)

text: z

text: G

text: θ

text: Moreover,

text: D∪G

text: is

text: the

text: maximal

text: domain

text: to

text: which

text: θ

text: can

text: be

text: extended

text: analytically.

text: θ

text: We

text: need

text: the

text: following

text: known

text: lemma.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DUALITY THEOREMS 6 | | Lemma 2.1. Let θ be an inner function with the Clark measure σ , α = 1. Then ρ(θ) = ρ(σ ). A point z ∈ T \ ρ(θ) belongs to supp σ if and oαnly if θ(z) = α. α ∈ {α} | (cid:48) |− Moreover, in the latter case we have z a(σ ) and σ z = θ (z) 1. α α (cid:90) Proof. As is easy to see from formula (3), we have α +− θ(z) = 1 +− ξ¯z dσ (ξ) + i Im α +− θ(0), z ∈ D ∪ G . (11) α θ(z) T 1 ξ¯z α α θ(0) θ Since θ is analytic on D ∪ G , a point z ∈ T \ ρ(θ) belongs to supp σ if and only θ α if θ(z) = α, and in the latter case there is no other points of supp σ in a small ∈ α neighbourhood of z. Hence z a(σ ) and we see from (11) that α { } (cid:48) − | (cid:48) |− σ z = (α¯zθ (z)) 1 = θ (z) 1. α It follows that T \ ρ(θ) ⊂ T \ ρ(σ ). For every z ∈ T \ ρ(σ ) either z is an isolated ∈ α α atom of σ or z / supp σ . In both cases formula (11) shows that the function θ admits tαhe analytic contαinuation from D to a small neighbourhood of z. Hence z ∈ T \ ρ(θ) and we have ρ(θ) = ρ(σ ). (cid:3) α The following result is in [3], see Theorem 1.11 and Remark 2 after its proof. Theorem (A. B. Aleksandrov). An inner function θ is one-component if and only if it satisﬁes the following conditions: (A1) m(ρ(θ)) = 0 and |θ(cid:48)| is unbounded on every open arc ∆ ⊂ T \ ρ(θ) such ∩ (cid:54) ∅ that ∆ ρ(θ) = ; (A2) θ satisﬁes the estimate |θ(cid:48)(cid:48)(ξ)| (cid:54) C|θ(cid:48)(ξ)|2 for all ξ ∈ T \ ρ(θ). − Essentially, we will show that conditions (a) (c) in Theorem 1 are equivalent to conditions (A1), (A2) above. Let θ be a one-component inner function and let σ be its Clark measure. By α Lemma 2.1 we have ρ(θ) = ρ(σ ). It was proved in [3] that m(ρ(θ)) = 0 and α σ (ρ(θ)) = 0. Hence σ is a discrete measure with isolated atoms and we have mα(supp σ ) = 0. Let ∆ αbe a connected component of the set T \ ρ(σ ) = T \ ρ(θ). α α By property (A1) the argument of θ on ∆ is a monotonic function unbounded near both endpoints of ∆. It follows that the arc ∆ contains inﬁnitely many points ξ k such that θ(ξ ) = α. Enumerate these points clockwise by integer numbers. We see from Lemmak2.1 that ξ ∈ a(σ ) for all k ∈ Z and every atom ξ has two neighbours k α k ξ − , ξ . This shows that the measure σ satisﬁes condition (a). The fact that k 1 k+1 α σ satisﬁes condition (b) follows from Lemma 5.1 of [7]. Now check condition (c). α ∈ (cid:90) Fix an atom ξ a(σ ). From (11) we see that 0 α − 1 = dσ−α(ξ) + c , z ∈ D ∪ G , (12) 1 α¯θ(z) T 1 ξ¯z α θ − where c = αθ(0)/(1 αθ(0)). Hence, α ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

text: THEOREMS

text: Lemma

text: Let

text: θ

text: be

text: an

text: inner

text: function

text: with

text: the

text: Clark

text: measure

text: σ

text: α

text: Then

text: ρ(θ)

text: ρ(σ

text: A

text: point

text: z

text: T

text: ρ(θ)

text: belongs

text: to

text: supp

text: σ

text: if

text: and

text: oαnly

text: if

text: θ(z)

text: α.

text: α

text: {α}

text: Moreover,

text: in

text: the

text: latter

text: case

text: we

text: have

text: z

text: a(σ

text: and

text: σ

text: z

text: θ

text: (z)

text: α

text: α

text: Proof.

text: As

text: is

text: easy

text: to

text: see

text: from

text: formula

text: we

text: have

text: α

text: θ(z)

text: ξ¯z

text: dσ

text: (ξ)

text: i

text: Im

text: α

text: z

text: D

text: G

text: α

text: θ(z)

text: T

text: ξ¯z

text: α

text: α

text: θ

text: Since

text: θ

text: is

text: analytic

text: on

text: D

text: G

text: a

text: point

text: z

text: T

text: ρ(θ)

text: belongs

text: to

text: supp

text: σ

text: if

text: and

text: only

text: θ

text: α

text: if

text: θ(z)

text: α,

text: and

text: in

text: the

text: latter

text: case

text: there

text: is

text: no

text: other

text: points

text: of

text: supp

text: σ

text: in

text: a

text: small

text: α

text: neighbourhood

text: of

text: z.

text: Hence

text: z

text: a(σ

text: and

text: we

text: see

text: from

text: that

text: α

text: σ

text: z

text: (α¯zθ

text: (z))

text: θ

text: (z)

text: α

text: It

text: follows

text: that

text: T

text: ρ(θ)

text: T

text: ρ(σ

text: For

text: every

text: z

text: T

text: ρ(σ

text: either

text: z

text: is

text: an

text: isolated

text: α

text: α

text: atom

text: of

text: σ

text: or

text: z

text: supp

text: σ

text: In

text: both

text: cases

text: formula

text: shows

text: that

text: the

text: function

text: θ

text: admits

text: tαhe

text: analytic

text: contαinuation

text: from

text: D

text: to

text: a

text: small

text: neighbourhood

text: of

text: z.

text: Hence

text: z

text: T

text: ρ(θ)

text: and

text: we

text: have

text: ρ(θ)

text: ρ(σ

text: α

text: The

text: following

text: result

text: is

text: in

text: see

text: Theorem

text: and

text: Remark

text: after

text: its

text: proof.

text: Theorem

text: (A.

text: B.

text: Aleksandrov).

text: An

text: inner

text: function

text: θ

text: is

text: one-component

text: if

text: and

text: only

text: if

text: it

text: satisﬁes

text: the

text: following

text: conditions:

text: m(ρ(θ))

text: and

text: is

text: unbounded

text: on

text: every

text: open

text: arc

text: T

text: ρ(θ)

text: such

text: that

text: ρ(θ)

text: θ

text: satisﬁes

text: the

text: estimate

text: for

text: all

text: ξ

text: T

text: ρ(θ).

text: Essentially,

text: we

text: will

text: show

text: that

text: conditions

text: (a)

text: (c)

text: in

text: Theorem

text: are

text: equivalent

text: to

text: conditions

text: above.

text: Let

text: θ

text: be

text: a

text: one-component

text: inner

text: function

text: and

text: let

text: σ

text: be

text: its

text: Clark

text: measure.

text: By

text: α

text: Lemma

text: we

text: have

text: ρ(θ)

text: ρ(σ

text: It

text: was

text: proved

text: in

text: that

text: m(ρ(θ))

text: and

text: α

text: σ

text: (ρ(θ))

text: Hence

text: σ

text: is

text: a

text: discrete

text: measure

text: with

text: isolated

text: atoms

text: and

text: we

text: have

text: mα(supp

text: σ

text: Let

text: αbe

text: a

text: connected

text: component

text: of

text: the

text: set

text: T

text: ρ(σ

text: T

text: ρ(θ).

text: α

text: α

text: By

text: property

text: the

text: argument

text: of

text: θ

text: on

text: is

text: a

text: monotonic

text: function

text: unbounded

text: near

text: both

text: endpoints

text: of

text: It

text: follows

text: that

text: the

text: arc

text: contains

text: inﬁnitely

text: many

text: points

text: ξ

text: k

text: such

text: that

text: θ(ξ

text: α.

text: Enumerate

text: these

text: points

text: clockwise

text: by

text: integer

text: numbers.

text: We

text: see

text: from

text: that

text: ξ

text: a(σ

text: for

text: all

text: k

text: Z

text: and

text: every

text: atom

text: ξ

text: has

text: two

text: neighbours

text: k

text: α

text: k

text: ξ

text: ξ

text: This

text: shows

text: that

text: the

text: measure

text: σ

text: satisﬁes

text: condition

text: (a).

text: The

text: fact

text: that

text: k

text: α

text: σ

text: satisﬁes

text: condition

text: (b)

text: follows

text: from

text: Lemma

text: of

text: Now

text: check

text: condition

text: (c).

text: α

text: Fix

text: an

text: atom

text: ξ

text: a(σ

text: From

text: we

text: see

text: that

text: α

text: dσ−α(ξ)

text: c

text: z

text: D

text: G

text: α¯θ(z)

text: T

text: ξ¯z

text: α

text: θ

text: where

text: c

text: Hence,

text: α

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DUALITY T(cid:18)HEOREMS (cid:19) 7 { } 1 − σ ξ (H 1)(ξ ) + c = lim − α− 0 . σα 0 α z→ξ0 1 α¯θ(z) 1 ξ¯0z Consider the analytic function kξ0 : z (cid:55)→ 11−−α¯ξ¯θ(cid:18)0(zz) on the domain(cid:19)D∪Gθ. We have 1 1 − 1 (H 1)(ξ ) + c = lim − σα 0 α z→ξ0 1(cid:48) ξ¯0z kξ0(z(cid:48)(cid:48)) kξ0(ξ0) (13) = −ξ0kξ0(ξ0) = − αθ(cid:48) (ξ0) . k2 (ξ ) 2θ (ξ )2 ξ0 0 0 From here and the estimate in (A2) we see that H 1 is bounded on a(σ ). Surprisingly simple relation (13) between the discrete Hiσlαbert transform H 1 aαnd the inner function θ is the key observation in the proof. σα − Let µ be a measure with properties (a) (c). Construct the inner function θ with the Clark measure σ = µ. To prove that θ is a one-component inner function α we will check conditions (A1) and (A2). By Lemma 2.1 we have ρ(θ) = ρ(σ ). Hence m(ρ(θ)) = 0 by property (a) of the measure σ . Let ∆ be an open arc ofαT such that ∆ ⊂ T \ ρ(θ) and ∆¯ ∩ ρ(θ) (cid:54)= ∅. α Then it follows from property (a) of the measure σ that ∆ contains inﬁnitely { } | α(cid:48) |− ∈ many atoms of σ . Since σ is ﬁnite and σ ξ = θ (ξ) 1 for every ξ a(σ ), | (cid:48)| α α α α the function θ cannot be bounded on ∆. This gives us condition (A1). Condition (A2) is more delicate. To check it we need the following lemma. Lemma 2.2. Assume that the Clark measure σ of an inner function θ has prop- − α ∈ ethretiesset(aD)ξ(κ()c=). {Tzh∈enCt:h|eξre−ezx|is(cid:54)tsκaσ(cid:12)(cid:12)(cid:12)αn{uξm}}beirsκc(cid:12)(cid:12)(cid:12)o>nta0insuedchinthDat∪foGrθeavnerdywξe haav(eσα) (cid:12) (cid:12) − 1 (cid:54) α θ(z) (cid:54) 2 { } − { } (14) 2σ ξ ξ z σ ξ α α ∈ for all z D (κ). ξ (cid:90)∈ Proof. Pick an atom ξ a(σ ) and rewrite formula (12) in the following form: 0 α { } − 1 = dσ−α(ξ) + σα− ξ0 + c , z ∈ D ∪ G . (15) 1 α¯θ(z) T\{ } 1 ξ¯z 1 ξ¯ z α θ (cid:12) (cid:12)ξ0 0 We have (cid:12)(cid:90) (cid:12) (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) | − | dσ (ξ) (cid:54) | | ξ z dσ (ξ) −α (H 1)(ξ ) + | 0− | · | −α |. T\{ } 1 ξ¯z σα 0 T\{ } ξ z ξ ξ ξ0 ξ0 0 | | (cid:54) ∗ − ∈ \ By property (c), (H 1)(ξ ) C . Put κ = (2B ) 1. For for ξ a(σ ) {ξ } and z ∈ D (κ∗) wσeαhave0 |ξ − zσ|α(cid:54) κ∗σ {ξ } (cid:54) |ξσα− ξ|/2 by property (bα) of 0 ξ0 0 α| −0 | (cid:62) | 0− |−| − | (cid:62) | − | the measure σ , which gives us the inequality ξ z ξ ξ ξ z 1 ξ ξ . α ∈ ∗ 0 0 2 0 It follows that for z D (κ ) we have ξ0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DUALITY

text: σ

text: ξ

text: (H

text: c

text: lim

text: α−

text: σα

text: α

text: z→ξ0

text: α¯θ(z)

text: ξ¯0z

text: Consider

text: the

text: analytic

text: function

text: z

text: on

text: the

text: We

text: have

text: (H

text: c

text: lim

text: σα

text: α

text: z→ξ0

text: ξ¯0z

text: (ξ

text: 2θ

text: (ξ

text: ξ0

text: From

text: here

text: and

text: the

text: estimate

text: in

text: we

text: see

text: that

text: H

text: is

text: bounded

text: on

text: a(σ

text: Surprisingly

text: simple

text: relation

text: between

text: the

text: discrete

text: Hiσlαbert

text: transform

text: H

text: aαnd

text: the

text: inner

text: function

text: θ

text: is

text: the

text: key

text: observation

text: in

text: the

text: proof.

text: σα

text: Let

text: µ

text: be

text: a

text: measure

text: with

text: properties

text: (a)

text: (c).

text: Construct

text: the

text: inner

text: function

text: θ

text: with

text: the

text: Clark

text: measure

text: σ

text: µ.

text: To

text: prove

text: that

text: θ

text: is

text: a

text: one-component

text: inner

text: function

text: α

text: we

text: will

text: check

text: conditions

text: and

text: By

text: Lemma

text: we

text: have

text: ρ(θ)

text: ρ(σ

text: Hence

text: m(ρ(θ))

text: by

text: property

text: (a)

text: of

text: the

text: measure

text: σ

text: Let

text: be

text: an

text: open

text: arc

text: ofαT

text: such

text: that

text: T

text: ρ(θ)

text: and

text: ρ(θ)

text: α

text: Then

text: it

text: follows

text: from

text: property

text: (a)

text: of

text: the

text: measure

text: σ

text: that

text: contains

text: inﬁnitely

text: many

text: atoms

text: of

text: σ

text: Since

text: σ

text: is

text: ﬁnite

text: and

text: σ

text: ξ

text: θ

text: (ξ)

text: for

text: every

text: ξ

text: a(σ

text: α

text: α

text: α

text: α

text: the

text: function

text: θ

text: cannot

text: be

text: bounded

text: on

text: This

text: gives

text: us

text: condition

text: Condition

text: is

text: more

text: delicate.

text: To

text: check

text: it

text: we

text: need

text: the

text: following

text: lemma.

text: Lemma

text: Assume

text: that

text: the

text: Clark

text: measure

text: σ

text: of

text: an

text: inner

text: function

text: θ

text: has

text: prop-

text: α

text: ethretiesset(aD)ξ(κ()c=).

text: haav(eσα)

text: α

text: θ(z)

text: 2σ

text: ξ

text: ξ

text: z

text: σ

text: ξ

text: α

text: α

text: for

text: all

text: z

text: D

text: (κ).

text: ξ

text: Proof.

text: Pick

text: an

text: atom

text: ξ

text: a(σ

text: and

text: rewrite

text: formula

text: in

text: the

text: following

text: form:

text: α

text: dσ−α(ξ)

text: σα−

text: ξ0

text: c

text: z

text: D

text: G

text: α¯θ(z)

text: T\{

text: ξ¯z

text: ξ¯

text: z

text: α

text: θ

text: We

text: have

text: dσ

text: (ξ)

text: ξ

text: z

text: dσ

text: (ξ)

text: −α

text: (H

text: 0−

text: −α

text: T\{

text: ξ¯z

text: σα

text: T\{

text: ξ

text: z

text: ξ

text: ξ

text: ξ0

text: ξ0

text: By

text: property

text: (c),

text: (H

text: C

text: Put

text: κ

text: For

text: for

text: ξ

text: a(σ

text: {ξ

text: and

text: z

text: D

text: (κ∗)

text: |ξ

text: κ∗σ

text: {ξ

text: |ξσα−

text: by

text: property

text: (bα)

text: of

text: ξ0

text: α|

text: −0

text: 0−

text: the

text: measure

text: σ

text: which

text: gives

text: us

text: the

text: inequality

text: ξ

text: z

text: ξ

text: ξ

text: ξ

text: z

text: ξ

text: ξ

text: α

text: It

text: follows

text: that

text: for

text: z

text: D

text: (κ

text: we

text: have

text: ξ0

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Contents 1. Introduction 1 2. Lattice model 2 3. Strong dynamics and chiral symmetry breaking 4 4. Axial symmetries in the continuum limit 6 5. Weakly gauging the SU(2) × SU(2) symmetries 8 6. Discussion 10 1. Introduction The idea that the Higgs mechanism can occur through the formation of fermionic con- densates is an attractive one when constructing many theories of Beyond Standard Model (BSM) physics and ﬁnds application in technicolor, composite Higgs models, tumbling and grand uniﬁcation schemes [1, 2, 3, 4]. Lattice realizations of these scenarios thus poten- tially give a rigorous setting for understanding how non-perturbative dynamics in models without elementary scalars can spontaneously break gauge symmetries and potentially can give us new tools to analyze such theories. In a lattice theory Elitzur’s theorem [5] guarantees that any condensate which is not invariant under the gauge symmetry must necessarily have vanishing expectation value. Instead, the gauge invariant way to understand spontaneous gauge symmetry breaking and the operation of the Higgs mechanism in such theories is that it proceeds via the † condensation of a gauge invariant four fermion operator of the form φ φ where φ is a fermion bilinear which carries a non-trivial representation of the gauge group. One can think of this bilinear as a composite Higgs ﬁeld. In this paper we construct a lattice realization of these ideas using a set of two massless staggered lattice fermions. Each of these ﬁelds can be reduced by simply restricting one ﬁeld to even parity sites and the other to sites of odd parity. The key observation is that in the absence of single site mass terms the kinetic terms for each of these reduced staggered fermions may be gauged independently. We will use this observation to construct a class of lattice technicolor theories in which the gauge forces factorize into strong and weakly in- teracting sectors with both reduced ﬁelds coupling identically to the strong interaction but diﬀering in their weak interactions [6]. In the limit in which the weak gauge couplings are – 1 – ##LTLine## ##LTLine##

text: Contents

text: Introduction

text: Lattice

text: model

text: Strong

text: dynamics

text: and

text: chiral

text: symmetry

text: breaking

text: Axial

text: symmetries

text: in

text: the

text: continuum

text: limit

text: Weakly

text: gauging

text: the

text: symmetries

text: Discussion

label: section

text: Introduction

text: The

text: idea

text: that

text: the

text: Higgs

text: mechanism

text: can

text: occur

text: through

text: the

text: formation

text: of

text: fermionic

text: con-

text: densates

text: is

text: an

text: attractive

text: one

text: when

text: constructing

text: many

text: theories

text: of

text: Beyond

text: Standard

text: Model

text: (BSM)

text: physics

text: and

text: ﬁnds

text: application

text: in

text: technicolor,

text: composite

text: Higgs

text: models,

text: tumbling

text: and

text: grand

text: uniﬁcation

text: schemes

text: Lattice

text: realizations

text: of

text: these

text: scenarios

text: thus

text: poten-

text: tially

text: give

text: a

text: rigorous

text: setting

text: for

text: understanding

text: how

text: non-perturbative

text: dynamics

text: in

text: models

text: without

text: elementary

text: scalars

text: can

text: spontaneously

text: break

text: gauge

text: symmetries

text: and

text: potentially

text: can

text: give

text: us

text: new

text: tools

text: to

text: analyze

text: such

text: theories.

text: In

text: a

text: lattice

text: theory

text: Elitzur’s

text: theorem

text: guarantees

text: that

text: any

text: condensate

text: which

text: is

text: not

text: invariant

text: under

text: the

text: gauge

text: symmetry

text: must

text: necessarily

text: have

text: vanishing

text: expectation

text: value.

text: Instead,

text: the

text: gauge

text: invariant

text: way

text: to

text: understand

text: spontaneous

text: gauge

text: symmetry

text: breaking

text: and

text: the

text: operation

text: of

text: the

text: Higgs

text: mechanism

text: in

text: such

text: theories

text: is

text: that

text: it

text: proceeds

text: via

text: the

text: condensation

text: of

text: a

text: gauge

text: invariant

text: four

text: fermion

text: operator

text: of

text: the

text: form

text: φ

text: φ

text: where

text: φ

text: is

text: a

text: fermion

text: bilinear

text: which

text: carries

text: a

text: non-trivial

text: representation

text: of

text: the

text: gauge

text: group.

text: One

text: can

text: think

text: of

text: this

text: bilinear

text: as

text: a

text: composite

text: Higgs

text: ﬁeld.

text: In

text: this

text: paper

text: we

text: construct

text: a

text: lattice

text: realization

text: of

text: these

text: ideas

text: using

text: a

text: set

text: of

text: two

text: massless

text: staggered

text: lattice

text: fermions.

text: Each

text: of

text: these

text: ﬁelds

text: can

text: be

text: reduced

text: by

text: simply

text: restricting

text: one

text: ﬁeld

text: to

text: even

text: parity

text: sites

text: and

text: the

text: other

text: to

text: sites

text: of

text: odd

text: parity.

text: The

text: key

text: observation

text: is

text: that

text: in

text: the

text: absence

text: of

text: single

text: site

text: mass

text: terms

text: the

text: kinetic

text: terms

text: for

text: each

text: of

text: these

text: reduced

text: staggered

text: fermions

text: may

text: be

text: gauged

text: independently.

text: We

text: will

text: use

text: this

text: observation

text: to

text: construct

text: a

text: class

text: of

text: lattice

text: technicolor

text: theories

text: in

text: which

text: the

text: gauge

text: forces

text: factorize

text: into

text: strong

text: and

text: weakly

text: in-

text: teracting

text: sectors

text: with

text: both

text: reduced

text: ﬁelds

text: coupling

text: identically

text: to

text: the

text: strong

text: interaction

text: but

text: diﬀering

text: in

text: their

text: weak

text: interactions

text: In

text: the

text: limit

text: in

text: which

text: the

text: weak

text: gauge

text: couplings

text: are

text: ##LTLine##

text: ##LTLine##


text: The Higgsed gauge ﬁelds will thus correspond to the broken generators τ = τ1 − τ2 (2.10) a a a with τ1 and τ2 the generators of the two weak groups respectively. In order to see such a Higgs phase, we need to add a small perturbation to the above (cid:88) (cid:16) (cid:17) action via an auxiliary ﬁeld φ. † δS = g φ(x)ψ (x)λ (x) + φ (x)λ−(x)ψ−(x) (2.11) + + x This ﬁeld φ(x) is a local ﬁeld which is needed to couple the two reduced fermions together and must transform under the gauge symmetry as a bifundamental in the weak groups in (cid:80) order that the perturbation is gauge invariant. To render the path integral well deﬁned after integration over φ(x) one must then also add a suitable action for φ(x). We choose † an additional simple term φ (x)φ(x). The eﬀect of these Yukawa terms is to add a x small gauge invariant four fermion interaction to the action that favors the conjectured symmetry breaking pattern. Spontaneous breaking of the global weak symmetries occurs iﬀ the fermion bilinear takes a vacuum expectation value in the thermodynamic limit as the auxiliary coupling is sent to zero. Finally to build the entire model requires the addition of Wilson plaquette terms for the three independent lattice gauge ﬁelds. In the rest of the paper will consider the case where the weak symmetry groups are SU(2) while the strong symmetry corresponds to SU(3). 3. Strong dynamics and chiral symmetry breaking Let us ﬁrst examine the case where the weak gauge coupling is set to zero and the weak symmetry is purely global. Such a scenario can be realized by freezing the auxiliary scalar φ and the gauge ﬁelds U and W to φ to the unit matrix. The auxiliary Yukawa term µ (cid:0)µ (cid:1) described above now acts as an explicit source term for the fermion bilinear Σ = ψ (x)λ−(x) + λ−(x)ψ−(x) (3.1) S + We will be interested in monitoring the corresponding single site condensate since a non- zero vacuum expectation value for this observable in the limit g → 0 would indicate a spontaneous breaking of SU(2) × SU(2) down to its diagonal SU(2) subgroup. However, this is not the only fermion bilinear one can add to the action; it is also possible to construct (cid:88) (cid:16) (cid:17) a single link operator Σ which is given on even parity sites by the expression1 L 1 † Σ = ψ (x)V (x)λ(x + µ) + V (x − µ)λ(x − µ) (3.2) L + µ 8 µ 1Our action also contains the corresponding odd site bilinear which is given by a similar expression. We observe that the vacuum expectation of the odd site operator matches its even site companion as one would expect – 4 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: Higgsed

text: gauge

text: ﬁelds

text: will

text: thus

text: correspond

text: to

text: the

text: broken

text: generators

text: τ

text: τ1

text: τ2

text: a

text: a

text: a

text: with

text: τ1

text: and

text: τ2

text: the

text: generators

text: of

text: the

text: two

text: weak

text: groups

text: respectively.

text: In

text: order

text: to

text: see

text: such

text: a

text: Higgs

text: phase,

text: we

text: need

text: to

text: add

text: a

text: small

text: perturbation

text: to

text: the

text: above

text: action

text: via

text: an

text: auxiliary

text: ﬁeld

text: φ.

text: δS

text: g

text: φ(x)ψ

text: (x)λ

text: (x)

text: φ

text: (x)λ−(x)ψ−(x)

text: x

text: This

text: ﬁeld

text: φ(x)

text: is

text: a

text: local

text: ﬁeld

text: which

text: is

text: needed

text: to

text: couple

text: the

text: two

text: reduced

text: fermions

text: together

text: and

text: must

text: transform

text: under

text: the

text: gauge

text: symmetry

text: as

text: a

text: bifundamental

text: in

text: the

text: weak

text: groups

text: in

text: order

text: that

text: the

text: perturbation

text: is

text: gauge

text: invariant.

text: To

text: render

text: the

text: path

text: integral

text: well

text: deﬁned

text: after

text: integration

text: over

text: φ(x)

text: one

text: must

text: then

text: also

text: add

text: a

text: suitable

text: action

text: for

text: φ(x).

text: We

text: choose

text: an

text: additional

text: simple

text: term

text: φ

text: (x)φ(x).

text: The

text: eﬀect

text: of

text: these

text: Yukawa

text: terms

text: is

text: to

text: add

text: a

text: x

text: small

text: gauge

text: invariant

text: four

text: fermion

text: interaction

text: to

text: the

text: action

text: that

text: favors

text: the

text: conjectured

text: symmetry

text: breaking

text: pattern.

text: Spontaneous

text: breaking

text: of

text: the

text: global

text: weak

text: symmetries

text: occurs

text: iﬀ

text: the

text: fermion

text: bilinear

text: takes

text: a

text: vacuum

text: expectation

text: value

text: in

text: the

text: thermodynamic

text: limit

text: as

text: the

text: auxiliary

text: coupling

text: is

text: sent

text: to

text: zero.

text: Finally

text: to

text: build

text: the

text: entire

text: model

text: requires

text: the

text: addition

text: of

text: Wilson

text: plaquette

text: terms

text: for

text: the

text: three

text: independent

text: lattice

text: gauge

text: ﬁelds.

text: In

text: the

text: rest

text: of

text: the

text: paper

text: will

text: consider

text: the

text: case

text: where

text: the

text: weak

text: symmetry

text: groups

text: are

text: while

text: the

text: strong

text: symmetry

text: corresponds

text: to

text: Strong

text: dynamics

text: and

text: chiral

text: symmetry

text: breaking

text: Let

text: us

text: ﬁrst

text: examine

text: the

text: case

text: where

text: the

text: weak

text: gauge

text: coupling

text: is

text: set

text: to

text: zero

text: and

text: the

text: weak

text: symmetry

text: is

text: purely

text: global.

text: Such

text: a

text: scenario

text: can

text: be

text: realized

text: by

text: freezing

text: the

text: auxiliary

text: scalar

text: φ

text: and

text: the

text: gauge

text: ﬁelds

text: U

text: and

text: W

text: to

text: φ

text: to

text: the

text: unit

text: matrix.

text: The

text: auxiliary

text: Yukawa

text: term

text: µ

text: described

text: above

text: now

text: acts

text: as

text: an

text: explicit

text: source

text: term

text: for

text: the

text: fermion

text: bilinear

text: Σ

text: ψ

text: (x)λ−(x)

text: λ−(x)ψ−(x)

text: S

text: We

text: will

text: be

text: interested

text: in

text: monitoring

text: the

text: corresponding

text: single

text: site

text: condensate

text: since

text: a

text: non-

text: zero

text: vacuum

text: expectation

text: value

text: for

text: this

text: observable

text: in

text: the

text: limit

text: g

text: would

text: indicate

text: a

text: spontaneous

text: breaking

text: of

text: down

text: to

text: its

text: diagonal

text: subgroup.

text: However,

text: this

text: is

text: not

text: the

text: only

text: fermion

text: bilinear

text: one

text: can

text: add

text: to

text: the

text: action;

text: it

text: is

text: also

text: possible

text: to

text: construct

text: a

text: single

text: link

text: operator

text: Σ

text: which

text: is

text: given

text: on

text: even

text: parity

text: sites

text: by

text: the

text: L

text: Σ

text: ψ

text: (x)V

text: (x)λ(x

text: µ)

text: V

text: (x

text: µ)λ(x

text: µ)

text: L

text: µ

text: µ

text: action

text: also

text: contains

text: the

text: corresponding

text: odd

text: site

text: bilinear

text: which

text: is

text: given

text: by

text: a

text: similar

text: expression.

text: We

text: observe

text: that

text: the

text: vacuum

text: expectation

text: of

text: the

text: odd

text: site

text: operator

text: matches

text: its

text: even

text: site

text: companion

text: as

text: one

text: would

text: expect

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 2: Σ vs β with g = 0.1 and L = 4 L S g with Σ approaching a constant as g is decreased before ﬁnally turning over to rapidly S approach zero for g = 0. Notice that this plateau region extends to smaller g as the volume increases consistent with a non-zero value of the condensate in the thermodynamic limit even for vanishing source g = 0. The results thus strongly support the idea that the global SU(2) × SU(2) symmetry is indeed spontaneously broken to its diagonal subgroup in the large volume continuum limit for non zero lattice spacing. At ﬁrst sight this is puzzling; the weak symmetries have a vector-like character in the lattice theory while the Vafa-Witten theorem would appear to prohibit the spontaneous breaking of such vector-like symmetries. In the next section we will show how to resolve this apparent contradiction. 4. Axial symmetries in the continuum limit Since each reduced staggered ﬁeld contributes 2 Dirac fermions in the continuum limit the theory will contain eight Dirac ﬁelds with global symmetry SU (8) × SU (8). As a result V A of strong interactions we expect that this symmetry will break according to the pattern SU (8) × SU (8) → SU (8). (4.1) V A V The SU(2) × SU(2) weak symmetries of the staggered lattice theory constitute a subgroup of this continuum global symmetry. The precise embedding of these weak symmetries in the continuum limit can be obtained by the following argument. We start by assuming – 6 – ##LTFigure## ##LTFigure##

text: Figure

text: Σ

label: caption

text: vs

text: β

text: with

text: g

text: and

text: L

text: L

text: S

text: g

text: with

text: Σ

text: approaching

text: a

text: constant

text: as

text: g

text: is

text: decreased

text: before

text: ﬁnally

text: turning

text: over

text: to

text: rapidly

text: S

text: approach

text: zero

text: for

text: g

text: Notice

text: that

text: this

text: plateau

text: region

text: extends

text: to

text: smaller

text: g

text: as

text: the

text: volume

text: increases

text: consistent

text: with

text: a

text: non-zero

text: value

text: of

text: the

text: condensate

text: in

text: the

text: thermodynamic

text: limit

text: even

text: for

text: vanishing

text: source

text: g

text: The

text: results

text: thus

text: strongly

text: support

text: the

text: idea

text: that

text: the

text: global

text: symmetry

text: is

text: indeed

text: spontaneously

text: broken

text: to

text: its

text: diagonal

text: subgroup

text: in

text: the

text: large

text: volume

text: continuum

text: limit

text: for

text: non

text: zero

text: lattice

text: spacing.

text: At

text: ﬁrst

text: sight

text: this

text: is

text: puzzling;

text: the

text: weak

text: symmetries

text: have

text: a

text: vector-like

text: character

text: in

text: the

text: lattice

text: theory

text: while

text: the

text: Vafa-Witten

text: theorem

text: would

text: appear

text: to

text: prohibit

text: the

text: spontaneous

text: breaking

text: of

text: such

text: vector-like

text: symmetries.

text: In

text: the

text: next

text: section

text: we

text: will

text: show

text: how

text: to

text: resolve

text: this

text: apparent

text: contradiction.

text: Axial

text: symmetries

text: in

text: the

text: continuum

text: limit

text: Since

text: each

text: reduced

text: staggered

text: ﬁeld

text: contributes

text: Dirac

text: fermions

text: in

text: the

text: continuum

text: limit

text: the

text: theory

text: will

text: contain

text: eight

text: Dirac

text: ﬁelds

text: with

text: global

text: symmetry

text: SU

text: SU

text: As

text: a

text: result

text: V

text: A

text: of

text: strong

text: interactions

text: we

text: expect

text: that

text: this

text: symmetry

text: will

text: break

text: according

text: to

text: the

text: pattern

text: SU

text: SU

text: SU

text: V

text: A

text: V

text: The

text: weak

text: symmetries

text: of

text: the

text: staggered

text: lattice

text: theory

text: constitute

text: a

text: subgroup

text: of

text: this

text: continuum

text: global

text: symmetry.

text: The

text: precise

text: embedding

text: of

text: these

text: weak

text: symmetries

text: in

text: the

text: continuum

text: limit

text: can

text: be

text: obtained

text: by

text: the

text: following

text: argument.

text: We

text: start

text: by

text: assuming

text: ##LTFigure##

text: ##LTFigure##


text: Standard universailty arguments tell us that it should be possible to change basis for our fermion ﬁelds to force this condensate to take the canonical ﬂavor symmetric form Σ = I . (cid:32) † (cid:33) 8 To accomplish this ﬁrst diagonalize M → P MP using the unitary transformation √1 (σ + σ ) 0 3 1 P = 2 0 √1 (σ + σ ) (4.6) 3 1 2 (cid:32) (cid:33) This results in a condensate of the form (cid:48) σ 0 3 Σ = (4.7) 0 σ 3 (cid:32) (cid:33) To transform this to the unit matrix we employ the non-anomalous chiral transformation (cid:48) (cid:48) M → QM Q with D 0 Q = † (4.8) (cid:32) 0 D (cid:33) where the 2 × 2 matrix 1 0 D = (4.9) 0 iγ 5 To ﬁnd the explicit form of the broken generators in this new basis we transform them † † according to the rule Q P τ P Q. It is straightforward to show that the broken generators (cid:32) (cid:33) a (cid:32) (cid:33) (cid:32) (cid:33) acquire an axial character in the new basis (we have reinserted the I at this point) 2 (cid:48) 0 −iσ (cid:48) 0 σ (cid:48) σ 0 1 1 2 τ = γ × I τ = γ × I τ = γ × I 1 5 iσ 0 2 2 5 σ 0 2 3 5 0 −σ 2 1 1 2 (4.10) This conﬁrms our earlier arguments that the broken generators do indeed correspond to axial symmetries in the continuum limit. We see that the staggered fermion action we use picks out a particular breaking direction corresponding to a speciﬁc embedding of the weak symmetries into the global symmetry group. The fact that the broken symmetries are axial in the continuum limit is a necessary condition, according to the Vafa-Witten theorem, for this broken phase of the lattice theory to survive the continuum limit2. One should contrast this with the situation in the continuum where the canonical chiral symmetry breaking pattern given in eqn. 4.1 gives rise to an inﬁnite number of degenerate vacua related by an SU(8) transformation. Depending on the vacuum that is picked any given SU(2) subgroup may be broken or left intact by the corresponding vacuum condensate. 5. Weakly gauging the SU(2) × SU(2) symmetries 2The reduced staggered theory has a sign problem for non zero lattice spacing. This allows a condensate that breaks vector symmetries to exist away from the continuum limit – 8 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Standard

text: universailty

text: arguments

text: tell

text: us

text: that

text: it

text: should

text: be

text: possible

text: to

text: change

text: basis

text: for

text: our

text: fermion

text: ﬁelds

text: to

text: force

text: this

text: condensate

text: to

text: take

text: the

text: canonical

text: ﬂavor

text: symmetric

text: form

text: Σ

text: I

text: To

text: accomplish

text: this

text: ﬁrst

text: diagonalize

text: M

text: P

text: MP

text: using

text: the

text: unitary

text: transformation

text: √1

text: (σ

text: σ

text: P

text: √1

text: (σ

text: σ

text: This

text: results

text: in

text: a

text: condensate

text: of

text: the

text: form

text: σ

text: Σ

text: σ

text: To

text: transform

text: this

text: to

text: the

text: unit

text: matrix

text: we

text: employ

text: the

text: non-anomalous

text: chiral

text: transformation

text: M

text: QM

text: Q

text: with

text: D

text: Q

text: D

text: where

text: the

text: matrix

text: D

text: iγ

text: To

text: ﬁnd

text: the

text: explicit

text: form

text: of

text: the

text: broken

text: generators

text: in

text: this

text: new

text: basis

text: we

text: transform

text: them

text: according

text: to

text: the

text: rule

text: Q

text: P

text: τ

text: P

text: Q.

text: It

text: is

text: straightforward

text: to

text: show

text: that

text: the

text: broken

text: generators

text: a

text: acquire

text: an

text: axial

text: character

text: in

text: the

text: new

text: basis

text: (we

text: have

text: reinserted

text: the

text: I

text: at

text: this

text: point)

text: −iσ

text: σ

text: σ

text: τ

text: γ

text: I

text: τ

text: γ

text: I

text: τ

text: γ

text: I

text: iσ

text: σ

text: −σ

text: This

text: conﬁrms

text: our

text: earlier

text: arguments

text: that

text: the

text: broken

text: generators

text: do

text: indeed

text: correspond

text: to

text: axial

text: symmetries

text: in

text: the

text: continuum

text: limit.

text: We

text: see

text: that

text: the

text: staggered

text: fermion

text: action

text: we

text: use

text: picks

text: out

text: a

text: particular

text: breaking

text: direction

text: corresponding

text: to

text: a

text: speciﬁc

text: embedding

text: of

text: the

text: weak

text: symmetries

text: into

text: the

text: global

text: symmetry

text: group.

text: The

text: fact

text: that

text: the

text: broken

text: symmetries

text: are

text: axial

text: in

text: the

text: continuum

text: limit

text: is

text: a

text: necessary

text: condition,

text: according

text: to

text: the

text: Vafa-Witten

text: theorem,

text: for

text: this

text: broken

text: phase

text: of

text: the

text: lattice

text: theory

text: to

text: survive

text: the

text: continuum

text: One

text: should

text: contrast

text: this

text: with

text: the

text: situation

text: in

text: the

text: continuum

text: where

text: the

text: canonical

text: chiral

text: symmetry

text: breaking

text: pattern

text: given

text: in

text: eqn.

text: gives

text: rise

text: to

text: an

text: inﬁnite

text: number

text: of

text: degenerate

text: vacua

text: related

text: by

text: an

text: transformation.

text: Depending

text: on

text: the

text: vacuum

text: that

text: is

text: picked

text: any

text: given

text: subgroup

text: may

text: be

text: broken

text: or

text: left

text: intact

text: by

text: the

text: corresponding

text: vacuum

text: condensate.

text: Weakly

text: gauging

text: the

text: symmetries

text: reduced

text: staggered

text: theory

text: has

text: a

text: sign

text: problem

text: for

text: non

text: zero

text: lattice

text: spacing.

text: This

text: allows

text: a

text: condensate

text: that

text: breaks

text: vector

text: symmetries

text: to

text: exist

text: away

text: from

text: the

text: continuum

text: limit

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 44 Lattice g = 0.1; β = β = 10.0 U W | − ψ − ¯λ ) †µ W µ U ( + λ + ¯ψ | Strong coupling, β s Figure 4: Absolute value of the four fermion condensate vs β for auxiliary Yukawa coupling g S = 0.1 We now examine the situation when the weak gauge coupling is switched on β = 10.0. (cid:88) W As a ﬁrst step, we analyzed the behavior of the gauge invariant four fermion term given by ∗ g < ψ (x)λ (x) > U (x)W (x) < λ−(x + µ)ψ−(x + µ) > . (5.1) + + µ µ µ This four fermion condensate, while strictly not an order parameter, is expected to be enhanced in the Higgs phase and this is indeed what we see in ﬁg. 4. Clearly for β > 6.0 S the condensate is small and approaches zero while for β < 6.0 it grows and approaches S a plateau within errors (we ﬁx the weak coupling β = 10.0 throughout the numerical W work). Notice that the plot in ﬁg. 4 was constructed with an auxiliary coupling g = 0.1. It is also very important to see how the condensate behaves for ﬁxed strong coupling β S as we vary g. Fig. 5 shows a plot at β = 5.5 as we scan in g. Clearly the value of the S condensate is insensitive to the auxiliary coupling until g becomes very small. We conclude that the magnitude of the four fermion condensate is determined primarily by the value of the strong coupling constant as expected. Further evidence that the lattice is indeed in a Higgs phase can be derived by plotting the appropriate weak Polyakov line that would correspond to the free energy of a static source transforming as a bi-fundamental under (cid:89) (cid:89) the weak symmetries. Explicitly we consider 1 ∗ P (x) = Tr U (x, t)Tr W (x, t) (5.2) W t t 4 t t – 9 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Lattice

text: g

text: β

text: β

text: U

text: W

text: ψ

text: ¯λ

text: †µ

text: W

text: µ

text: U

text: λ

text: ¯ψ

text: Strong

text: coupling,

text: β

text: s

text: Figure

text: Absolute

text: value

text: of

text: the

text: four

text: fermion

text: condensate

text: vs

text: β

text: for

text: auxiliary

text: Yukawa

text: coupling

text: g

text: S

text: We

text: now

text: examine

text: the

text: situation

text: when

text: the

text: weak

text: gauge

text: coupling

text: is

text: switched

text: on

text: β

text: W

text: As

text: a

text: ﬁrst

text: step,

text: we

text: analyzed

text: the

text: behavior

text: of

text: the

text: gauge

text: invariant

text: four

text: fermion

text: term

text: given

text: by

text: g

text: ψ

text: (x)λ

text: (x)

text: U

text: (x)W

text: (x)

text: λ−(x

text: µ)ψ−(x

text: µ)

text: µ

text: µ

text: µ

text: This

text: four

text: fermion

text: condensate,

text: while

text: strictly

text: not

text: an

text: order

text: parameter,

text: is

text: expected

text: to

text: be

text: enhanced

text: in

text: the

text: Higgs

text: phase

text: and

text: this

text: is

text: indeed

text: what

text: we

text: see

text: in

text: ﬁg.

text: Clearly

text: for

text: β

text: S

text: the

text: condensate

text: is

text: small

text: and

text: approaches

text: zero

text: while

text: for

text: β

text: it

text: grows

text: and

text: approaches

text: S

text: a

text: plateau

text: within

text: errors

text: (we

text: ﬁx

text: the

text: weak

text: coupling

text: β

text: throughout

text: the

text: numerical

text: W

text: work).

text: Notice

text: that

text: the

text: plot

text: in

text: ﬁg.

text: was

text: constructed

text: with

text: an

text: auxiliary

text: coupling

text: g

text: It

text: is

text: also

text: very

text: important

text: to

text: see

text: how

text: the

text: condensate

text: behaves

text: for

text: ﬁxed

text: strong

text: coupling

text: β

text: S

text: as

text: we

text: vary

text: g.

text: Fig.

text: shows

text: a

text: plot

text: at

text: β

text: as

text: we

text: scan

text: in

text: g.

text: Clearly

text: the

text: value

text: of

text: the

text: S

text: condensate

text: is

text: insensitive

text: to

text: the

text: auxiliary

text: coupling

text: until

text: g

text: becomes

text: very

text: small.

text: We

text: conclude

text: that

text: the

text: magnitude

text: of

text: the

text: four

text: fermion

text: condensate

text: is

text: determined

text: primarily

text: by

text: the

text: value

text: of

text: the

text: strong

text: coupling

text: constant

text: as

text: expected.

text: Further

text: evidence

text: that

text: the

text: lattice

text: is

text: indeed

text: in

text: a

text: Higgs

text: phase

text: can

text: be

text: derived

text: by

text: plotting

text: the

text: appropriate

text: weak

text: Polyakov

text: line

text: that

text: would

text: correspond

text: to

text: the

text: free

text: energy

text: of

text: a

text: static

text: source

text: transforming

text: as

text: a

text: bi-fundamental

text: under

text: the

text: weak

text: symmetries.

text: Explicitly

text: we

text: consider

text: P

text: (x)

text: Tr

text: U

text: (x,

text: t)Tr

text: W

text: (x,

text: t)

text: W

text: t

text: t

text: t

text: t

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 speciﬁc integrals in the Fourier plane that con- the complex wavevector k. This new property nect the real (Re ε) and imaginary (Im ε) parts extends the possibility to use the analyticity of permittivity (for instance) [8, 9]. In other for all angles of incidence from 0 to 90 de- words, it is well established that Im ε at one fre- grees. Next, a second property, denominated quency can be deduced from the values of Re ε by “passivity property”, shows that the sum at all other frequencies, and conversely. This is a of the reﬂection coeﬃcient with a phase shift, − general result for signals so-called causal signals. exp[ iβ◦(ω, k)d] + r(ω, k), cannot vanish in an Hence the amplitude reﬂection from a multilayer appropriate domain of the complex frequency should also follow the Kramers-Kronig criterion, and wavevector. This passivity property makes since the reﬂected ﬁeld can also be written as the it possible to apply the complex logarithm with- result of a linear ﬁlter, where the characteristic out alteration of the analytic properties with re- function is a double inverse Fourier transform spect to the complex frequency and wavevector. (over temporal and spatial frequency) of the re- They are used to propose alternative solutions to ﬂection coeﬃcient. In this case the consequence retrieve the phase of reﬂection and transmission is that the spectral phase of the stack can be coeﬃcients. In particular, since the quantity { } retrieved from the intensity spectral properties 1 + exp[iβ◦(ω, k)d] r(ω, k) cannot vanish, the of the same stack. Several authors have worked proposed alternative solutions for the reﬂection on this topic in various situations of multilayers coeﬃcient do not use Blaschke factors [11, 16]. [10, 11]. However, as pointed out in the litera- II. BACKGROUND ture, serious diﬃculties remain and result from Throughout this article an orthonormal basis the existence of zeros in reﬂection spectra since R is used: every vector x in 3 is described by its they produce branch points and cuts when the three components x , x and x (see Fig. 1). We 1 2 3 complex logarithm is used [12–14] to separate start with the usual Helmholtz equation in non the phase from the modulus. In addition, to magnetic, isotropic and linear media, and in the our knowledge, a rigorous proof of the possibil- absence of sources. The time-harmonic electric ity to use Kramers-Kronig relation at oblique ﬁeld E(x, ω) is the solution of incidence has not been established. In this paper, we use the techniques derived − ∇× ∇× ω2µ◦ε(x, ω)E(x, ω) E(x, ω) = 0 . (1) in [15] to show new properties of reﬂection and transmission coeﬃcients of multilayered stacks. ∇× First, the usual analytic properties with respect where is the curl operator, ω is the to the complex frequency z, currently stated temporal pulsation resulting from the Fourier at normal incidence [11, 16], are generalized to decomposition with respect to time, µ◦ is the

text: speciﬁc

text: integrals

text: in

text: the

text: Fourier

text: plane

text: that

text: con-

text: the

text: complex

text: wavevector

text: k.

text: This

text: new

text: property

text: nect

text: the

text: real

text: (Re

text: ε)

text: and

text: imaginary

text: (Im

text: ε)

text: parts

text: extends

text: the

text: possibility

text: to

text: use

text: the

text: analyticity

text: of

text: permittivity

text: (for

text: instance)

text: In

text: other

text: for

text: all

text: angles

text: of

text: incidence

text: from

text: to

text: de-

text: words,

text: it

text: is

text: well

text: established

text: that

text: Im

text: ε

text: at

text: one

text: fre-

text: grees.

text: Next,

text: a

text: second

text: property,

text: denominated

text: quency

text: can

text: be

text: deduced

text: from

text: the

text: values

text: of

text: Re

text: ε

text: by

text: “passivity

text: property”,

text: shows

text: that

text: the

text: sum

text: at

text: all

text: other

text: frequencies,

text: and

text: conversely.

text: This

text: is

text: a

text: of

text: the

text: reﬂection

text: coeﬃcient

text: with

text: a

text: phase

text: shift,

text: general

text: result

text: for

text: signals

text: so-called

text: causal

text: signals.

text: exp[

text: iβ◦(ω,

text: k)d]

text: r(ω,

text: k),

text: cannot

text: vanish

text: in

text: an

text: Hence

text: the

text: amplitude

text: reﬂection

text: from

text: a

text: multilayer

text: appropriate

text: domain

text: of

text: the

text: complex

text: frequency

text: should

text: also

text: follow

text: the

text: Kramers-Kronig

text: criterion,

text: and

text: wavevector.

text: This

text: passivity

text: property

text: makes

text: since

text: the

text: reﬂected

text: ﬁeld

text: can

text: also

text: be

text: written

text: as

text: the

text: it

text: possible

text: to

text: apply

text: the

text: complex

text: logarithm

text: with-

text: result

text: of

text: a

text: linear

text: ﬁlter,

text: where

text: the

text: characteristic

text: out

text: alteration

text: of

text: the

text: analytic

text: properties

text: with

text: re-

text: function

text: is

text: a

text: double

text: inverse

text: Fourier

text: transform

text: spect

text: to

text: the

text: complex

text: frequency

text: and

text: wavevector.

text: (over

text: temporal

text: and

text: spatial

text: frequency)

text: of

text: the

text: re-

text: They

text: are

text: used

text: to

text: propose

text: alternative

text: solutions

text: to

text: ﬂection

text: coeﬃcient.

text: In

text: this

text: case

text: the

text: consequence

text: retrieve

text: the

text: phase

text: of

text: reﬂection

text: and

text: transmission

text: is

text: that

text: the

text: spectral

text: phase

text: of

text: the

text: stack

text: can

text: be

text: coeﬃcients.

text: In

text: particular,

text: since

text: the

text: quantity

text: retrieved

text: from

text: the

text: intensity

text: spectral

text: properties

text: exp[iβ◦(ω,

text: k)d]

text: r(ω,

text: k)

text: cannot

text: vanish,

text: the

text: of

text: the

text: same

text: stack.

text: Several

text: authors

text: have

text: worked

text: proposed

text: alternative

text: solutions

text: for

text: the

text: reﬂection

text: on

text: this

text: topic

text: in

text: various

text: situations

text: of

text: multilayers

text: coeﬃcient

text: do

text: not

text: use

text: Blaschke

text: factors

text: However,

text: as

text: pointed

text: out

text: in

text: the

text: litera-

text: II.

text: BACKGROUND

text: ture,

text: serious

text: diﬃculties

text: remain

text: and

text: result

text: from

text: Throughout

text: this

text: article

text: an

text: orthonormal

text: basis

text: the

text: existence

text: of

text: zeros

text: in

text: reﬂection

text: spectra

text: since

text: R

text: is

text: used:

text: every

text: vector

text: x

text: in

text: is

text: described

text: by

text: its

text: they

text: produce

text: branch

text: points

text: and

text: cuts

text: when

text: the

text: three

text: components

text: x

text: x

text: and

text: x

text: (see

text: Fig.

text: We

text: complex

text: logarithm

text: is

text: used

text: to

text: separate

text: start

text: with

text: the

text: usual

text: Helmholtz

text: equation

text: in

text: non

text: the

text: phase

text: from

text: the

text: modulus.

text: In

text: addition,

text: to

text: magnetic,

text: isotropic

text: and

text: linear

text: media,

text: and

text: in

text: the

text: our

text: knowledge,

text: a

text: rigorous

text: proof

text: of

text: the

text: possibil-

text: absence

text: of

text: sources.

text: The

text: time-harmonic

text: electric

text: ity

text: to

text: use

text: Kramers-Kronig

text: relation

text: at

text: oblique

text: ﬁeld

text: E(x,

text: ω)

text: is

text: the

text: solution

text: of

text: incidence

text: has

text: not

text: been

text: established.

text: In

text: this

text: paper,

text: we

text: use

text: the

text: techniques

text: derived

text: ω2µ◦ε(x,

text: ω)E(x,

text: ω)

text: E(x,

text: ω)

text: in

text: to

text: show

text: new

text: properties

text: of

text: reﬂection

text: and

text: transmission

text: coeﬃcients

text: of

text: multilayered

text: stacks.

text: First,

text: the

text: usual

text: analytic

text: properties

text: with

text: respect

text: where

text: is

text: the

text: curl

text: operator,

text: ω

text: is

text: the

text: to

text: the

text: complex

text: frequency

text: z,

text: currently

text: stated

text: temporal

text: pulsation

text: resulting

text: from

text: the

text: Fourier

text: at

text: normal

text: incidence

text: are

text: generalized

text: to

text: decomposition

text: with

text: respect

text: to

text: time,

text: µ◦

text: is

text: the


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