text: 12 instead of three wave mixing. Another potential practical diﬃculty with the AC Stark eﬀect detection method is that the three driving ﬁelds producing the AC Stark eﬀect must be present when detecting the molecular absorption or → free induction decay of the a d transition. This will require that the frequency of this transition to be suﬃciently separated that one can use ﬁlters to selectively attenuate these driving ﬁelds. Another potential concern of the AC Stark measurement is that one will split the zero ﬁeld peak into multiple M components, and this will reduce the signal to noise ratio. IV. ACKNOWLEDGMENTS The author would like to thank Brooks Pate for encouragement and helpful discussions. ∗ Electronic address: \markcomm{footercolor}{lehmann@virginia.edu} 1 K. Mislo, Introduction to Stereochemistry (Dover, Mineola, New York, 2002). 2 L. N. Johnson, European Review 13, 77 (2005). 3 P. Kral, I. Thanopulos, M. Shapiro, and D. Cohen, Physical Review Letters 90, 033001 (2003). 4 P. Kral, I. Thanopulos, and M. Shapiro, Physical Review A 72, 020303 (2005). 5 D. Gerbasi, M. Shapiro, and P. Brumer, Journal of Chemical Physics 124, 074315 (2006). 6 E. Hirota, Proceedings of the Japan Academy Series B-Physical and Biological Sciences 88, 120 (2012). 7 D. Patterson, M. Schnell, and J. M. Doyle, Nature 497, 475 (2013). 8 D. Patterson and J. M. Doyle, Physical Review Letters 111, 023008 (2013). 9 J. H. Shirley, Phys. Rev. 138, B979 (1965). 10 R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (John Wiley & Sons, Ithaca, NY, 1988). 11 C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955). ##LTLine## ##LTLine## ##LTLine##

text: instead

text: of

text: three

text: wave

text: mixing.

text: Another

text: potential

text: practical

text: diﬃculty

text: with

text: the

text: AC

text: Stark

text: eﬀect

text: detection

text: method

text: is

text: that

text: the

text: three

text: driving

text: ﬁelds

text: producing

text: the

text: AC

text: Stark

text: eﬀect

text: must

text: be

text: present

text: when

text: detecting

text: the

text: molecular

text: absorption

text: or

text: free

text: induction

text: decay

text: of

text: the

text: a

text: d

text: transition.

text: This

text: will

text: require

text: that

text: the

text: frequency

text: of

text: this

text: transition

text: to

text: be

text: suﬃciently

text: separated

text: that

text: one

text: can

text: use

text: ﬁlters

text: to

text: selectively

text: attenuate

text: these

text: driving

text: ﬁelds.

text: Another

text: potential

text: concern

text: of

text: the

text: AC

text: Stark

text: measurement

text: is

text: that

text: one

text: will

text: split

text: the

text: zero

text: ﬁeld

text: peak

text: into

text: multiple

text: M

text: components,

text: and

text: this

text: will

text: reduce

text: the

text: signal

text: to

text: noise

text: ratio.

text: IV.

text: ACKNOWLEDGMENTS

text: The

text: author

text: would

text: like

text: to

text: thank

text: Brooks

text: Pate

text: for

text: encouragement

text: and

text: helpful

text: discussions.

text: Electronic

text: address:

text: K.

text: Mislo,

text: Introduction

text: to

text: Stereochemistry

text: (Dover,

text: Mineola,

text: New

text: York,

text: L.

text: N.

text: Johnson,

text: European

text: Review

text: P.

text: Kral,

text: I.

text: Thanopulos,

text: M.

text: Shapiro,

text: and

text: D.

text: Cohen,

text: Physical

text: Review

text: Letters

text: P.

text: Kral,

text: I.

text: Thanopulos,

text: and

text: M.

text: Shapiro,

text: Physical

text: Review

text: A

text: D.

text: Gerbasi,

text: M.

text: Shapiro,

text: and

text: P.

text: Brumer,

text: Journal

text: of

text: Chemical

text: Physics

text: E.

text: Hirota,

text: Proceedings

text: of

text: the

text: Japan

text: Academy

text: Series

text: B-Physical

text: and

text: Biological

text: Sciences

text: D.

text: Patterson,

text: M.

text: Schnell,

text: and

text: J.

text: M.

text: Doyle,

text: Nature

text: D.

text: Patterson

text: and

text: J.

text: M.

text: Doyle,

text: Physical

text: Review

text: Letters

text: J.

text: H.

text: Shirley,

text: Phys.

text: Rev.

text: R.

text: N.

text: Zare,

text: Angular

text: Momentum:

text: Understanding

text: Spatial

text: Aspects

text: in

text: Chemistry

text: and

text: Physics

text: (John

text: Wiley

text: Sons,

text: Ithaca,

text: NY,

text: C.

text: H.

text: Townes

text: and

text: A.

text: L.

text: Schawlow,

text: Microwave

text: Spectroscopy

text: (McGraw-Hill,

text: New

text: York,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 sets of transitions. For the case where the three coupled transitions are all rotational transition in a given vibronic state, each transition will have a sign determined by the sign of one of the components of the permanent electrical dipole moment projection onto a diﬀerent principle axis of the inertial tensor. In assigning the inertial axes to the molecule, one is free to pick the sign of two of the components of the permanent dipole moment to be positive, but then the sign of the third component will be ﬁxed by the requirement that one has a right handed coordinate system. Thus, if we have a mixture of a pair of enantiomers, there will be a diﬀerent sign of the third order AC Stark shifts of the energy levels while they are driven by the ﬁelds with the properties we invoked. The AC Stark eﬀect proposed here, will produce a pair of peaks that are shifted in equal and opposite directions from the positions of the peaks ↔ without the closed cycle. For example, if one is observing the frequency of the a d transition, then the third order Stark shift of state a can be observed by a shift in this resonance frequency when the ﬁeld at ω is switched on. Note 2 that ω is far from resonance with any transition coming directly from state a. 2 The second order Stark terms can be expected to be larger than the third for the case where perturbation theory is reasonably accurate. However, by selecting the signs of the two detunings to the two virtual levels from a state of interest, one can, for this simple three state model, arrange to cancel the second order shifts. For example, if we | | −| | select detunings and ﬁeld amplitudes such that Ω 2∆ω = Ω 2∆ω , the net second order AC Stark shift of state ab c ac b a will be zero, leaving only the enantiometric distinguishing third order terms. As will be demonstrated below, this does not hold when the (2J + 1) spatial degeneracy of states is considered. Going to 5’th order in pertu(cid:18)rbation theory, w(cid:19)e (cid:18)can write the en(cid:19)ergy of dressed state a as: | | | | | | (cid:18)− Ω 2 Ω (cid:19)2(cid:18) Ω 2 (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) E /¯h = ω ab + ac 1 + bc a a ∆ω ∆ω ∆ω ∆ω b c b c (cid:60) | | | | | | 2 (Ω Ω Ω ) Ω 2 − 1 1 Ω 2 Ω 2 + ac cb ba 1 + cb + ab + ac (6) ∆ω ∆ω ∆ω ∆ω ∆ω ∆ω ∆ω ∆ω b c b c b c b c | | −| | Thus, the 4’th order terms are the same for the two enantiomers and that selecting Ω 2∆ω = Ω 2∆ω , cancels ab c ac b both the 2’nd and 4’th order terms. The 5’th order terms are, like the 3’rd order terms, of opposite sign for a pair of enantiomers. If we select parameters to cancel out the 2’nd order term, then ∆ω ∆ω < 0 and the 5’th order term b | c | ∼ | | will cancel part of the 3’rd order contribution. This suggests that the optimal size of Ω 2 ∆ω ∆ω . This I have × bc b c conﬁrmed with numerical calculations with the 3 3 H . Figure 1 shows the AC Stark shifts of level a as a function ds − − of Ω for ∆ω = 1000, Ω = 100. The two curves are for ∆ω = 1000, Ω = 100 and ∆ω = 4000, Ω = 200, bc b ab (cid:112) c ac c ac selected such that the second order AC Stark shift is zero. In both cases, the range of AC Stark shift (as a function ± of Ω ) is 10, which is the same as the individual contributions to the second order Stark shifts from levels b and c. bc | | | | In both cases, the maximum shifts occur when Ω = ∆ω ∆ω . bc b c II. EFFECT OF SPATIAL DEGENERACY Transitions in gas phase molecules have degeneracies that arise from the rotational symmetry of space.10 These can be distinguished by specifying the projection of the total angular moment, with M quantum number, on some axis, usually the Z axis. Like for the static Stark eﬀect, this degeneracy will be at least partially lifted by the second and third order, chiral AC Stark eﬀects. In general, to account for this eﬀect, we must use degenerate perturbation theory × (cid:48) and diagonalize the (2J + 1) (2J + 1) matrix of M, M values for the level with total rotational angular momentum ↔ quantum number J. If we probe the spectrum of a transition J J where we can neglect the AC Stark eﬀect in f state J , the AC Stark shift and splitting pattern of state J will be directly observed in the spectrum. We can expand f the asymmetric top states in terms of symmetric top wavefunctions using:11 (cid:88) | (cid:105) J | (cid:105) J, τ, M = J, K, M A (7) K,τ − K= J where τ = 0 . . . 2J labels the diﬀerent rotational energy levels for ﬁxed J in increasing energy order. The more commonly used asymmetric rotor rotational state labels K and K can be determined using K =ﬂoor((τ + 1)/2) − a ≤ b a and K =ﬂoor((2J τ)/2) where ﬂoor(x) is the largest integer x. The transition dipole matrix element between c | (cid:48) (cid:48) (cid:48) | (cid:104) | | (cid:48) (cid:48) (cid:48)(cid:105) symmetric top states J , K , M > and J, K, M > is given by (E µ /2) J, τ, M φ J , τ , M , where the integral is G α α,G known as a direction cosine matrix element as it is the matrix element of the dot product of a unit vector aligned with the G = X, Y, Z axis of the laboratory axes system, and the unit vector along the α = A, B, C principle axes of the moment of inertia tensor of the molecule, in order of increasing moment of inertia. Townes & Schawlow11 give explicit (cid:104) | | (cid:48) (cid:48) (cid:48)(cid:105) (cid:104) | | (cid:48)(cid:105) (cid:104) | | (cid:48) (cid:48)(cid:105) (cid:104) | | (cid:48) (cid:48)(cid:105) expressions for these, using a notation J, K, M φ J , K , M = J φ J J, K φ J , K J, M φ J , M . I α,G J α α,G ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: sets

text: of

text: transitions.

text: For

text: the

text: case

text: where

text: the

text: three

text: coupled

text: transitions

text: are

text: all

text: rotational

text: transition

text: in

text: a

text: given

text: vibronic

text: state,

text: each

text: transition

text: will

text: have

text: a

text: sign

text: determined

text: by

text: the

text: sign

text: of

text: one

text: of

text: the

text: components

text: of

text: the

text: permanent

text: electrical

text: dipole

text: moment

text: projection

text: onto

text: a

text: diﬀerent

text: principle

text: axis

text: of

text: the

text: inertial

text: tensor.

text: In

text: assigning

text: the

text: inertial

text: axes

text: to

text: the

text: molecule,

text: one

text: is

text: free

text: to

text: pick

text: the

text: sign

text: of

text: two

text: of

text: the

text: components

text: of

text: the

text: permanent

text: dipole

text: moment

text: to

text: be

text: positive,

text: but

text: then

text: the

text: sign

text: of

text: the

text: third

text: component

text: will

text: be

text: ﬁxed

text: by

text: the

text: requirement

text: that

text: one

text: has

text: a

text: right

text: handed

text: coordinate

text: system.

text: Thus,

text: if

text: we

text: have

text: a

text: mixture

text: of

text: a

text: pair

text: of

text: enantiomers,

text: there

text: will

text: be

text: a

text: diﬀerent

text: sign

text: of

text: the

text: third

text: order

text: AC

text: Stark

text: shifts

text: of

text: the

text: energy

text: levels

text: while

text: they

text: are

text: driven

text: by

text: the

text: ﬁelds

text: with

text: the

text: properties

text: we

text: invoked.

text: The

text: AC

text: Stark

text: eﬀect

text: proposed

text: here,

text: will

text: produce

text: a

text: pair

text: of

text: peaks

text: that

text: are

text: shifted

text: in

text: equal

text: and

text: opposite

text: directions

text: from

text: the

text: positions

text: of

text: the

text: peaks

text: without

text: the

text: closed

text: cycle.

text: For

text: example,

text: if

text: one

text: is

text: observing

text: the

text: frequency

text: of

text: the

text: a

text: d

text: transition,

text: then

text: the

text: third

text: order

text: Stark

text: shift

text: of

text: state

text: a

text: can

text: be

text: observed

text: by

text: a

text: shift

text: in

text: this

text: resonance

text: frequency

text: when

text: the

text: ﬁeld

text: at

text: ω

text: is

text: switched

text: on.

text: Note

text: that

text: ω

text: is

text: far

text: from

text: resonance

text: with

text: any

text: transition

text: coming

text: directly

text: from

text: state

text: a.

text: The

text: second

text: order

text: Stark

text: terms

text: can

text: be

text: expected

text: to

text: be

text: larger

text: than

text: the

text: third

text: for

text: the

text: case

text: where

text: perturbation

text: theory

text: is

text: reasonably

text: accurate.

text: However,

text: by

text: selecting

text: the

text: signs

text: of

text: the

text: two

text: detunings

text: to

text: the

text: two

text: virtual

text: levels

text: from

text: a

text: state

text: of

text: interest,

text: one

text: can,

text: for

text: this

text: simple

text: three

text: state

text: model,

text: arrange

text: to

text: cancel

text: the

text: second

text: order

text: shifts.

text: For

text: example,

text: if

text: we

text: select

text: detunings

text: and

text: ﬁeld

text: amplitudes

text: such

text: that

text: Ω

text: 2∆ω

text: Ω

text: 2∆ω

text: the

text: net

text: second

text: order

text: AC

text: Stark

text: shift

text: of

text: state

text: ab

text: c

text: ac

text: b

text: a

text: will

text: be

text: zero,

text: leaving

text: only

text: the

text: enantiometric

text: distinguishing

text: third

text: order

text: terms.

text: As

text: will

text: be

text: demonstrated

text: below,

text: this

text: does

text: not

text: hold

text: when

text: the

text: spatial

text: degeneracy

text: of

text: states

text: is

text: considered.

text: Going

text: to

text: order

text: in

text: theory,

text: write

text: the

text: of

text: dressed

text: state

text: a

text: as:

text: Ω

text: Ω

text: Ω

text: E

text: /¯h

text: ω

text: ab

text: ac

text: bc

text: a

text: a

text: ∆ω

text: ∆ω

text: ∆ω

text: ∆ω

text: b

text: c

text: b

text: c

text: (Ω

text: Ω

text: Ω

text: Ω

text: Ω

text: Ω

text: ac

text: cb

text: ba

text: cb

text: ab

text: ac

text: ∆ω

text: ∆ω

text: ∆ω

text: ∆ω

text: ∆ω

text: ∆ω

text: ∆ω

text: ∆ω

text: b

text: c

text: b

text: c

text: b

text: c

text: b

text: c

text: Thus,

text: the

text: order

text: terms

text: are

text: the

text: same

text: for

text: the

text: two

text: enantiomers

text: and

text: that

text: selecting

text: Ω

text: 2∆ω

text: Ω

text: 2∆ω

text: cancels

text: ab

text: c

text: ac

text: b

text: both

text: the

text: and

text: order

text: terms.

text: The

text: order

text: terms

text: are,

text: like

text: the

text: order

text: terms,

text: of

text: opposite

text: sign

text: for

text: a

text: pair

text: of

text: enantiomers.

text: If

text: we

text: select

text: parameters

text: to

text: cancel

text: out

text: the

text: order

text: term,

text: then

text: ∆ω

text: ∆ω

text: and

text: the

text: order

text: term

text: b

text: c

text: will

text: cancel

text: part

text: of

text: the

text: order

text: contribution.

text: This

text: suggests

text: that

text: the

text: optimal

text: size

text: of

text: Ω

text: ∆ω

text: ∆ω

text: This

text: I

text: have

text: bc

text: b

text: c

text: conﬁrmed

text: with

text: numerical

text: calculations

text: with

text: the

text: H

text: Figure

text: shows

text: the

text: AC

text: Stark

text: shifts

text: of

text: level

text: a

text: as

text: a

text: function

text: ds

text: of

text: Ω

text: for

text: ∆ω

text: Ω

text: The

text: two

text: curves

text: are

text: for

text: ∆ω

text: Ω

text: and

text: ∆ω

text: Ω

text: bc

text: b

text: ab

text: c

text: ac

text: c

text: ac

text: selected

text: such

text: that

text: the

text: second

text: order

text: AC

text: Stark

text: shift

text: is

text: zero.

text: In

text: both

text: cases,

text: the

text: range

text: of

text: AC

text: Stark

text: shift

text: (as

text: a

text: function

text: of

text: Ω

text: is

text: which

text: is

text: the

text: same

text: as

text: the

text: individual

text: contributions

text: to

text: the

text: second

text: order

text: Stark

text: shifts

text: from

text: levels

text: b

text: and

text: c.

text: bc

text: In

text: both

text: cases,

text: the

text: maximum

text: shifts

text: occur

text: when

text: Ω

text: ∆ω

text: ∆ω

text: bc

text: b

text: c

text: II.

text: EFFECT

text: OF

text: SPATIAL

text: DEGENERACY

text: Transitions

text: in

text: gas

text: phase

text: molecules

text: have

text: degeneracies

text: that

text: arise

text: from

text: the

text: rotational

text: symmetry

text: of

text: These

text: can

text: be

text: distinguished

text: by

text: specifying

text: the

text: projection

text: of

text: the

text: total

text: angular

text: moment,

text: with

text: M

text: quantum

text: number,

text: on

text: some

text: axis,

text: usually

text: the

text: Z

text: axis.

text: Like

text: for

text: the

text: static

text: Stark

text: eﬀect,

text: this

text: degeneracy

text: will

text: be

text: at

text: least

text: partially

text: lifted

text: by

text: the

text: second

text: and

text: third

text: order,

text: chiral

text: AC

text: Stark

text: eﬀects.

text: In

text: general,

text: to

text: account

text: for

text: this

text: eﬀect,

text: we

text: must

text: use

text: degenerate

text: perturbation

text: theory

text: and

text: diagonalize

text: the

text: matrix

text: of

text: M,

text: M

text: values

text: for

text: the

text: level

text: with

text: total

text: rotational

text: angular

text: momentum

text: quantum

text: number

text: J.

text: If

text: we

text: probe

text: the

text: spectrum

text: of

text: a

text: transition

text: J

text: J

text: where

text: we

text: can

text: neglect

text: the

text: AC

text: Stark

text: eﬀect

text: in

text: f

text: state

text: J

text: the

text: AC

text: Stark

text: shift

text: and

text: splitting

text: pattern

text: of

text: state

text: J

text: will

text: be

text: directly

text: observed

text: in

text: the

text: spectrum.

text: We

text: can

text: expand

text: f

text: the

text: asymmetric

text: top

text: states

text: in

text: terms

text: of

text: symmetric

text: top

text: wavefunctions

text: J

text: J,

text: τ,

text: M

text: J,

text: K,

text: M

text: A

text: K,τ

text: K=

text: J

text: where

text: τ

text: labels

text: the

text: diﬀerent

text: rotational

text: energy

text: levels

text: for

text: ﬁxed

text: J

text: in

text: increasing

text: energy

text: order.

text: The

text: more

text: commonly

text: used

text: asymmetric

text: rotor

text: rotational

text: state

text: labels

text: K

text: and

text: K

text: can

text: be

text: determined

text: using

text: K

text: =ﬂoor((τ

text: a

text: b

text: a

text: and

text: K

text: where

text: ﬂoor(x)

text: is

text: the

text: largest

text: integer

text: x.

text: The

text: transition

text: dipole

text: matrix

text: element

text: between

text: c

text: symmetric

text: top

text: states

text: J

text: K

text: M

text: and

text: J,

text: K,

text: M

text: is

text: given

text: by

text: (E

text: µ

text: J,

text: τ,

text: M

text: φ

text: J

text: τ

text: M

text: where

text: the

text: integral

text: is

text: G

text: α

text: α,G

text: known

text: as

text: a

text: direction

text: cosine

text: matrix

text: element

text: as

text: it

text: is

text: the

text: matrix

text: element

text: of

text: the

text: dot

text: product

text: of

text: a

text: unit

text: vector

text: aligned

text: with

text: the

text: G

text: X,

text: Y,

text: Z

text: axis

text: of

text: the

text: laboratory

text: axes

text: system,

text: and

text: the

text: unit

text: vector

text: along

text: the

text: α

text: A,

text: B,

text: C

text: principle

text: axes

text: of

text: the

text: moment

text: of

text: inertia

text: tensor

text: of

text: the

text: molecule,

text: in

text: order

text: of

text: increasing

text: moment

text: of

text: inertia.

text: Townes

text: give

text: explicit

text: expressions

text: for

text: these,

text: using

text: a

text: notation

text: J,

text: K,

text: M

text: φ

text: J

text: K

text: M

text: J

text: φ

text: J

text: J,

text: K

text: φ

text: J

text: K

text: J,

text: M

text: φ

text: J

text: M

text: I

text: α,G

text: J

text: α

text: α,G

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: tfihS kratS CA tfihS kratS CA 8 − shifts for the same transitions when ∆ω = ∆ω = 0, Ω = Ω = 100 and Ω = 100i, 0, +100i. In the cases with b c ab ac bc (cid:54) zero detunings of the 3-cycle transitions, the ∆ω spectrum is symmetric for Ω = 0. This is lifted for Ω = 0 but ad bc bc it is symmetric for simultaneous change in the signs of ∆ω and Ω . ad bc 0 -50 -100 -150 -200 -1500 -1000 -500 0 500 1000 1500 Im( ) Ω 23 FIG. 5: Plot of AC Stark Shifts of the J = 2 state as function of Ω with J = J = 1, Ω = 100, Ω = 200, ∆ω = ∆ω = 1000 1 bc 2 3 ab ac b c 25 20 15 10 5 0 -5 -10 -1500 -1000 -500 0 500 1000 1500 Im( ) Ω 23 FIG. 6: Plot of AC Stark Shifts of the J = 2 state as function of Ω with J = J = 1, Ω = 100, Ω = 200, ∆ω = − 1 bc 2 3 ab ac b 1000, ∆ω = 1000 c ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: tfihS

font: default

text: kratS

text: CA

text: tfihS

text: kratS

text: CA

text: shifts

text: for

text: the

text: same

text: transitions

text: when

text: ∆ω

text: ∆ω

text: Ω

text: Ω

text: and

text: Ω

text: In

text: the

text: cases

text: with

text: b

text: c

text: ab

text: ac

text: bc

text: zero

text: detunings

text: of

text: the

text: transitions,

text: the

text: ∆ω

text: spectrum

text: is

text: symmetric

text: for

text: Ω

text: This

text: is

text: lifted

text: for

text: Ω

text: but

text: ad

text: bc

text: bc

text: it

text: is

text: symmetric

text: for

text: simultaneous

text: change

text: in

text: the

text: signs

text: of

text: ∆ω

text: and

text: Ω

text: ad

text: bc

text: Im(

text: Ω

text: FIG.

text: Plot

text: of

text: AC

text: Stark

text: Shifts

text: of

text: the

text: J

text: state

text: as

text: function

text: of

text: Ω

text: with

text: J

text: J

text: Ω

text: Ω

text: ∆ω

text: ∆ω

text: bc

text: ab

text: ac

text: b

text: c

text: Im(

text: Ω

text: FIG.

text: Plot

text: of

text: AC

text: Stark

text: Shifts

text: of

text: the

text: J

text: state

text: as

text: function

text: of

text: Ω

text: with

text: J

text: J

text: Ω

text: Ω

text: ∆ω

text: bc

text: ab

text: ac

text: b

text: ∆ω

text: c

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: noitisnarT 1 + J >- J rof ytisnetnI a a 10 4 3.5 3 2.5 2 1.5 1 0.5 0 -40 -30 -20 -10 0 AC Stark Shift ↔ FIG. 9: Simulated J = 2 3 spectrum displaying AC Stark shifts for J = J = 1, Ω = 100, Ω = 200, ∆ω = ∆ω = 1000. a − 2 3 ab ac b c The spectra are oﬀset for clarity, with (from bottom to top) Ω = 1200i, 0, +1200i bc 2 1.5 1 0.5 0 -150 -100 -50 0 50 100 150 Δω ad ↔ FIG. 10: Simulated J = 2 3 spectrum displaying AC Stark shifts for J = J = 1, Ω = 100, Ω = 100, ∆ω = ∆ω = 0. a − 2 3 ab ac b c The spectra are oﬀset for clarity, with (from bottom to top) Ω = 100i, 0, +100i bc ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: noitisnarT

text: J

text: J

text: rof

text: ytisnetnI

text: a

text: a

text: AC

text: Stark

text: Shift

label: caption

text: FIG.

text: Simulated

text: J

text: spectrum

text: displaying

text: AC

text: Stark

text: shifts

text: for

text: J

text: J

text: Ω

text: Ω

text: ∆ω

text: ∆ω

text: a

text: ab

text: ac

text: b

text: c

text: The

text: spectra

text: are

text: oﬀset

text: for

text: clarity,

text: with

text: (from

text: bottom

text: to

text: top)

text: Ω

text: bc

text: Δω

text: ad

text: FIG.

text: Simulated

text: J

text: spectrum

text: displaying

text: AC

text: Stark

text: shifts

text: for

text: J

text: J

text: Ω

text: Ω

text: ∆ω

text: ∆ω

text: a

text: ab

text: ac

text: b

text: c

text: The

text: spectra

text: are

text: oﬀset

text: for

text: clarity,

text: with

text: (from

text: bottom

text: to

text: top)

text: Ω

text: bc

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: NONLINEAR FOURIER TRANSFORMS 11 We ﬁrst consider [X] . Our ﬁrst goal is to show that Xˆ is well-deﬁned and invertible 2 for k large enough. { } ≥ Claim(cid:40)1. X (x) m+1 are C1 functions of x 0 satisfying j 1 ∈ ∞ ∩ ∞ ∞ (1 + x)nX (x) L1([0, )) L ([0, )), j (cid:48) ∈ ∞ j = 1, . . . , m + 1. (3.27) (1 + x)nX (x) L1([0, )), j Proof of Claim 1. The assumption (3.3) implies | | ≤ C ≥ u(i)(x) , x 0, i = 0, 1, . . . , m. (3.28) (1 + x)n ≥ (cid:90) (cid:12)Indeed, if i = 0(cid:12), 1, . . . , m, and x (cid:12) 0, then (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ≤ | | x (cid:48) − (cid:48) (cid:48) (cid:48) (cid:48) (1 + x)nu(i)(x) u(i)(0) + n(1 + x )n 1u(i)(x ) + (1 + x )nu(i+1)(x ) dx < C. 0 (3.29)   Let S referto the statement j ∈ − ∞ ∈ − ∞ X(o) Cm+2 j([0, )), X(d) Cm+3 j([0, )), j ∈ ∞ j − (1 + x)n∂iX(o) L1([0, )) for i = 0, 1, . . . , m + 2 j, j ∈ ∞ − (1 + x)n∂iX(d) L1([0, )) for i = 0, 1, . . . , m + 3 j. (cid:90) j (cid:90) (cid:90) By (3.28), ∞ ∞ ∞ (cid:48) (cid:48) ≤ (cid:48) (cid:48) (cid:48) ≤ (cid:48) | (cid:48) | (cid:48) → (1 + x)n+1 u2(x )dx (1 + x )n+1u2(x )dx C (1 + x ) u(x ) dx 0 x x x → ∞ (cid:90) (cid:90) as x . Consequently, an integration by parts yields ∞ ∞ (cid:107) (cid:90) (cid:107) ≤ (cid:90) (cid:48) (cid:48) (1 + x)nX(d) ∞ C (1 + x)n u2(x )dx dx 1 L1([0, )) 0 x ∞ ∞ ≤ (1 + x)n+1 ≤ | | ∞ C + C u2(x)dx C + C (1 + x) u(x) dx < . (3.30) n + 1 0 0 Using the estimate (3.30) and the expression (3.10) for X , we conclude that S holds. 1 ≤ ≤ 1 Similar estimates together with the relations (3.8) imply that if 1 j m and S holds, { } { } j then S also holds. Thus, by induction, S m+1 hold. This shows that X m+1 are j+1 ∈ j j=∞1 j 1 C1 functions satisfying (1 + x)n∂iXj(x) L1([0, )) for i = 0, 1 and j = 1, . . . , m +(cid:53)1. The boundedness of (1 + x)nX (x) follows by an estimate analogous to (3.29). j − ∈ C | | ≥ Claim 2. There exists a K > 0 such that Xˆ(x, k) 1 exists for all k with k K. Moreover, letting(cid:0)A = ikσ + U and (cid:1) 3 − ≥ | | ≥ Aˆ(x, k) = Xˆ (x, k) + ikXˆ(x, k)σ Xˆ(x, k) 1, x 0, k K, (3.31) x 3 − the diﬀerence ∆(x, k) = A(x, k) Aˆ(x, k) satisﬁes ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: We

text: ﬁrst

text: consider

text: [X]

text: Our

text: ﬁrst

text: goal

text: is

text: to

text: show

text: that

text: Xˆ

text: is

text: well-deﬁned

text: and

text: invertible

text: for

text: k

text: large

text: enough.

label: equation

text: X

text: (x)

text: are

text: functions

text: of

text: x

text: satisfying

text: j

text: x)nX

text: (x)

text: L

text: j

text: j

text: m

text: x)nX

text: (x)

text: j

text: Proof

text: of

text: Claim

text: The

text: assumption

text: implies

text: C

text: u(i)(x)

text: x

text: i

text: m.

text: x)n

text: if

text: i

text: m,

text: and

text: x

text: then

text: x

text: x)nu(i)(x)

text: x

text: )n

text: x

text: dx

text: C.

text: Let

text: S

text: referto

text: the

text: statement

text: j

text: X(o)

text: X(d)

text: j

text: j

text: x)n∂iX(o)

text: for

text: i

text: m

text: j,

text: j

text: x)n∂iX(d)

text: for

text: i

text: m

text: j.

text: j

text: By

text: )dx

text: x

text: )dx

text: C

text: x

text: u(x

text: dx

text: x

text: x

text: x

text: as

text: x

text: Consequently,

text: an

text: integration

text: by

text: parts

text: yields

text: x)nX(d)

text: C

text: x)n

text: )dx

text: dx

text: x

text: C

text: C

text: C

text: C

text: x)

text: u(x)

text: dx

text: n

text: Using

text: the

text: estimate

text: and

text: the

text: expression

text: for

text: X

text: we

text: conclude

text: that

text: S

text: holds.

text: Similar

text: estimates

text: together

text: with

text: the

text: relations

text: imply

text: that

text: if

text: j

text: m

text: and

text: S

text: holds,

text: j

text: then

text: S

text: also

text: holds.

text: Thus,

text: by

text: induction,

text: S

text: hold.

text: This

text: shows

text: that

text: X

text: are

text: j

text: j

text: functions

text: satisfying

text: x)n∂iXj(x)

text: for

text: i

text: and

text: j

text: m

text: The

text: boundedness

text: of

text: x)nX

text: (x)

text: follows

text: by

text: an

text: estimate

text: analogous

text: to

text: j

text: C

text: Claim

text: There

text: exists

text: a

text: K

text: such

text: that

text: Xˆ(x,

text: k)

text: exists

text: for

text: all

text: k

text: with

text: k

text: K.

text: Moreover,

text: ikσ

text: U

text: and

text: Aˆ(x,

text: k)

text: Xˆ

text: (x,

text: k)

text: ikXˆ(x,

text: k)σ

text: Xˆ(x,

text: k)

text: x

text: k

text: K,

text: x

text: the

text: diﬀerence

text: ∆(x,

text: k)

text: A(x,

text: k)

text: Aˆ(x,

text: k)

text: satisﬁes

text: ##LTLine##

text: ##LTLine##


text: NONLINEAR FO(cid:18)URIER TRANSF(cid:19)ORMS 35 where − (cid:48) (cid:48) e8ik3(t t ) 0 − (cid:48) E(t, t , k) = Wˆ (t, k) Wˆ 1(t , k). 0 1 As in the proof of Claim 8, the estimates | (cid:48) | | | | (cid:48) − | ≤ (cid:48) ≤ ∞ ∈ ∂jE(t, t , k) < C k 2j(1 + t t )j, 0 t t < , k D¯K, j = 0, 1, . . . , n, k + and | | ≤ ≥ ∈ Ψ (t, k) C, t 0, k D¯K, 0 + together with (4.44) yield | − | ≤ Cte|k|Cmt+1 ≥ ∈ Ψ(t, k) Ψ (t, k) | | , t 0, k D¯K. (4.55) 0 k m+1 + Equations (4.45b) and (4.55) prove the second column of (4.13c) for j = 0. Proceeding as in the proof of Claim 8, (4.13c) follows also for j = 1, . . . , n. (cid:50) ∈ We let S(k) = T (0, k) and (cid:32)deﬁne the spect(cid:33)ral functions A(k) and B(k) for k D¯ by + A(k¯) B(k) ∈ S(k) = , k (D¯−, D¯ ). (4.56) λB(k¯) A(k) + { } ≥ ≥ Theorem 4.3. Suppose g (t) 2 satisfy (4.3) for some integers m 1 and n 1. Then j 0 the spectral functions A(k) and B(k) have the following properties: ∈ ∈ (a) A(k) and B(k) are continuous for k D¯ and analytic for k D . { } + + (b) There exist complex constants A , B m such that(cid:18) (cid:19) j j 1 A · · · A (cid:18) (cid:19)1 1 m A(k) = 1 + + + + O , k km km+1 B · · · B 1 1 m B(k) = + + + O , (4.57a) k km km+1 → ∞ ∈ uniformly as k with k D¯ . + (c) For j = 1, . . . n, the derivatives A(j)(k) and B(j)(k) have continuous extensions to ∈ k D¯ and (cid:18) (cid:19) (cid:18) (cid:19) + dj (cid:18) A · · · (cid:19)A (cid:18) 1(cid:19) A(j)(k) = 1 + 1 + + m + O − , dkk k km km+1 2j dj B · · · B 1 B(j)(k) = 1 + + m + O − , (4.57b) dkj k km km+1 2j → ∞ ∈ uniformly as k with k D¯ . + ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: where

text: t

text: E(t,

text: t

text: k)

text: Wˆ

text: (t,

text: k)

text: Wˆ

text: k).

text: As

text: in

text: the

text: proof

text: of

text: Claim

text: the

text: estimates

text: ∂jE(t,

text: t

text: k)

text: C

text: k

text: t

text: t

text: )j,

text: t

text: t

text: k

text: D¯K,

text: j

text: n,

text: k

text: and

text: Ψ

text: (t,

text: k)

text: C,

text: t

text: k

text: D¯K,

text: together

text: with

text: yield

text: Ψ(t,

text: k)

text: Ψ

text: (t,

text: k)

text: t

text: k

text: D¯K.

text: k

text: Equations

text: and

text: prove

text: the

text: second

text: column

text: of

text: for

text: j

text: Proceeding

text: as

text: in

text: the

text: proof

text: of

text: Claim

text: follows

text: also

text: for

text: j

text: n.

text: We

text: let

text: S(k)

text: T

text: k)

text: and

text: the

text: functions

text: A(k)

text: and

text: B(k)

text: for

text: k

text: D¯

text: by

text: A(k¯)

text: B(k)

text: S(k)

text: k

text: (D¯−,

text: D¯

text: λB(k¯)

text: A(k)

text: Theorem

text: Suppose

text: g

text: (t)

text: satisfy

text: for

text: some

text: integers

text: m

text: and

text: n

text: Then

text: j

text: the

text: spectral

text: functions

text: A(k)

text: and

text: B(k)

text: have

text: the

text: following

text: properties:

text: (a)

text: A(k)

text: and

text: B(k)

text: are

text: continuous

text: for

text: k

text: D¯

text: and

text: analytic

text: for

text: k

text: D

text: (b)

text: There

text: exist

text: complex

text: constants

text: A

text: B

text: m

text: such

text: j

text: j

text: A

text: A

text: m

text: A(k)

text: O

text: k

text: km

text: B

text: B

text: m

text: B(k)

text: O

text: k

text: km

text: uniformly

text: as

text: k

text: with

text: k

text: D¯

text: (c)

text: For

text: j

text: n,

text: the

text: derivatives

text: A(j)(k)

text: and

text: B(j)(k)

text: have

text: continuous

text: extensions

text: to

text: k

text: D¯

text: and

text: dj

text: A

text: A(j)(k)

text: m

text: O

text: dkk

text: k

text: km

text: dj

text: B

text: B

text: B(j)(k)

text: m

text: O

text: dkj

text: k

text: km

text: uniformly

text: as

text: k

text: with

text: k

text: D¯

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: ##LTLine##


text: NONLINEAR FOURIER TRANSFORMS 39 ≥ ≥ Claim 2. (Vanishing Lemma) Let x 0 and t 0. Suppose N(x, t, k) is a solution of · the homogeneous(cid:40)L2-RH problem determined by (Γ, J(x, t, )), i.e. · ∈ C \ N(x, t, ) E˙ 2( Γ), ∈ (5.9) N (x, t, k) = N−(x, t, k)J(x, t, k) for a.e. k Γ. + Then N vanishes identically. − Proof of Claim 2. For λ = 1, this holds by assumption. Thus suppose λ = 1. We write N(k) := N(x, t, k(cid:90)) and let G(k) = N(k)N(cid:90)(k¯)T . Applying Lemma A.6 with m = 2 ∈ C \ and n = 0, we obtain G E˙ 1( Γ) and G (k)dk = 0, G−(k)dk = 0. + ∂D3(cid:90) (cid:90) ∂D4 Adding these equations, we ﬁnd − − G dk (G G−)dk = 0. + + R ∂D4 Since G (k) = N (k)N−(k¯)T = N−(k)J(x, t, k)N−(k¯)T + + = N−(k)J(x, t, k¯)T N−(k¯)T = G−(k), k ∈ Γ, (cid:90) (cid:90) this gives 0 = G (k)dk = N−(k)J(x, t, k)N−(k)T dk. + R R ∈ R For k , J(x, t, k) is a Hermitian matrix with (11) entry 1 and determinant one; by R Sylvester’s criterion it is positive deﬁnite. It follows that N− = 0 a.e. on . But then R N = N−J also vanishes a.e. on . + ∪ R ∪ Let B be a small open ball contained in(cid:90)D D centered at some nonzero real r 2 3 number. The function N˜(k) deﬁned by 1 N(s) N˜(k) = − ds 2πi s k (cid:90) ∂B(cid:90)r ∈ \ R ∈ ∩ C is analytic in B and equals N(k) for k B . Indeed, if k B , then r r r + 1 N (s) 1 N−(s) N˜(k) = +− ds + − ds = N(k) + 0, 2πi ∂(Br∩C+) s k 2πi ∂(Br∩C−) s k ∈ ∩ C and a similar argument applies if k B −. We infer that N(k) s analytic in B . ∩ R r r Since N = 0 on B , it follows by analytic continuation that N vanishes identically ∈ ∪ R ∪ r ∪ for k D2 D3. But∈thCen\ N± = 0 on ∂D1 ∂D4 so by a similar argument we ﬁn(cid:53)d that N vanishes for all k Γ. − The global relation (5.3) implies that h(0) = b(0) and r(0) = 0.1 It follows that a(0) unless b(0) = 0, the matrix J is not continuous at k = 0. In order to obtain a jump 1The symmetries (3.62) and (4.58) imply that the values of a, b, A, B at k = 0 are real. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: Claim

text: (Vanishing

text: Lemma)

text: Let

text: x

text: and

text: t

text: Suppose

text: N(x,

text: t,

text: k)

text: is

text: a

text: solution

text: of

text: the

text: problem

text: determined

text: by

text: (Γ,

text: J(x,

text: t,

text: i.e.

text: C

text: N(x,

text: t,

text: E˙

text: Γ),

text: N

text: (x,

text: t,

text: k)

text: N−(x,

text: t,

text: k)J(x,

text: t,

text: k)

text: for

text: a.e.

text: k

text: Γ.

text: Then

text: N

text: vanishes

text: identically.

text: Proof

text: of

text: Claim

text: For

text: λ

text: this

text: holds

text: by

text: assumption.

text: Thus

text: suppose

text: λ

text: We

text: write

text: N(k)

text: N(x,

text: t,

text: and

text: let

text: G(k)

text: Applying

text: Lemma

text: with

text: m

text: C

text: and

text: n

text: we

text: obtain

text: G

text: E˙

text: Γ)

text: and

text: G

text: (k)dk

text: G−(k)dk

text: Adding

text: these

text: equations,

text: we

text: ﬁnd

text: G

text: dk

text: (G

text: G−)dk

text: R

text: Since

text: G

text: (k)

text: N

text: (k)N−(k¯)T

text: N−(k)J(x,

text: t,

text: k)N−(k¯)T

text: N−(k)J(x,

text: t,

text: k¯)T

text: N−(k¯)T

text: G−(k),

text: k

text: Γ,

text: this

text: gives

text: G

text: (k)dk

text: N−(k)J(x,

text: t,

text: k)N−(k)T

text: dk.

text: R

text: R

text: R

text: For

text: k

text: J(x,

text: t,

text: k)

text: is

text: a

text: Hermitian

text: matrix

text: with

text: entry

text: and

text: determinant

text: one;

text: by

text: R

text: Sylvester’s

text: criterion

text: it

text: is

text: positive

text: deﬁnite.

text: It

text: follows

text: that

text: N−

text: a.e.

text: on

text: But

text: then

text: R

text: N

text: N−J

text: also

text: vanishes

text: a.e.

text: on

text: R

text: Let

text: B

text: be

text: a

text: small

text: open

text: ball

text: contained

text: D

text: centered

text: at

text: some

text: nonzero

text: real

text: r

text: number.

text: The

text: function

text: N˜(k)

text: deﬁned

text: by

text: N(s)

text: N˜(k)

text: ds

text: s

text: k

text: R

text: C

text: is

text: analytic

text: in

text: B

text: and

text: equals

text: N(k)

text: for

text: k

text: B

text: Indeed,

text: if

text: k

text: B

text: then

text: r

text: r

text: r

text: N

text: (s)

text: N−(s)

text: N˜(k)

text: ds

text: ds

text: N(k)

text: ∂(Br∩C+)

text: s

text: k

text: ∂(Br∩C−)

text: s

text: k

text: C

text: and

text: a

text: similar

text: argument

text: applies

text: if

text: k

text: B

text: We

text: infer

text: that

text: N(k)

text: s

text: analytic

text: in

text: B

text: R

text: r

text: r

text: Since

text: N

text: on

text: B

text: it

text: follows

text: by

text: analytic

text: continuation

text: that

text: N

text: vanishes

text: identically

text: R

text: r

text: for

text: k

text: But∈thCen\

text: N±

text: on

text: so

text: by

text: a

text: similar

text: argument

text: we

text: that

text: N

text: vanishes

text: for

text: all

text: k

text: Γ.

text: The

text: global

text: relation

text: implies

text: that

text: and

text: It

text: follows

text: that

text: unless

text: the

text: matrix

text: J

text: is

text: not

text: continuous

text: at

text: k

text: In

text: order

text: to

text: obtain

text: a

text: jump

text: symmetries

text: and

text: imply

text: that

text: the

text: values

text: of

text: a,

text: b,

text: A,

text: B

text: at

text: k

text: are

text: real.

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: NONLINEAR FOURIER TRANSFORMS 43 (cid:88) 3 m (x, t) − → ∞ ∈ m(x, t, k) = I + j + O(k 4), k , k W, (5.20) kj (cid:90)j=1 ∈ where the error term is uniform with respect to arg k [α, β] and − 1 − − m (x, t) = (µ(w+ + w ))(x, t, s)sj 1ds, j = 1, 2, 3. j 2πi Γ (cid:90) We infer that the nontangential limit in (5.5) exists and that ∠ − 1 − u(x, t) = 2i lim (km(x, t, k)) = (µ(w+ + w )) (x, t, s)ds. (5.21) →∞ 12 12 k π Γ Similarly, since the L2-norms of µ , µ as well as the L1 and L2 norms of (s2 + x t − (cid:40) − s−3 )(w+ + w ) and (1(cid:80)+ −s )(w+ + w ) are bounded, k s x k s t − m = 2 ∂xmj(x,t) + O(k 3), → ∞ ∈ x j=−1 kj k , k W. (5.22) m = O(k 1), t Substituting (5.20) and (5.22) into (5.19) the terms of O(k) and O(1) yield − F = iσ , F = i[σ , m ]. 1 3 0 3 1 − − The symmetry J(x, t, k) = σ J(x, t, k) 1σ , where j = 1 if λ = 1 and j = 2 if λ = − − j j 1, implies that σ m(x, t,(cid:40) k)σ satisﬁes the same L2-RH problem as m(x, t, k); so by j j − uniqueness m(x, t, k) = σ m(x, t, k)σ . Hence j j − ( 1)jσ m (x, t)σ , λ = 1, 1 j 1 m (x, t) = − − j = 1, 2, 3. j ( 1)jσ m (x, t, )σ , λ = 1, 2 j 2 − We give the remainder of the proof of Claim 4 in the case of λ = 1; the case of λ = 1 is similar. Assume λ = 1. Writing m = a σ + b σ + c σ + d I, where a , b , c , d are scalar- j j 1 j 2 j 3 j j j j j valued functions of (x, t), we ﬁnd a = d = 0 for j odd and b = c = 0 for j even. We  j j j j infer that    m (x, t) = b (x, t)σ + c (x, t)σ , 1 1 2 1 3 m (x, t) = a (x, t)σ + d (x, t)I, (5.23) 2 2 1 2 m (x, t) = b (x, t)σ + c (x, t)σ . 3 3 2 3 3 − − − Hence b = u/2 and F = i[σ , m ] = 2b σ ; thus 1 0 3 1 1 1 F + kF = ikσ + u(x, t)σ . 0 1 3 1 Recalling (5.19), this proves that − m ik[σ , m] = Um. (5.24) x 3 − − Substituting (5.20) and (5.22) into (5(cid:0).24), the terms of O(k 1) and O(k (cid:1)2) yield u − iu −1 − a = x (m ) , b = u3 + 4u(m ) + 2iu (m ) u . (5.25) 2 1 11 3 2 11 x 1 11 xx 4 2 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: m

text: (x,

text: t)

text: m(x,

text: t,

text: k)

text: I

text: j

text: O(k

text: k

text: k

text: W,

text: kj

text: where

text: the

text: error

text: term

text: is

text: uniform

text: with

text: respect

text: to

text: arg

text: k

text: [α,

text: β]

text: and

text: m

text: (x,

text: t)

text: (µ(w+

text: w

text: ))(x,

text: t,

text: s)sj

text: j

text: j

text: Γ

text: We

text: infer

text: that

text: the

text: nontangential

text: limit

text: in

text: exists

text: and

text: that

text: u(x,

text: t)

text: lim

text: (km(x,

text: t,

text: k))

text: (µ(w+

text: w

text: (x,

text: t,

text: s)ds.

text: k

text: π

text: Γ

text: Similarly,

text: since

text: the

text: of

text: µ

text: µ

text: as

text: well

text: as

text: the

text: and

text: norms

text: of

text: x

text: t

text: )(w+

text: w

text: and

text: −s

text: )(w+

text: w

text: are

text: bounded,

text: k

text: s

text: x

text: k

text: s

text: t

text: m

text: ∂xmj(x,t)

text: O(k

text: x

text: kj

text: k

text: k

text: W.

text: m

text: O(k

text: t

text: Substituting

text: and

text: into

text: the

text: terms

text: of

text: O(k)

text: and

text: yield

text: F

text: iσ

text: F

text: i[σ

text: m

text: The

text: symmetry

text: J(x,

text: t,

text: k)

text: σ

text: J(x,

text: t,

text: k)

text: 1σ

text: where

text: j

text: if

text: λ

text: and

text: j

text: if

text: λ

text: j

text: j

text: implies

text: that

text: σ

text: m(x,

text: k)σ

text: satisﬁes

text: the

text: same

text: problem

text: as

text: m(x,

text: t,

text: k);

text: so

text: by

text: j

text: j

text: uniqueness

text: m(x,

text: t,

text: k)

text: σ

text: m(x,

text: t,

text: k)σ

text: Hence

text: j

text: j

text: m

text: (x,

text: t)σ

text: λ

text: j

text: m

text: (x,

text: t)

text: j

text: j

text: m

text: (x,

text: t,

text: )σ

text: λ

text: j

text: We

text: give

text: the

text: remainder

text: of

text: the

text: proof

text: of

text: Claim

text: in

text: the

text: case

text: of

text: λ

text: the

text: case

text: of

text: λ

text: is

text: similar.

text: Assume

text: λ

text: Writing

text: m

text: a

text: σ

text: b

text: σ

text: c

text: σ

text: d

text: I,

text: where

text: a

text: b

text: c

text: d

text: are

text: scalar-

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: valued

text: functions

text: of

text: (x,

text: t),

text: we

text: ﬁnd

text: a

text: d

text: for

text: j

text: odd

text: and

text: b

text: c

text: for

text: j

text: even.

text: We

text: j

text: j

text: j

text: j

text: infer

text: that

text: m

text: (x,

text: t)

text: b

text: (x,

text: t)σ

text: c

text: (x,

text: t)σ

text: m

text: (x,

text: t)

text: a

text: (x,

text: t)σ

text: d

text: (x,

text: t)I,

text: m

text: (x,

text: t)

text: b

text: (x,

text: t)σ

text: c

text: (x,

text: t)σ

text: Hence

text: b

text: and

text: F

text: i[σ

text: m

text: σ

text: thus

text: F

text: kF

text: ikσ

text: u(x,

text: t)σ

text: Recalling

text: this

text: proves

text: that

text: m

text: ik[σ

text: m]

text: Um.

text: x

text: Substituting

text: and

text: into

text: the

text: terms

text: of

text: O(k

text: and

text: O(k

text: yield

text: u

text: iu

text: −1

text: a

text: x

text: (m

text: b

text: (m

text: u

text: x

text: xx

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 44 NONLINEAR FOURIER TRANSFORMS − C \ { } Since m I and m belong to 2( Γ), there exist functions g (x, t) 4 such that the t (cid:88) j 1 function g deﬁned by 4 1 − − g (x, t) g(x, t, k) = ψ ψ 1 j (k + i)4 t (k + i)j (cid:88) j=1 4 1 − − − g (x, t) = [m 4ik3mσ ]m 1 j (k + i)4 t 3 (k + i)j j=1 C \ lies in E˙ 1( Γ). The jump condition (5.11) for m implies that g = g− a.e. on Γ and + { } hence that g vanishes identically. We conclude that there exist functions G (x, t) 3 such j 0 that (cid:18) (cid:19) (cid:88) 3 − ∈ C \ m 4ik3mσ = kjG (x, t) m, k Γ. (5.26) t 3 j j=0 Substituting (5.20) and (5.22) into (5.26) the terms of O(kj), j = 0, . . . , 3, yield − − − − − − G = 4iσ , G = 4im σ G m , G = 4im σ G m G m , 3 3 2 1 3 3 (cid:80)1 1 2 3 3 2 2 1 − − − − G = 4im σ G m G m G m . 0 3 3 3 3 2 2 1 1 − (cid:53) Equations (5.23) and (5.25) now show that 3 kjG = 4ik3σ + V. j=0 j 3 ≥ ≥ Claim 5. u(x, t) is C3 in(cid:90) x 0 and C1 in t 0. (cid:90) Proof of Claim 5. By (5.21), we have 1 − 1 u(x, t) = ((µ I)w+) (x, t, s)ds + (w+) (x, t, s)ds. 12 12 π π Γ Γ The claim follows from the diﬀerentiability properties of the maps in (5.13) and (5.14a) as well as the fact that the map (cid:55)→ · ∞ × ∞ → (x, t) w+(x, t, ) : [0, ) [0, ) L1(Γ) (cid:53) is C3 in x and C1 in t. ≥ ≥ Claim 6. u(x, t) satisﬁes the mKdV equation (1.1) for x 0 and t 0. Proof of Claim 6. Equations (5.17) and Claim 5 imply that m is C1 in t and that x ≥ m≥t is C1 in x. Hence the mixed partials mxt and mtx exist and are equal for x 0 an(cid:53)d t 0. The compatibility of (5.17) implies that u satisﬁes (1.1). ≥ Claim 7. u(x, 0) = u (x) for x 0. 0 Proof of Claim 7. Consider the x-part (3.1) with potential u(x) given by u (x). Let 0 X(x, k) and Y (x, k) denote the associated eigenfunctions deﬁned in Theorem 3.1. The (cid:16) (cid:17) results of Section imply that the function m(x)(x, k) deﬁned by   (cid:18) (cid:19)  [Y (x,k)]1 , [X(x, k)] , Im k < 0, a(k) 2 m(x)(x, k) = (5.27) [X(x, k)] , [Y (x,k)]2 , Im k > 0, 1 a(k¯) satisﬁes the L2-RH problem ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: C

text: Since

text: m

text: I

text: and

text: m

text: belong

text: to

text: Γ),

text: there

text: exist

text: functions

text: g

text: (x,

text: t)

text: such

text: that

text: the

text: t

text: j

text: function

text: g

text: deﬁned

text: by

text: g

text: (x,

text: t)

text: g(x,

text: t,

text: k)

text: ψ

text: ψ

text: j

text: (k

text: t

text: (k

text: i)j

text: g

text: (x,

text: t)

text: [m

text: ]m

text: j

text: (k

text: t

text: (k

text: i)j

text: C

text: lies

text: in

text: E˙

text: Γ).

text: The

text: jump

text: condition

text: for

text: m

text: implies

text: that

text: g

text: g−

text: a.e.

text: on

text: Γ

text: and

text: hence

text: that

text: g

text: vanishes

text: identically.

text: We

text: conclude

text: that

text: there

text: exist

text: functions

text: G

text: (x,

text: t)

text: such

text: j

text: that

text: C

text: m

text: kjG

text: (x,

text: t)

text: m,

text: k

text: Γ.

text: t

text: j

text: Substituting

text: and

text: into

text: the

text: terms

text: of

text: O(kj),

text: j

text: yield

text: G

text: G

text: σ

text: G

text: m

text: G

text: σ

text: G

text: m

text: G

text: m

text: G

text: σ

text: G

text: m

text: G

text: m

text: G

text: m

text: Equations

text: and

text: now

text: show

text: that

text: kjG

text: V.

text: j

text: Claim

text: u(x,

text: t)

text: is

text: x

text: and

text: in

text: t

text: Proof

text: of

text: Claim

text: By

text: we

text: have

text: u(x,

text: t)

text: ((µ

text: I)w+)

text: (x,

text: t,

text: s)ds

text: (w+)

text: (x,

text: t,

text: s)ds.

text: π

text: π

text: Γ

text: Γ

text: The

text: claim

text: follows

text: from

text: the

text: diﬀerentiability

text: properties

text: of

text: the

text: maps

text: in

text: and

text: as

text: well

text: as

text: the

text: fact

text: that

text: the

text: map

text: (x,

text: t)

text: w+(x,

text: t,

text: is

text: in

text: x

text: and

text: in

text: t.

text: Claim

text: u(x,

text: t)

text: satisﬁes

text: the

text: mKdV

text: equation

text: for

text: x

text: and

text: t

text: Proof

text: of

text: Claim

text: Equations

text: and

text: Claim

text: imply

text: that

text: m

text: is

text: in

text: t

text: and

text: that

text: x

text: m≥t

text: is

text: in

text: x.

text: Hence

text: the

text: mixed

text: partials

text: mxt

text: and

text: mtx

text: exist

text: and

text: are

text: equal

text: for

text: x

text: t

text: The

text: compatibility

text: of

text: implies

text: that

text: u

text: satisﬁes

text: Claim

text: u(x,

text: u

text: (x)

text: for

text: x

text: Proof

text: of

text: Claim

text: Consider

text: the

text: x-part

text: with

text: potential

text: u(x)

text: given

text: by

text: u

text: (x).

text: Let

text: X(x,

text: k)

text: and

text: Y

text: (x,

text: k)

text: denote

text: the

text: associated

text: eigenfunctions

text: deﬁned

text: in

text: Theorem

text: The

text: results

text: of

text: Section

text: imply

text: that

text: the

text: function

text: m(x)(x,

text: k)

text: deﬁned

text: by

text: [Y

text: [X(x,

text: k)]

text: Im

text: k

text: a(k)

text: m(x)(x,

text: k)

text: [X(x,

text: k)]

text: [Y

text: Im

text: k

text: a(k¯)

text: satisﬁes

text: the

text: problem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: NONLINEAR FOURIER TRANSFORMS 45 D D 3 D 3 4 Σ D D 1 D 2 2 {D } Figure 2. The contour Σ and the deformed domains 4 in the complex k-plane. j 1      · ∈ C \ R m(x)(x, ) I + E˙ 2( ),    − 1 b(k) e2ikx m(x)(x, k) = m(−x)(x, k) a(k¯) for a.e. k ∈ R. (5.28) + λb(k¯)e−2ikx | 1 | a(k) a(k) 2 Moreover, by (3.15a) and the explicit expression (3.10) for X , we have 1 − u (x) = 2i lim (km(x)(x, k)) , (5.29) 0 →∞ 12 k ≤ where the limit is taken along any direction in Im k 0. On the other hand, in view of Claim 1, − ∈ ∩ ∞ ∈ ∩ ∞ h(k)e 2ikx E˙ 2(D ) E (D ), h(k¯)e2ikx E˙ 2(D ) E (D ). 2 2 3 3 Hence, Lemma A.5 together with the expression (5.2) for J show that the function  M(x)(x, k) deﬁned by  (cid:32) (cid:33)    ∈ ∪ M(x, 0, k), k D D , 1 4  (cid:32) 1 (cid:33)0 ∈  M(x)(x, k) = M(x, 0, k) λh(k)e−2ikx 1 , k D2, 1 h(k¯)e2ikx ∈ M(x, 0, k) , k D , 3 0 1 also satisﬁes (5.28). By uniqueness, M(x) = m(x)≥. Comparing the deﬁnition (5.5) (cid:53)of u(x, t) with (5.29), we obtain u(x, 0) = u (x) for x 0. 0 ≥ Claim 8. u(0, t) = g (t) and u (0, t) = g (t) for t 0. 0 x → 1 → ∞ ∈ Proof of Claim 8. Since A(k) 1 uniformly as k , k D¯ , we can deﬁne {D } D 3 deformed domains 4 so that A(k) is nonzero in ¯ , see Figure 2. We let Σ = R ∪ D ∩ D ∪ D ∩jD1 3 D ( ¯ ¯ ) ( ¯ ¯ ) denote the contour separating the ’s oriented as in Figure 1 2 D3 4 j → 2. We choose the ’s so that Σ is invariant under the involution k k¯. j ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: D

text: D

text: D

text: Σ

text: D

text: D

text: D

text: {D

text: Figure

text: The

text: contour

text: Σ

text: and

text: the

text: deformed

text: domains

text: in

text: the

text: complex

text: k-plane.

text: j

text: C

text: R

text: m(x)(x,

text: I

text: E˙

text: b(k)

text: m(x)(x,

text: k)

text: m(−x)(x,

text: k)

text: a(k¯)

text: for

text: a.e.

text: k

text: R.

text: a(k)

text: a(k)

text: Moreover,

text: by

text: and

text: the

text: explicit

text: expression

text: for

text: X

text: we

text: have

text: u

text: (x)

text: lim

text: (km(x)(x,

text: k))

text: k

text: where

text: the

text: limit

text: is

text: taken

text: along

text: any

text: direction

text: in

text: Im

text: k

text: On

text: the

text: other

text: hand,

text: in

text: view

text: of

text: Claim

text: h(k)e

text: E˙

text: E

text: (D

text: E˙

text: E

text: (D

text: Hence,

text: Lemma

text: together

text: with

text: the

text: expression

text: for

text: J

text: show

text: that

text: the

text: function

text: M(x)(x,

text: k)

text: deﬁned

text: by

text: M(x,

text: k),

text: k

text: D

text: D

text: M(x)(x,

text: k)

text: M(x,

text: k)

text: k

text: M(x,

text: k)

text: k

text: D

text: also

text: satisﬁes

text: By

text: uniqueness,

text: M(x)

text: m(x)≥.

text: Comparing

text: the

text: deﬁnition

text: u(x,

text: t)

text: with

text: we

text: obtain

text: u(x,

text: u

text: (x)

text: for

text: x

text: Claim

text: t)

text: g

text: (t)

text: and

text: u

text: t)

text: g

text: (t)

text: for

text: t

text: x

text: Proof

text: of

text: Claim

text: Since

text: A(k)

text: uniformly

text: as

text: k

text: k

text: D¯

text: we

text: can

text: deﬁne

text: {D

text: D

text: deformed

text: domains

text: so

text: that

text: A(k)

text: is

text: nonzero

text: in

text: see

text: Figure

text: We

text: let

text: Σ

text: R

text: D

text: D

text: D

text: D

text: denote

text: the

text: contour

text: separating

text: the

text: ’s

text: oriented

text: as

text: in

text: Figure

text: j

text: We

text: choose

text: the

text: ’s

text: so

text: that

text: Σ

text: is

text: invariant

text: under

text: the

text: involution

text: k

text: k¯.

text: j

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 48 NONLINEAR FOURIER TRANSFORMS Remark 5.4. Let us comment on the existence of a vanishing lemma for the RH problem (5.4). Zhou showed in [38] (see Theorem 9.3 of [38]) that a homogeneous RH problem with a Schwarz reﬂection invariant contour has only the trivial solution provided that the jump matrix J(k) atisﬁes (a) J(k) = J(k¯)T for k ∈ Γ \ R and (b) Re J(k) is positive R − deﬁnite on .2 In the focusing (i.e. λ = 1) case, the jump matrix (5.2) satisﬁes these conditions, so there exists a vanishing lemma. In the defocusing (i.e. λ = 1) case, we have not been able to establish a vanishing lemma. Let us explain the main diﬃculties. Let λ = 1. Then the condition (a) is not satisﬁed; instead the jump matrix J satisﬁes J(k) = σ J(k¯)T σ . This implies that if M(k) is a solution of the homogeneous 3 3 RH problem, then the function M(k)σ M(k¯)T has no jump across Γ; hence it vanishes 3 identically. In particular, M−(k)J(k)σ M−(k)T = M−(k)σ J(k)T M−(k)T = 0, k ∈ R. 3 3 However, since no linear combination of Jσ and σ J¯T is positive deﬁnite, we cannot 3 3 conclude that M− = 0. More generally, proceeding as in the appendix of [20], we may consider a sectionally analytic function G(k) deﬁned by G(k) = M(k)H (k)M(k¯)T , k ∈ D , j = 1, . . . , 4, j j × where H (k) is any 2 2-matrix valued function which is analytic and bounded in D . j j The requirement that (cid:18)G be continuous acr(cid:19)oss ∂D1(cid:18)enforces the conditio(cid:19)n J(k)H1(k) = H (k)J(k¯)T on ∂D , i.e. 2 1 1 0 1 0 H (k) = − H (k) − . 2 h(k)e 2ikx+8ik3t 1 1 h(k)e 2ikx+8ik3t 1 Since we want H to be analytic and bounded in D we assume that H is traceless and 2 2 1 lower triangular. Similarly, continuity across ∂D enforces the condition J(k)H (k) = 4 3 H (k)J(k¯)T on ∂D ,(cid:90)so we need to(cid:90) assume that H is traceless and upper triangular. 4 4 4 Then H and H are also traceless and lower resp. upper triangular. The equations 2 3 (cid:90) (cid:90) 0 = G (k)dk = M−(k)J(k)H (k)M−(k)T dk, + 3 R R 0 = G−(k)dk = M−(k)H (k)J(k)T M−(k)T dk, 2 (cid:90) R R imply that M−XM−T dk = 0, X := c JH + c H T JT + c H JT + c JH T , 1 3 2 3 3 2 4 2 R { } { } ∈ R for any complex constants c 4. We seek c 4 such that X is positive deﬁnite for k . j 1 j 1 Since X is positive deﬁnite iﬀ Re X is positive deﬁnite, it is enough to consider positive deﬁniteness of T T Y := d Re (JH ) + d Im (JH ) + d Re (H J ) + d Im (H J ), 1 3 2 3 3 2 4 2 { } where d 4 are real coeﬃcients. The matrix Y is positive deﬁnite iﬀ the inequalities j 1 { } Y > 0 and det Y > 0 are satisﬁed. However, it appears that no real numbers d 4 and 11 j 1 traceless lower resp. upper triangular matrices H (k) and H (k) fulﬁll these inequalities. 2 3 2The real and imaginary parts of a square matrix A are deﬁned by Re A = 1(A + A¯T) and Im A = − 2 1 (A A¯T). 2i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: Remark

text: Let

text: us

text: comment

text: on

text: the

text: existence

text: of

text: a

text: vanishing

text: lemma

text: for

text: the

text: RH

text: problem

text: Zhou

text: showed

text: in

text: (see

text: Theorem

text: of

text: that

text: a

text: homogeneous

text: RH

text: problem

text: with

text: a

text: Schwarz

text: reﬂection

text: invariant

text: contour

text: has

text: only

text: the

text: trivial

text: solution

text: provided

text: that

text: the

text: jump

text: matrix

text: J(k)

text: atisﬁes

text: (a)

text: J(k)

text: J(k¯)T

text: for

text: k

text: Γ

text: R

text: and

text: (b)

text: Re

text: J(k)

text: is

text: positive

text: R

text: deﬁnite

text: on

text: In

text: the

text: focusing

text: (i.e.

text: λ

text: case,

text: the

text: jump

text: matrix

text: satisﬁes

text: these

text: conditions,

text: so

text: there

text: exists

text: a

text: vanishing

text: lemma.

text: In

text: the

text: defocusing

text: (i.e.

text: λ

text: case,

text: we

text: have

text: not

text: been

text: able

text: to

text: establish

text: a

text: vanishing

text: lemma.

text: Let

text: us

text: explain

text: the

text: main

text: diﬃculties.

text: Let

text: λ

text: Then

text: the

text: condition

text: (a)

text: is

text: not

text: satisﬁed;

text: instead

text: the

text: jump

text: matrix

text: J

text: satisﬁes

text: J(k)

text: σ

text: J(k¯)T

text: σ

text: This

text: implies

text: that

text: if

text: M(k)

text: is

text: a

text: solution

text: of

text: the

text: homogeneous

text: RH

text: problem,

text: then

text: the

text: function

text: M(k)σ

text: M(k¯)T

text: has

text: no

text: jump

text: across

text: Γ;

text: hence

text: it

text: vanishes

text: identically.

text: In

text: particular,

text: M−(k)J(k)σ

text: M−(k)T

text: M−(k)σ

text: J(k)T

text: M−(k)T

text: k

text: R.

text: However,

text: since

text: no

text: linear

text: combination

text: of

text: Jσ

text: and

text: σ

text: J¯T

text: is

text: positive

text: deﬁnite,

text: we

text: cannot

text: conclude

text: that

text: M−

text: More

text: generally,

text: proceeding

text: as

text: in

text: the

text: appendix

text: of

text: we

text: may

text: consider

text: a

text: sectionally

text: analytic

text: function

text: G(k)

text: deﬁned

text: by

text: G(k)

text: M(k)H

text: (k)M(k¯)T

text: k

text: D

text: j

text: j

text: j

text: where

text: H

text: (k)

text: is

text: any

text: valued

text: function

text: which

text: is

text: analytic

text: and

text: bounded

text: in

text: D

text: j

text: j

text: The

text: requirement

text: that

text: be

text: continuous

text: the

text: H

text: (k)J(k¯)T

text: on

text: ∂D

text: i.e.

text: H

text: (k)

text: H

text: (k)

text: h(k)e

text: h(k)e

text: Since

text: we

text: want

text: H

text: to

text: be

text: analytic

text: and

text: bounded

text: in

text: D

text: we

text: assume

text: that

text: H

text: is

text: traceless

text: and

text: lower

text: triangular.

text: Similarly,

text: continuity

text: across

text: ∂D

text: enforces

text: the

text: condition

text: J(k)H

text: (k)

text: H

text: (k)J(k¯)T

text: on

text: ∂D

text: we

text: need

text: assume

text: that

text: H

text: is

text: traceless

text: and

text: upper

text: triangular.

text: Then

text: H

text: and

text: H

text: are

text: also

text: traceless

text: and

text: lower

text: resp.

text: upper

text: triangular.

text: The

text: equations

text: G

text: (k)dk

text: M−(k)J(k)H

text: (k)M−(k)T

text: dk,

text: R

text: R

text: G−(k)dk

text: M−(k)H

text: (k)J(k)T

text: M−(k)T

text: dk,

text: R

text: R

text: imply

text: that

text: M−XM−T

text: dk

text: X

text: c

text: JH

text: c

text: H

text: T

text: JT

text: c

text: H

text: JT

text: c

text: JH

text: T

text: R

text: R

text: for

text: any

text: complex

text: constants

text: c

text: We

text: seek

text: c

text: such

text: that

text: X

text: is

text: positive

text: deﬁnite

text: for

text: k

text: j

text: j

text: Since

text: X

text: is

text: positive

text: deﬁnite

text: iﬀ

text: Re

text: X

text: is

text: positive

text: deﬁnite,

text: it

text: is

text: enough

text: to

text: consider

text: positive

text: deﬁniteness

text: of

text: T

text: T

text: Y

text: d

text: Re

text: (JH

text: d

text: Im

text: (JH

text: d

text: Re

text: (H

text: J

text: d

text: Im

text: (H

text: J

text: where

text: d

text: are

text: real

text: coeﬃcients.

text: The

text: matrix

text: Y

text: is

text: positive

text: deﬁnite

text: iﬀ

text: the

text: inequalities

text: j

text: Y

text: and

text: det

text: Y

text: are

text: satisﬁed.

text: However,

text: it

text: appears

text: that

text: no

text: real

text: numbers

text: d

text: and

text: j

text: traceless

text: lower

text: resp.

text: upper

text: triangular

text: matrices

text: H

text: (k)

text: and

text: H

text: (k)

text: fulﬁll

text: these

text: inequalities.

text: real

text: and

text: imaginary

text: parts

text: of

text: a

text: square

text: matrix

text: A

text: are

text: deﬁned

text: by

text: Re

text: A

text: A¯T)

text: and

text: Im

text: A

text: (A

text: A¯T).

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:2) (cid:3) NONLINEAR FOURIER TRANSFORMS 7 (cid:26) (cid:90) 1 (cid:2) − − Z (x) = 5λu2u + 4(Z ) u + 2i(λu3 u )(Z ) + 8iu(Z ) u σ σ 4 x 2 22 x xx 1 22 3 22 xxx 3 λ 16 x λu − (cid:27) + I 5λu2u(cid:3) + 4(Z ) u + 2i(λu3 u )(Z ) x 2 22 x xx 1 22 16 0 − (cid:48) 3u4(0) − λ λu2(0) + 8iu(Z ) u dx + u(0)u (0) + x , (3.11) 3 22 xxx xx 16 8 16 and (cid:26) (cid:90) (cid:27) − iu(0) W (x) = σ , 1 λ 2 − λiu(W ) − λi x (cid:48) u (0) W (x) = 1 12 I σ σ (W ) u2dx + x , 2 3 λ 1 12 2 2 4 0 (cid:26) (cid:90) (cid:27) − λ − (cid:2) (cid:3) W (x) = ( 2iu(W ) + (W ) u )σ 3 2 12 1 12 x 3 4 − λ x − (cid:48) − iλ σ u 2iu(W ) (W ) u dx + iu3(0) u (0) , λ 2 12 1 12 x xx 4 2 0 (cid:26) (cid:90) − iλ (cid:2) − (cid:3) W (x) = (λu3(W ) + 4u(W ) + 2i(W ) u (W ) u )I 4 1 12 3 12 2 12 x 1 12 xx 8 − i x (cid:27) − (cid:48) σ σ u u3(W ) + λ(4u(W ) + 2i(W ) u (W ) u ) dx 3 λ 1 12 3 12 2 12 x 1 12 xx 8 0 − i 3iλu2(0)u (0) + u (0) . (3.12) x xxx 2 If u(x) has a ﬁnite degree of regularity and decay, only ﬁnitely many coeﬃcients { } X , Z , W are well-deﬁned. The following result, whose proof is given in Section , j j j → ∞ describes the behavior of X and Y as k . ≥ ≥ Theorem 3.2. Suppose u(x) satisﬁes (3.3) for some integers m 1 and n 1. Then, → ∞ as k , X and Y coincide to order m with X and Y respectively in the formal formal following sense: The functions (cid:18) (cid:19) X (x) · · · X (x) Xˆ(x, k) = I + 1 + + m+1 , (3.13) k km+1 Z (x) · · · Z (x) W (k) · · · W (x) − Yˆ (x, k) = I + 1 + + m+1 + 1 + + m+1 e 2ikxσ3, (3.14) k km+1 k km+1 (cid:12) (cid:12) are we(cid:12)ll-de(cid:0)ﬁned an(cid:1)d(cid:12)there exists a K > 0 such that (cid:12) (cid:12) (cid:12) (cid:12) ∂j − ≤ C ≥ X Xˆ | | − , x 0, j = 0, 1, . . . , n, (3.15a) ∂kj k m+1(1 + x)n j ∈ C C | | for al(cid:12)l k (¯ , ¯−(cid:12)) with k > K, and (cid:12) (cid:0) + (cid:1)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ∂j − ≤ C(1 + x)j+2e|k|Cmx+1 ≥ Y Yˆ | | , x 0, j = 0, 1, . . . , n, (3.15b) ∂kj k m+1 ∈ C C | | for all k (¯−, ¯ ) with k > K, and + ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NONLINEAR

text: FOURIER

text: TRANSFORMS

text: Z

text: (x)

text: u

text: u

text: )(Z

text: u

text: σ

text: σ

text: x

text: x

text: xx

text: xxx

text: λ

text: x

text: λu

text: I

text: u

text: u

text: )(Z

text: x

text: x

text: xx

text: λ

text: u

text: dx

text: x

text: xxx

text: xx

text: and

text: W

text: (x)

text: σ

text: λ

text: λiu(W

text: λi

text: x

text: u

text: W

text: (x)

text: I

text: σ

text: σ

text: (W

text: x

text: λ

text: λ

text: W

text: (x)

text: (W

text: u

text: )σ

text: x

text: λ

text: x

text: iλ

text: σ

text: u

text: (W

text: u

text: dx

text: u

text: λ

text: x

text: xx

text: iλ

text: W

text: (x)

text: u

text: (W

text: u

text: )I

text: x

text: xx

text: i

text: x

text: σ

text: σ

text: u

text: u

text: (W

text: u

text: dx

text: λ

text: x

text: xx

text: i

text: u

text: x

text: xxx

text: If

text: u(x)

text: has

text: a

text: ﬁnite

text: degree

text: of

text: regularity

text: and

text: decay,

text: only

text: ﬁnitely

text: many

text: coeﬃcients

text: X

text: Z

text: W

text: are

text: well-deﬁned.

text: The

text: following

text: result,

text: whose

text: proof

text: is

text: given

text: in

text: Section

text: j

text: j

text: j

text: describes

text: the

text: behavior

text: of

text: X

text: and

text: Y

text: as

text: k

text: Theorem

text: Suppose

text: u(x)

text: satisﬁes

text: for

text: some

text: integers

text: m

text: and

text: n

text: Then,

text: as

text: k

text: X

text: and

text: Y

text: coincide

text: to

text: order

text: m

text: with

text: X

text: and

text: Y

text: respectively

text: in

text: the

text: formal

text: formal

text: following

text: sense:

text: The

text: functions

text: X

text: (x)

text: X

text: (x)

text: Xˆ(x,

text: k)

text: I

text: k

text: Z

text: (x)

text: Z

text: (x)

text: W

text: (k)

text: W

text: (x)

text: Yˆ

text: (x,

text: k)

text: I

text: e

text: k

text: k

text: are

text: exists

text: a

text: K

text: such

text: that

text: ∂j

text: C

text: X

text: Xˆ

text: x

text: j

text: n,

text: ∂kj

text: k

text: x)n

text: j

text: C

text: C

text: for

text: k

text: with

text: k

text: K,

text: and

text: ∂j

text: Y

text: Yˆ

text: x

text: j

text: n,

text: ∂kj

text: k

text: C

text: C

text: for

text: all

text: k

text: with

text: k

text: K,

text: and

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: )Ve( ygrenE )Ve( ygrenE 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 −4 −4 −5 −5 L X W K A H L A K M Γ Γ Γ Γ (a) GaAs (b) InN FIG. 1. Band structure of cubic GaAs and wurtzite InN, normalized so that the Fermi level is set to zero at the conduction band minimum. R2 > 0.99 in all segments ﬁtted. We ﬁt these polynomials at diﬀerent segments of the band structure and carefully choose only the ones that result in the maximum R2 and minimum discontinuity where the polynomials meet. This results in very smooth calculated group velocities, and, subsequently, other transport properties. IV. RESULTS AND DISCUSSIONS A. ab initio Calculated Parameter Inputs to the Transport Model The computed band structures of GaAs and InN are shown in Figure 1. We have cal- culated a GW 0 band structure, which starts from the wavefunctions previously computed using the GGA-PBE functional, as shown in Figure 1a. The band structures used in previous semi-empirical models3,8 express the energy of the conduction band as a function of the distance from the Γ point. Instead, we calculate the ab initio band structure in a three-dimensional grid around the CBM, and then average the energy values of the k-points that share the same distance from the Γ point (Figure 2). For · both GaAs and InN, the ab initio and k p band structures agree well at small k-points; however, they diverge at larger k-points. This directly impacts the group velocity of the electrons and, ultimately, the transport properties – particularly at higher temperatures where higher energy electrons have nonzero occupation. We have also calculated a GGA+U48 band structure, with U values taken from the pub- 14 ##LTFigure## ##LTFigure##

text: )Ve(

text: ygrenE

text: )Ve(

text: ygrenE

text: −1

text: −1

text: −2

text: −2

text: −3

text: −3

text: −4

text: −4

text: −5

text: −5

text: L

text: X

text: W

text: K

text: A

text: H

text: L

text: A

text: K

text: M

text: Γ

text: Γ

text: Γ

text: Γ

text: (a)

text: GaAs

text: (b)

text: InN

text: FIG.

text: Band

text: structure

text: of

text: cubic

text: GaAs

text: and

text: wurtzite

text: InN,

text: normalized

text: so

text: that

text: the

text: Fermi

text: level

text: is

text: set

text: to

text: zero

text: at

text: the

text: conduction

text: band

text: minimum.

text: in

text: all

text: segments

text: ﬁtted.

text: We

text: ﬁt

text: these

text: polynomials

text: at

text: diﬀerent

text: segments

text: of

text: the

text: band

text: structure

text: and

text: carefully

text: choose

text: only

text: the

text: ones

text: that

text: result

text: in

text: the

text: maximum

text: and

text: minimum

text: discontinuity

text: where

text: the

text: polynomials

text: meet.

text: This

text: results

text: in

text: very

text: smooth

text: calculated

text: group

text: velocities,

text: and,

text: subsequently,

text: other

text: transport

text: properties.

text: IV.

text: RESULTS

text: AND

text: DISCUSSIONS

text: A.

text: ab

text: initio

text: Calculated

text: Parameter

text: Inputs

text: to

text: the

text: Transport

text: Model

text: The

text: computed

text: band

text: structures

text: of

text: GaAs

text: and

text: InN

text: are

text: shown

text: in

text: Figure

text: We

text: have

text: cal-

text: culated

text: a

text: GW

text: band

text: structure,

text: which

text: starts

text: from

text: the

text: wavefunctions

text: previously

text: computed

text: using

text: the

text: GGA-PBE

text: functional,

text: as

text: shown

text: in

text: Figure

text: The

text: band

text: structures

text: used

text: in

text: previous

text: semi-empirical

text: express

text: the

text: energy

text: of

text: the

text: conduction

text: band

text: as

text: a

text: function

text: of

text: the

text: distance

text: from

text: the

text: Γ

text: point.

text: Instead,

text: we

text: calculate

text: the

text: ab

text: initio

text: band

text: structure

text: in

text: a

text: three-dimensional

text: grid

text: around

text: the

text: CBM,

text: and

text: then

text: average

text: the

text: energy

text: values

text: of

text: the

text: k-points

text: that

text: share

text: the

text: same

text: distance

text: from

text: the

text: Γ

text: point

text: (Figure

text: For

text: both

text: GaAs

text: and

text: InN,

text: the

text: ab

text: initio

text: and

text: k

text: p

text: band

text: structures

text: agree

text: well

text: at

text: small

text: k-points;

text: however,

text: they

text: diverge

text: at

text: larger

text: k-points.

text: This

text: directly

text: impacts

text: the

text: group

text: velocity

text: of

text: the

text: electrons

text: and,

text: ultimately,

text: the

text: transport

text: properties

text: particularly

text: at

text: higher

text: temperatures

text: where

text: higher

text: energy

text: electrons

text: have

text: nonzero

text: occupation.

text: We

text: have

text: also

text: calculated

text: a

text: band

text: structure,

text: with

text: U

text: values

text: taken

text: from

text: the

text: pub-

text: ##LTFigure##

text: ##LTFigure##


text: TABLE III. Inputs to the transport model, as calculated ab initio compared to experimentally measured values. The bolded numbers are used in our transport property calculations; note that not all appear in the ﬁnal expressions listed in Table I. GaAs InN Parameter a ab initio Exp. ab initio Exp. c (nm) 0.562 0.575 3 0.565 0.569 3 l ω (THz) 8.16 8.73 3 17.83 3 17.65 po (cid:15) 12.18 12.91 3 11.42 10.3 8 0 (cid:15)∞ 10.32 10.91 3 6.24 6.7 8 E (eV) 6.04 8.6 3 4.46 3.6 3 D ∗ m 0.053-0.066 b 0.0636-0.082 53,54 0.062, 0.071 (GW) 0.05-0.08 55–60 ε (eV) 0.96, 1.19 (GW) 1.424 6 0.50 0.675-0.7 55,57,61 g lished literature49,50, as shown in Figure 1. For InN, GGA+U correctly yields an s-like conduction band and a band gap of 0.5 eV, which is comparable to the self-interaction cor- rected band gap of 0.58 eV reported by Furthmu¨ller et al.51 and the experimental values of 0.675-0.7 eV55,57,61 (Table III). We include DFT+U calculations only to show the feasibility of these less-expensive methods, in the case of more complex semiconductor materials for which a GW calculations is too expensive. Also, DFT usually suﬀers from vastly underesti- mating the eﬀective mass1,2, and the introduction of the ﬁtting parameter U may reduce the predictability of the ab initio model as a whole. Therefore, we stress that all reported trans- port properties are calculated here using the parameter-free GW band structures, unless otherwise stated. Although we do not directly use the value of the electron eﬀective mass in the transport property expressions, we see that the calculated eﬀective mass of 0.062 for InN is consistent with the previously calculated eﬀective mass (0.066) using an empirical pseudopotential62, and well within the range (0.05-0.08) measured experimentally55–60. We also show the calculated phonon band structure and density of states of these two compounds in Figure 8. For GaAs, the calculated PO-phonon frequency of 8.16 THz is shown in Figure 8a. For InN, the calculated optical phonon frequency of 17.83 THz is close to the 17.65 THz value reported by Bungaro et al.55,63. We have listed all the parameters 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: III.

text: Inputs

text: to

text: the

text: transport

text: model,

text: as

text: calculated

text: ab

text: initio

text: compared

text: to

text: experimentally

text: measured

text: values.

text: The

text: bolded

text: numbers

text: are

text: used

text: in

text: our

text: transport

text: property

text: calculations;

text: note

text: that

text: not

text: all

text: appear

text: in

text: the

text: ﬁnal

text: expressions

text: listed

text: in

text: Table

text: I.

text: GaAs

label: table

text: InN

text: Parameter

text: a

text: ab

text: initio

text: Exp.

text: ab

text: initio

text: Exp.

text: c

text: (nm)

text: l

text: ω

text: (THz)

text: po

text: E

text: (eV)

text: D

text: m

text: b

text: (GW)

text: ε

text: (eV)

text: (GW)

text: g

text: lished

text: as

text: shown

text: in

text: Figure

text: For

text: InN,

text: GGA+U

text: correctly

text: yields

text: an

text: s-like

text: conduction

text: band

text: and

text: a

text: band

text: gap

text: of

text: eV,

text: which

text: is

text: comparable

text: to

text: the

text: self-interaction

text: cor-

text: rected

text: band

text: gap

text: of

text: eV

text: reported

text: by

text: Furthmu¨ller

text: et

text: and

text: the

text: experimental

text: values

text: of

text: (Table

text: III).

text: We

text: include

text: DFT+U

text: calculations

text: only

text: to

text: show

text: the

text: feasibility

text: of

text: these

text: less-expensive

text: methods,

text: in

text: the

text: case

text: of

text: more

text: complex

text: semiconductor

text: materials

text: for

text: which

text: a

text: GW

text: calculations

text: is

text: too

text: expensive.

text: Also,

text: DFT

text: usually

text: suﬀers

text: from

text: vastly

text: underesti-

text: mating

text: the

text: eﬀective

text: and

text: the

text: introduction

text: of

text: the

text: ﬁtting

text: parameter

text: U

text: may

text: reduce

text: the

text: predictability

text: of

text: the

text: ab

text: initio

text: model

text: as

text: a

text: whole.

text: Therefore,

text: we

text: stress

text: that

text: all

text: reported

text: trans-

text: port

text: properties

text: are

text: calculated

text: here

text: using

text: the

text: parameter-free

text: GW

text: band

text: structures,

text: unless

text: otherwise

text: stated.

text: Although

text: we

text: do

text: not

text: directly

text: use

text: the

text: value

text: of

text: the

text: electron

text: eﬀective

text: mass

text: in

text: the

text: transport

text: property

text: expressions,

text: we

text: see

text: that

text: the

text: calculated

text: eﬀective

text: mass

text: of

text: for

text: InN

text: is

text: consistent

text: with

text: the

text: previously

text: calculated

text: eﬀective

text: mass

text: using

text: an

text: empirical

text: and

text: well

text: within

text: the

text: range

text: measured

text: We

text: also

text: show

text: the

text: calculated

text: phonon

text: band

text: structure

text: and

text: density

text: of

text: states

text: of

text: these

text: two

text: compounds

text: in

text: Figure

text: For

text: GaAs,

text: the

text: calculated

text: PO-phonon

text: frequency

text: of

text: THz

text: is

text: shown

text: in

text: Figure

text: For

text: InN,

text: the

text: calculated

text: optical

text: phonon

text: frequency

text: of

text: THz

text: is

text: close

text: to

text: the

text: THz

text: value

text: reported

text: by

text: Bungaro

text: et

text: We

text: have

text: listed

text: all

text: the

text: parameters

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: )K/Vµ( tneiciffeoC kcebeeS -1000 exp Nii = 1 x n -800 Nii = 10 x n Pisarenko -600 -400 -200 1013 1015 1017 1019 Carrier Concentration (cm-3) FIG. 5. Calculated and experimental4 GaAs Seebeck coeﬃcient, at diﬀerent ratios of the ion- ized impurity concentration, N , to the electron concentration, n. For the Pisarenko plot, using ii − ∗ Equation 14, we have used values of r = 1 for acoustic phonons, and m = 0.063. 2 best agreement with experiment, through Pisarenko behavior, is only achievable by choosing ∗ either m = 0.11 or r = 0.35, both of these values are far from experimental measurements and thus lack physical meaning.     ∗ (cid:39) k 5 2 (2πm k T )3/2 B B S + r + ln (14) e 2 h3n C. Model Validation on InN In order to further evaluate the accuracy of our model and its applicability to more com- plicated semiconductors, we also calculate the mobility and Seebeck coeﬃcient (Figure 6) of three experimentally synthesized and characterized InN samples by Miller et al.8. These calculations are more challenging due to the reported presence of linear charged dislocations in the crystal structure8,65–67, due to the processing conditions employed. For each sample at a given carrier concentration, as shown in Table V, we change the concentration of dis- locations, N , until the calculated mobility values match the experimental measurements. dis The ﬁtted N (Table V) is within the range of measured concentrations from transmission dis electron microscopy analysis (TEM)8, which conﬁrms that the limiting mechanism is indeed scattering from dislocation lines. As shown in Figure 6a and 6b, while there is an excellent agreement between the cal- 20 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )K/Vµ(

text: tneiciffeoC

text: kcebeeS

text: exp

text: Nii

text: x

text: n

text: Nii

text: x

text: n

text: Pisarenko

text: Carrier

text: Concentration

text: FIG.

text: Calculated

text: and

text: GaAs

text: Seebeck

text: coeﬃcient,

text: at

text: diﬀerent

text: ratios

text: of

text: the

text: ion-

text: ized

text: impurity

text: concentration,

text: N

text: to

text: the

text: electron

text: concentration,

text: n.

text: For

text: the

text: Pisarenko

text: plot,

text: using

text: ii

text: Equation

text: we

text: have

text: used

text: values

text: of

text: r

text: for

text: acoustic

text: phonons,

text: and

text: m

text: best

text: agreement

text: with

text: experiment,

text: through

text: Pisarenko

text: behavior,

text: is

text: only

text: achievable

text: by

text: choosing

text: either

text: m

text: or

text: r

text: both

text: of

text: these

text: values

text: are

text: far

text: from

text: experimental

text: measurements

text: and

text: thus

text: lack

text: physical

text: meaning.

text: k

text: k

text: T

text: B

text: B

text: S

text: r

text: ln

text: e

text: C.

text: Model

text: Validation

text: on

text: InN

text: In

text: order

text: to

text: further

text: evaluate

text: the

text: accuracy

text: of

text: our

text: model

text: and

text: its

text: applicability

text: to

text: more

text: com-

text: plicated

text: semiconductors,

text: we

text: also

text: calculate

text: the

text: mobility

text: and

text: Seebeck

text: coeﬃcient

text: (Figure

text: of

text: three

text: experimentally

text: synthesized

text: and

text: characterized

text: InN

text: samples

text: by

text: Miller

text: et

text: These

text: calculations

text: are

text: more

text: challenging

text: due

text: to

text: the

text: reported

text: presence

text: of

text: linear

text: charged

text: dislocations

text: in

text: the

text: crystal

text: due

text: to

text: the

text: processing

text: conditions

text: employed.

text: For

text: each

text: sample

text: at

text: a

text: given

text: carrier

text: concentration,

text: as

text: shown

text: in

text: Table

text: V,

text: we

text: change

text: the

text: concentration

text: of

text: dis-

text: locations,

text: N

text: until

text: the

text: calculated

text: mobility

text: values

text: match

text: the

text: experimental

text: measurements.

text: dis

text: The

text: ﬁtted

text: N

text: (Table

text: V)

text: is

text: within

text: the

text: range

text: of

text: measured

text: concentrations

text: from

text: transmission

text: dis

text: electron

text: microscopy

text: analysis

text: which

text: conﬁrms

text: that

text: the

text: limiting

text: mechanism

text: is

text: indeed

text: scattering

text: from

text: dislocation

text: lines.

text: As

text: shown

text: in

text: Figure

text: and

text: while

text: there

text: is

text: an

text: excellent

text: agreement

text: between

text: the

text: cal-

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:104)M N (cid:105) P P But then we recognize that , A = where is the Hermitian line bundle on Y M− N described in (2.6) for the Hermitian line bundles 1 and on X. This shows that there is an isometry ∼ (cid:104)M N (cid:105) (cid:104)M− N (cid:105) , A = 1, . D The usual Deligne pairing is bilinear, see ([Del87, 6.2]). So we have for example: ∼ (cid:104)L ⊗ L(cid:48) M(cid:105) (cid:104)L M(cid:105) ⊗ (cid:104)L(cid:48) M(cid:105) O , = , Z , . X D D D This isomorphism of line bundles becomes an isometry if we equip each line bundle with the Deligne metric, see ([Del87, 6.5]). Using this we immediately see that there is an isometry ∼ (cid:104)O M(cid:105) O , = . X D Y L(cid:48) L− Using these facts for = 1 we see that there is an isometry (cid:104)L− M(cid:105) ∼ (cid:104)L M(cid:105) (−1) 1, = , . D D Putting everything together we get an isometry (cid:104)M N (cid:105) ∼ (cid:104)M− N (cid:105) ∼ (cid:104)M N (cid:105) (−1) , A = 1, = , . D D A The questions remains, if we can somehow compare this -Deligne pairing with the one E given by Borek in ([Bor08, Section 2.3]). If is a Hermitian locally free sheaf, then he replaces the Hermitian metric on the associated vector bundle E on the Riemann surface E S by an automorphism β of the sheaf R, ([Bor08, Deﬁnition 2.2.4]). The Quillen metric is replaced by the determinant of the action of β on the Ext-groups, ([Bor08, Section 2.3.2]). Is there any relationship between these two approaches? Furthermore one can ask the following questions: • A Is there a Riemann-Roch type formula for the -Deligne pairing? • A How does the -determinant of the cohomology behave with respect to Serre duality? Do we have to adjust the chosen metrics to get better results? • A Can we ﬁnd arithmetic -Chern classes to get a similar formula for the ﬁrst arith- A metic Chern class of the -Deligne pairing, like for the usual Deligne pairing, see ([GS90, Theorem 4.10.1.])(cid:98): (cid:98) (cid:98) (cid:104)L M(cid:105) L L c ( , ) = π∗(c ( ).c ( )). 1 D 1 1 References [Bor08] T. Borek. Arakelov theory of noncommutative arithmetic curves and surfaces. PhD thesis, ETH Zu¨rich, 2008. [Del87] P. Deligne. Le d´eterminant de la cohomologie. Contemporary Mathematics, 67:93–177, 1987. [Gro68] A. Grothendieck. Le groupe de Brauer III: Exemples et compl´ements. In Dix Exposes sur la Cohomologie des Schemas, pages 88–188. North-Holland, 1968. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: P

text: P

text: But

text: then

text: we

text: recognize

text: that

text: A

text: where

text: is

text: the

text: Hermitian

text: line

text: bundle

text: on

text: Y

text: M−

text: N

text: described

text: in

text: for

text: the

text: Hermitian

text: line

text: bundles

text: and

text: on

text: X.

text: This

text: shows

text: that

text: there

text: is

text: an

text: isometry

text: N

text: N

text: A

text: D

text: The

text: usual

text: Deligne

text: pairing

text: is

text: bilinear,

text: see

text: So

text: we

text: have

text: for

text: example:

text: O

text: Z

text: X

text: D

text: D

text: D

text: This

text: isomorphism

text: of

text: line

text: bundles

text: becomes

text: an

text: isometry

text: if

text: we

text: equip

text: each

text: line

text: bundle

text: with

text: the

text: Deligne

text: metric,

text: see

text: Using

text: this

text: we

text: immediately

text: see

text: that

text: there

text: is

text: an

text: isometry

text: O

text: X

text: D

text: Y

text: L−

text: Using

text: these

text: facts

text: for

text: we

text: see

text: that

text: there

text: is

text: an

text: isometry

text: D

text: D

text: Putting

text: everything

text: together

text: we

text: get

text: an

text: isometry

text: N

text: N

text: N

text: A

text: D

text: D

text: A

text: The

text: questions

text: remains,

text: if

text: we

text: can

text: somehow

text: compare

text: this

text: -Deligne

text: pairing

text: with

text: the

text: one

text: E

text: given

text: by

text: Borek

text: in

text: Section

text: If

text: is

text: a

text: Hermitian

text: locally

text: free

text: sheaf,

text: then

text: he

text: replaces

text: the

text: Hermitian

text: metric

text: on

text: the

text: associated

text: vector

text: bundle

text: E

text: on

text: the

text: Riemann

text: surface

text: E

text: S

text: by

text: an

text: automorphism

text: β

text: of

text: the

text: sheaf

text: R,

text: Deﬁnition

text: The

text: Quillen

text: metric

text: is

text: replaced

text: by

text: the

text: determinant

text: of

text: the

text: action

text: of

text: β

text: on

text: the

text: Ext-groups,

text: Section

text: Is

text: there

text: any

text: relationship

text: between

text: these

text: two

text: approaches?

text: Furthermore

text: one

text: can

text: ask

text: the

text: following

text: questions:

text: A

text: Is

text: there

text: a

text: Riemann-Roch

text: type

text: formula

text: for

text: the

text: -Deligne

text: pairing?

text: A

text: How

text: does

text: the

text: -determinant

text: of

text: the

text: cohomology

text: behave

text: with

text: respect

text: to

text: Serre

text: duality?

text: Do

text: we

text: have

text: to

text: adjust

text: the

text: chosen

text: metrics

text: to

text: get

text: better

text: results?

text: A

text: Can

text: we

text: ﬁnd

text: arithmetic

text: -Chern

text: classes

text: to

text: get

text: a

text: similar

text: formula

text: for

text: the

text: ﬁrst

text: arith-

text: A

text: metic

text: Chern

text: class

text: of

text: the

text: -Deligne

text: pairing,

text: like

text: for

text: the

text: usual

text: Deligne

text: pairing,

text: see

text: Theorem

text: L

text: L

text: c

text: π∗(c

text: ).c

text: D

text: References

text: T.

text: Borek.

text: Arakelov

text: theory

text: of

text: noncommutative

text: arithmetic

text: curves

text: and

text: surfaces.

text: PhD

text: thesis,

text: ETH

text: Zu¨rich,

text: P.

text: Deligne.

text: Le

text: d´eterminant

text: de

text: la

text: cohomologie.

text: Contemporary

text: Mathematics,

text: A.

text: Grothendieck.

text: Le

text: groupe

text: de

text: Brauer

text: III:

text: Exemples

text: et

text: compl´ements.

text: In

text: Dix

text: Exposes

text: sur

text: la

text: Cohomologie

text: des

text: Schemas,

text: pages

text: North-Holland,

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [GS90] H. Gillet and C. Soul´e. Characteristic classes for algberaic vector bundles with Hermitian metric I. The Annals of Mathematics, 131(1):163–203, 1990. [Liu02] Q. Liu. Algebraic Geometry and Arithmetic Curves. Oxford University Press, 2002. [Sou89] C. Soul´e. G´eom´etrie d’Arakelov des surfaces arithm´etiques. S´eminaire Bourbaki, 31:327–343, 1988/1989. [Sou92] C. Soul´e. Lectures on Arakelov geometry. Cambridge University Press, 1992. Written with D. Abramovich, J.-F. Burnol & J. Kramer. [Vor90] A.A. Voronov. A uniﬁed approach to string scattering amplitudes. Communica- tions in Mathematical Physics, 131(1):179–218, 1990. [Wen99] L. Weng. Ω-Admissible theory. Proceedings of the London Mathematical Society, 79(3):481–510, 1999. 9

label: reference

text: H.

text: Gillet

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