text: 62 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI description of the nonlinearities which are allowed for such constructions. Also, the notions of balanced, ﬂexible and closable networks can be extended to higher di- mensions in a rather natural way. However, the construction of examples becomes quite diﬃcult and we believed that this was not worth the eﬀort. References [1] H. Berestycki and P.L. Lions, Nonlinear scalar ﬁeld equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. [2] H. Berestycki and P.L. Lions, Nonlinear scalar ﬁeld equations, II, Arch. Rat. Mech. Anal. 82 (1981), 347-375. [3] T. Bartsch and M. Willem, Inﬁnitely Many Radial Solutions of a Semilinear Elliptic Problem on RN, Arch. Rational Mech. Anal., 124 (1993), 261-276. [4] T. Bartsch and M. Willem, Inﬁnitely many radial solutions of a semilinear elliptic problem on RN, Arch. Rat. Mech. Anal. 124 (1993), 261-276. [5] E.N. Dancer, New solutions of equations on Rn. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 no. 3-4, 535-563 (2002). [6] M. del Pino, M. Kowalcyzk, F. Pacard and J. Wei, The Toda system and multiple-end so- lutions of autonomous planar elliptic problems, Advances in Mathematics 224 (2010), 1462- 1516. [7] M. Jleli and F. Pacard, An end-to-end construction for compactconstant mean curvature surfaces. Paciﬁc Journal of Maths, 221, no. 1, (2005) 81-108. [8] B. Gidas, W.M. Ni and L. Nirenberg , Symmetry and related properties via the maximun principle, Comm. Math. Phys. 68 (1979), 209-243. [9] N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three space, Ann. of Math. 131 (1990), 239-330. [10] N. Kapouleas, Compact constant mean curvature surfaces, J. Diﬀerential Geometry, 33, (1991) 683-715. [11] C. Breinier and N. Kapouleas, Embedded constant mean curvature surfaces in euclidean three space, arXiv:1210.3394 [math.DG]. − [12] M.K. Kwong, Uniqueness of positive solutions of ∆u u + up = 0 in Rn, Arch. Rational Mech. Anal., 105 (1989), 243-266. [13] M. K. Kwong and L. Zhang, Uniqueness of the positive solution of ∆u + f(u) = 0 in an annulus, Diﬀ. Int. Eqns. 4 , no. 3, (1991), 583-599. [14] S. Lorca and P. Ubilla, Symmetric and nonsymmetric solutions for an elliptic equation on R n, Nonlinear Anal. 58, (2004), 961-968. R [15] A. Malchiodi, Some neqw entire solutions of semilinear elliptic equations in n, Adv. Math. 221, no. 6, (2009), 1843-1909. [16] M. Musso, F. Pacard and J.C. Wei, Finite energy sign changing solution with dihedral sym- metry for the stationary non linear schrodinger equation, Journal of European Mathematical Society. [17] W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70, no. 2, (1993), 247-281. [18] S. Santra and J. Wei, New entire positive solution for the nonlinear Schr¨odinger equation : coexistence of fronts and bumps, American J. Math. [19] M. Struwe, Multiple solutions of diﬀerential equations without the Palais-Smale condition, Math. Ann. 261, (1982), 399-412. [20] M. Traizet, An embedded minimal surface with no symmetries. Journal of Diﬀ. Geom. 60, 103-153 (2002).

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: description

text: of

text: the

text: nonlinearities

text: which

text: are

text: allowed

text: for

text: such

text: constructions.

text: Also,

text: the

text: notions

text: of

text: balanced,

text: ﬂexible

text: and

text: closable

text: networks

text: can

text: be

text: extended

text: to

text: higher

text: di-

text: mensions

text: in

text: a

text: rather

text: natural

text: way.

text: However,

text: the

text: construction

text: of

text: examples

text: becomes

text: quite

text: diﬃcult

text: and

text: we

text: believed

text: that

text: this

text: was

text: not

text: worth

text: the

text: eﬀort.

text: References

text: H.

text: Berestycki

text: and

text: P.L.

text: Lions,

text: Nonlinear

text: scalar

text: ﬁeld

text: equations.

text: I.

text: Existence

text: of

text: a

text: ground

text: state,

text: Arch.

text: Rational

text: Mech.

text: Anal.,

text: H.

text: Berestycki

text: and

text: P.L.

text: Lions,

text: Nonlinear

text: scalar

text: ﬁeld

text: equations,

text: II,

text: Arch.

text: Rat.

text: Mech.

text: Anal.

text: T.

text: Bartsch

text: and

text: M.

text: Willem,

text: Inﬁnitely

text: Many

text: Radial

text: Solutions

text: of

text: a

text: Semilinear

text: Elliptic

text: Problem

text: on

text: RN,

text: Arch.

text: Rational

text: Mech.

text: Anal.,

text: T.

text: Bartsch

text: and

text: M.

text: Willem,

text: Inﬁnitely

text: many

text: radial

text: solutions

text: of

text: a

text: semilinear

text: elliptic

text: problem

text: on

text: RN,

text: Arch.

text: Rat.

text: Mech.

text: Anal.

text: E.N.

text: Dancer,

text: New

text: solutions

text: of

text: equations

text: on

text: Rn.

text: Ann.

text: Scuola

text: Norm.

text: Sup.

text: Pisa

text: Cl.

text: Sci.

text: no.

text: M.

text: del

text: Pino,

text: M.

text: Kowalcyzk,

text: F.

text: Pacard

text: and

text: J.

text: Wei,

text: The

text: Toda

text: system

text: and

text: multiple-end

text: so-

text: lutions

text: of

text: autonomous

text: planar

text: elliptic

text: problems,

text: Advances

text: in

text: Mathematics

text: M.

text: Jleli

text: and

text: F.

text: Pacard,

text: An

text: end-to-end

text: construction

text: for

text: compactconstant

text: mean

text: curvature

text: surfaces.

text: Paciﬁc

text: Journal

text: of

text: Maths,

text: no.

text: B.

text: Gidas,

text: W.M.

text: Ni

text: and

text: L.

text: Nirenberg

text: Symmetry

text: and

text: related

text: properties

text: via

text: the

text: maximun

text: principle,

text: Comm.

text: Math.

text: Phys.

text: N.

text: Kapouleas,

text: Complete

text: constant

text: mean

text: curvature

text: surfaces

text: in

text: Euclidean

text: three

text: space,

text: Ann.

text: of

text: Math.

text: N.

text: Kapouleas,

text: Compact

text: constant

text: mean

text: curvature

text: surfaces,

text: J.

text: Diﬀerential

text: Geometry,

text: C.

text: Breinier

text: and

text: N.

text: Kapouleas,

text: Embedded

text: constant

text: mean

text: curvature

text: surfaces

text: in

text: euclidean

text: three

text: space,

text: [math.DG].

text: M.K.

text: Kwong,

text: Uniqueness

text: of

text: positive

text: solutions

text: of

text: ∆u

text: u

text: up

text: in

text: Rn,

text: Arch.

text: Rational

text: Mech.

text: Anal.,

text: M.

text: K.

text: Kwong

text: and

text: L.

text: Zhang,

text: Uniqueness

text: of

text: the

text: positive

text: solution

text: of

text: ∆u

text: f(u)

text: in

text: an

text: annulus,

text: Diﬀ.

text: Int.

text: Eqns.

text: no.

text: S.

text: Lorca

text: and

text: P.

text: Ubilla,

text: Symmetric

text: and

text: nonsymmetric

text: solutions

text: for

text: an

text: elliptic

text: equation

text: on

text: R

text: n,

text: Nonlinear

text: Anal.

text: R

text: A.

text: Malchiodi,

text: Some

text: neqw

text: entire

text: solutions

text: of

text: semilinear

text: elliptic

text: equations

text: in

text: n,

text: Adv.

text: Math.

text: no.

text: M.

text: Musso,

text: F.

text: Pacard

text: and

text: J.C.

text: Wei,

text: Finite

text: energy

text: sign

text: changing

text: solution

text: with

text: dihedral

text: sym-

text: metry

text: for

text: the

text: stationary

text: non

text: linear

text: schrodinger

text: equation,

text: Journal

text: of

text: European

text: Mathematical

text: Society.

text: W.

text: M.

text: Ni

text: and

text: I.

text: Takagi,

text: Locating

text: the

text: peaks

text: of

text: least-energy

text: solutions

text: to

text: a

text: semilinear

text: Neumann

text: problem,

text: Duke

text: Math.

text: J.

text: no.

text: S.

text: Santra

text: and

text: J.

text: Wei,

text: New

text: entire

text: positive

text: solution

text: for

text: the

text: nonlinear

text: Schr¨odinger

text: equation

text: coexistence

text: of

text: fronts

text: and

text: bumps,

text: American

text: J.

text: Math.

text: M.

text: Struwe,

text: Multiple

text: solutions

text: of

text: diﬀerential

text: equations

text: without

text: the

text: Palais-Smale

text: condition,

text: Math.

text: Ann.

text: M.

text: Traizet,

text: An

text: embedded

text: minimal

text: surface

text: with

text: no

text: symmetries.

text: Journal

text: of

text: Diﬀ.

text: Geom.


text: 2 Section III gives the theoretical framework of DTQWs | (cid:105) on 1D chain with moving and swapping shift operators, where ψ(0) is the initial state. Finally, we obtain the and determines the matrix form of the various shift op- probability distribution, erators and evolution operators in the Hilbert space. In (cid:88) (cid:88) Sec. IV, we show the analytical results for the moving shift operator. We determine the eigenvalues and eigen- d |(cid:104) | (cid:105)| d |(cid:104) | | (cid:105)| P(i, t) = i, J ψ(t) 2 = i, J Uˆt ψ(0) 2. (4) states of the evolution operator and obtain the long-time averaged probability. In Sec. V, we show the analytical J=1 J=1 results for the swapping shift operator. We determine | (cid:105) the eigenvalues and eigenstates of the evolution operator Sup| pose (cid:105)the(cid:48)ei∈genvalue e(cid:48)qu∈ation of Uˆ is Uˆ Ψi(cid:48),J(cid:48) = and calculate the long-time averaged probability. We also u| i(cid:48),J(cid:48) Ψ(cid:105) i(cid:48),J(cid:48) (i [1, N], J [1, d]), where ui(cid:48),J(cid:48) and show±that the eigenstates corresponding to the eigenval- Ψi(cid:48),J(cid:48) are the eigenvalues and orthonormalized eigen- ues 1 play an important role in the quantum dynamics. states of the evolution operator Uˆ. Then Eq. (4) can be We compare the dynamical diﬀerence between the two written as, shift operators. Conclusions and discussions are given in (cid:88) (cid:12)(cid:12) (cid:88) (cid:88) (cid:12)(cid:12) the last part, Sec. VI. (cid:12) (cid:12) II. DEFINITION OF DISCRETE-TIME d d N (cid:104) | (cid:105)(cid:104) | (cid:105) 2 QUANTUM WALKS P(i, t) = (cid:88) (cid:88) (cid:88) uti(cid:48),J(cid:48) i, J Ψi(cid:48),J(cid:48) Ψi(cid:48),J(cid:48) ψ(0) (cid:48) (cid:48) J=1 J =1 i =1 Discrete-time quantum walk was ﬁrst introduced by d ∗ (cid:104) | (cid:105)(cid:104) | (cid:105) Mayer and Aharonov et al. in Ref. [24, 25]. DTQW = uti(cid:48),J(cid:48)ui(cid:48)t(cid:48),J(cid:48)(cid:48) i, J Ψi(cid:48),J(cid:48) Ψi(cid:48),J(cid:48) ψ(0) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) takes place in a discrete position space, with a unitary J=1 i ,J i ,J (cid:104) | (cid:105)(cid:104) | (cid:105) evolution of coin toss and position shift in discrete time ψ(0) Ψ (cid:48)(cid:48) (cid:48)(cid:48) Ψ (cid:48)(cid:48) (cid:48)(cid:48) i, J . i ,J i ,J steps. Here, we review the deﬁnition of DTQWs on d- (5) regular graph, which is a regular graph each vertex has † exactly d edges. Noting that Uˆ is a unitary operator, i.e., UˆUˆ = Iˆ | | DTQHW on d-regular graph happenHs on the coin Hilbert (which leads to u = 1), the long time averages of P(i, t) space c and posiHtion HHilb⊗eHrt space p, the total Hilbert can be written as, space is given by = [1]. If the d-regular graph (cid:88) c p has N vertices, the position and coin Hilbert space are H {| (cid:105) } H {| (cid:105) d1,e2n,o.t.e.,dd}a.s Thpe=coini ﬂ:iip=op1e,r2a,t.o..r, NCˆ ,andc p=osHitJion: sJhi=ft χ(i) = T(cid:88)l→im∞(cid:88)T1 (cid:88)T P(i, t) operator Sˆ are applied to the total state in at each t=0 H time step [1]. The coin ﬂip operation Cˆ (acting on ) is d (cid:104) | (cid:105)(cid:104) | (cid:105) c = i, J Ψ (cid:48) (cid:48) Ψ (cid:48) (cid:48) ψ(0) the quantum equivalent of randomly choosing which way i ,J i ,J (cid:88) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) the particle will move, then the position-shift operation J=1 i ,J i ,J Sˆ moves the particle according to the coin state, trans- (cid:104) | (cid:105)(cid:104) | (cid:105) 1 T ∗ ferring the particle into the new superposition state in ψ((cid:88)0) Ψ(cid:88)i(cid:48)(cid:48),J(cid:48)(cid:88)(cid:48) Ψi(cid:48)(cid:48),J(cid:48)(cid:48) i, J Tl→im∞ T (ui(cid:48),J(cid:48)ui(cid:48)(cid:48),J(cid:48)(cid:48))t position space. For every vertex, all the outgoing edges t=0 (cid:48) are labeled as 1, 2, . . . , d. Let us call eJ an edge e = (i, i ) d − (cid:104) | (cid:105)(cid:104) | (cid:105) i which on i’s end is labelled by J. The conditional shift = δ(ui(cid:48),J(cid:48) ui(cid:48)(cid:48),J(cid:48)(cid:48)) i, J Ψi(cid:48),J(cid:48) Ψi(cid:48),J(cid:48) ψ(0) operation Sˆ moves the particle from i to i(cid:48) if the edge J=1 i(cid:48),J(cid:48) i(cid:48)(cid:48),J(cid:48)(cid:48) → (cid:48) (cid:26) (cid:104) | (cid:105)(cid:104) | (cid:105) (i i ) is labeled by J on i’s side [1]: ψ(0) Ψ (cid:48)(cid:48) (cid:48)(cid:48) Ψ (cid:48)(cid:48) (cid:48)(cid:48) i, J , i ,J i ,J (6) | (cid:48) (cid:105) (cid:48) Sˆ|i, J(cid:105) = i , J , if eJi = (i, i ), (1) where δ(u (cid:48) (cid:48) − u (cid:48)(cid:48) (cid:48)(cid:48)) takes value 1 if u (cid:48) (cid:48) = u (cid:48)(cid:48) (cid:48)(cid:48) 0, otherwise. i ,J i ,J i ,J i ,J and equals to 0 otherwise. In the above equation, we can see that the limit distribution χ(i) depends on the The evolution of the system at each step of the walk is governed by the total operator, eigenvalues and eigens{tates o|f(cid:48)th∈e evolutio(cid:48)n∈operato}r Uˆ. If all the eigenvalues u (cid:48) (cid:48) i [1, N], J [1, d] are i ,J ⊗ diﬀerent (i.e., all the eigenvalues are not degenerated), Uˆ = Sˆ(Iˆp Cˆ), (2) the above Equation can be simpliﬁed as, (cid:88) (cid:88) H where Iˆ is the identity operator in . Thus the total p d |(cid:104) | (cid:105)| · |(cid:104) | (cid:105)| state after t steps is given by, χ(i) = i, J Ψ (cid:48) (cid:48) 2 Ψ (cid:48) (cid:48) ψ(0) 2. (7) i ,J i ,J (cid:48) (cid:48) J=1 i ,J | (cid:105) | (cid:105) ψ(t) = Uˆt ψ(0) , (3) ##LTLine## ##LTLine##

label: paragraph

text: Section

text: III

text: gives

text: the

text: theoretical

text: framework

text: of

text: DTQWs

text: on

text: chain

text: with

text: moving

text: and

text: swapping

text: shift

text: operators,

text: where

text: is

text: the

text: initial

text: state.

text: Finally,

text: we

text: obtain

text: the

text: and

text: determines

text: the

text: matrix

text: form

text: of

text: the

text: various

text: shift

text: op-

text: probability

text: distribution,

text: erators

text: and

text: evolution

text: operators

text: in

text: the

text: Hilbert

text: space.

text: In

label: equation

text: Sec.

text: IV,

text: we

text: show

text: the

text: analytical

text: results

text: for

text: the

text: moving

text: shift

text: operator.

text: We

text: determine

text: the

text: eigenvalues

text: and

text: eigen-

text: d

text: d

text: P(i,

text: t)

text: i,

text: J

text: ψ(t)

text: i,

text: J

text: Uˆt

text: states

text: of

text: the

text: evolution

text: operator

text: and

text: obtain

text: the

text: long-time

text: averaged

text: probability.

text: In

text: Sec.

text: V,

text: we

text: show

text: the

text: analytical

text: results

text: for

text: the

text: swapping

text: shift

text: operator.

text: We

text: determine

text: the

text: eigenvalues

text: and

text: eigenstates

text: of

text: the

text: evolution

text: operator

text: Sup|

text: pose

text: of

text: Uˆ

text: is

text: Uˆ

text: and

text: calculate

text: the

text: long-time

text: averaged

text: probability.

text: We

text: also

text: u|

text: (i

text: N],

text: J

text: d]),

text: where

text: and

text: show±that

text: the

text: eigenstates

text: corresponding

text: to

text: the

text: eigenval-

text: are

text: the

text: eigenvalues

text: and

text: orthonormalized

text: eigen-

text: ues

text: play

text: an

text: important

text: role

text: in

text: the

text: quantum

text: dynamics.

text: states

text: of

text: the

text: evolution

text: operator

text: Uˆ.

text: Then

text: Eq.

text: can

text: be

text: We

text: compare

text: the

text: dynamical

text: diﬀerence

text: between

text: the

text: two

text: written

text: as,

text: shift

text: operators.

text: Conclusions

text: and

text: discussions

text: are

text: given

text: in

text: the

text: last

text: part,

text: Sec.

text: VI.

text: II.

text: DEFINITION

text: OF

text: DISCRETE-TIME

text: d

text: d

text: N

text: QUANTUM

text: WALKS

text: P(i,

text: t)

text: i,

text: J

text: J

text: i

text: Discrete-time

text: quantum

text: walk

text: was

text: ﬁrst

text: introduced

text: by

text: d

text: Mayer

text: and

text: Aharonov

text: et

text: al.

text: in

text: Ref.

text: DTQW

text: i,

text: J

text: takes

text: place

text: in

text: a

text: discrete

text: position

text: space,

text: with

text: a

text: unitary

text: i

text: ,J

text: i

text: ,J

text: evolution

text: of

text: coin

text: toss

text: and

text: position

text: shift

text: in

text: discrete

text: time

text: Ψ

text: Ψ

text: i,

text: J

text: i

text: ,J

text: i

text: ,J

text: steps.

text: Here,

text: we

text: review

text: the

text: deﬁnition

text: of

text: DTQWs

text: on

text: d-

text: regular

text: graph,

text: which

text: is

text: a

text: regular

text: graph

text: each

text: vertex

text: has

text: exactly

text: d

text: edges.

text: Noting

text: that

text: Uˆ

text: is

text: a

text: unitary

text: operator,

text: i.e.,

text: UˆUˆ

text: Iˆ

text: DTQHW

text: on

text: d-regular

text: graph

text: happenHs

text: on

text: the

text: coin

text: Hilbert

text: (which

text: leads

text: to

text: u

text: the

text: long

text: time

text: averages

text: of

text: P(i,

text: t)

text: space

text: c

text: and

text: posiHtion

text: HHilb⊗eHrt

text: space

text: p,

text: the

text: total

text: Hilbert

text: can

text: be

text: written

text: as,

text: space

text: is

text: given

text: by

text: If

text: the

text: d-regular

text: graph

text: c

text: p

text: has

text: N

text: vertices,

text: the

text: position

text: and

text: coin

text: Hilbert

text: space

text: are

text: H

text: H

text: Thpe=coini

text: NCˆ

text: ,andc

text: p=osHitJion:

text: sJhi=ft

text: χ(i)

text: P(i,

text: t)

text: operator

text: Sˆ

text: are

text: applied

text: to

text: the

text: total

text: state

text: in

text: at

text: each

text: H

text: time

text: step

text: The

text: coin

text: ﬂip

text: operation

text: Cˆ

text: (acting

text: on

text: is

text: d

text: c

text: i,

text: J

text: Ψ

text: Ψ

text: the

text: quantum

text: equivalent

text: of

text: randomly

text: choosing

text: which

text: way

text: i

text: ,J

text: i

text: ,J

text: the

text: particle

text: will

text: move,

text: then

text: the

text: position-shift

text: operation

text: i

text: ,J

text: i

text: ,J

text: Sˆ

text: moves

text: the

text: particle

text: according

text: to

text: the

text: coin

text: state,

text: trans-

text: T

text: ferring

text: the

text: particle

text: into

text: the

text: new

text: superposition

text: state

text: in

text: i,

text: J

text: Tl→im∞

text: T

text: position

text: space.

text: For

text: every

text: vertex,

text: all

text: the

text: outgoing

text: edges

text: are

text: labeled

text: as

text: d.

text: Let

text: us

text: call

text: eJ

text: an

text: edge

text: e

text: (i,

text: i

text: d

text: i

text: which

text: on

text: i’s

text: end

text: is

text: labelled

text: by

text: J.

text: The

text: conditional

text: shift

text: i,

text: J

text: operation

text: Sˆ

text: moves

text: the

text: particle

text: from

text: i

text: to

text: if

text: the

text: edge

text: (i

text: i

text: is

text: labeled

text: by

text: J

text: on

text: i’s

text: side

text: Ψ

text: Ψ

text: i,

text: J

text: i

text: ,J

text: i

text: ,J

text: Sˆ|i,

text: i

text: J

text: if

text: eJi

text: (i,

text: i

text: where

text: δ(u

text: u

text: takes

text: value

text: if

text: u

text: u

text: otherwise.

text: i

text: ,J

text: i

text: ,J

text: i

text: ,J

text: i

text: ,J

text: and

text: equals

text: to

text: otherwise.

text: In

text: the

text: above

text: equation,

text: we

text: can

text: see

text: that

text: the

text: limit

text: distribution

text: χ(i)

text: depends

text: on

text: the

text: The

text: evolution

text: of

text: the

text: system

text: at

text: each

text: step

text: of

text: the

text: walk

text: is

text: governed

text: by

text: the

text: total

text: operator,

text: eigenvalues

text: and

text: eigens{tates

text: Uˆ.

text: If

text: all

text: the

text: eigenvalues

text: u

text: i

text: N],

text: J

text: d]

text: are

text: i

text: ,J

text: diﬀerent

text: (i.e.,

text: all

text: the

text: eigenvalues

text: are

text: not

text: degenerated),

text: Uˆ

text: Sˆ(Iˆp

text: Cˆ),

text: the

text: above

text: Equation

text: can

text: be

text: simpliﬁed

text: as,

text: H

text: where

text: Iˆ

text: is

text: the

text: identity

text: operator

text: in

text: Thus

text: the

text: total

text: p

text: d

text: state

text: after

text: t

text: steps

text: is

text: given

text: by,

text: χ(i)

text: i,

text: J

text: Ψ

text: Ψ

text: i

text: ,J

text: i

text: ,J

text: i

text: ,J

text: ψ(t)

text: Uˆt

text: ##LTLine##

text: ##LTLine##


text: 3 In order to calculate the analytical expressions for Pst(ait,ets) |Ψan(cid:48)d (cid:48)χ(cid:105)(oi)f,thaell etvhoeluteiiognenovpaeluraestoruiU(cid:48)ˆ,J(cid:48)araenrdeqeuiigreend-. Cˆp = Iˆp ⊗ Cˆ =|1(cid:105)(cid:104)(cid:88)1| ⊗ |R(cid:105)(cid:104)R| + |N(cid:105)(cid:104)N| ⊗ |L(cid:105)(cid:104)L| i ,J − In the following we will use Eq. (7) to derive the limit N 1 | (cid:105)(cid:104) | ⊗ (10) probability distribution of DTQWs on 1D chain, and + i i Cˆ. put emphasis on the calculation of the eigenvalues and i=2 eigenstates of Uˆ. III. DTQWS ON 1D CHAIN WITH THE In the Hilbert space, the above coin operator Cˆp can MOVING AND SWAPPING SHIFT OPERATORS be represented as the following matrix,   In this section, we will continue the calculation for ··· ··· DTQWs on 1D chain. For a 1D chain system composed  2 N−1 N 1 2 N−1  { } 2  a 0 0 0 b 0  of N nodes, which are labeled as i : i = 1, 2, ..., N , each   node 1 < i < N is connected two nearest neighbors (i−1 ...  ... ... ... ...    and i+1) except the leftmost node 1 and rightmost node −  0 a 0 0 0 b  − N 1  · · · · · ·  Nde.noT(cid:105)thede Hasil|b2e,rLt(cid:105)s,p|3a,cLe(cid:105)h,a· ·s·2,N|N, L2(cid:105)b,a|1s,eRv(cid:105)e,c|t2o,rRs,(cid:105)w, ·h·i·ch, |Nar−e Cˆp = N1  00 · · · 00 10 01 −00 · · · 00 , (11) 1, R . We will derive the matrix form for the coin oper- 2 b 0 0 0 a 0 ator and shift operator in the Hilbert space, and obtain ... ... ... ... ... the matrix representation for the evolution operator. We − also determine the eigenvalues and eigenstates of Uˆ based N−1 0 b 0 0 0 a on the matrix representation. A. Coin operator Cˆ and initial state where the fo|ur(cid:105)(cid:104)squ| a|re(cid:105)s(cid:104)co|rr|esp(cid:105)(cid:104)ond| t|o (cid:105)t(cid:104)he|orthonormal- ized basis set L L , L R , R L , R R respectively. In this paper, we will use this coin operator to do ana- The probability distribution is related to the coin op- | (cid:105) lytical calculations. erator Cˆ and initial state ψ(0) . In the scientiﬁc com- | (cid:105) | (cid:105)⊗| (cid:105) For the initial state ψ(0) = i C , the initial posi- munity, the coin ﬂip operator Cˆ can be of various forms, 0 0 tion i can be at the two end points and the center nodes, for instance, the most commonly used Grover coin and 0 | (cid:105) and the corresponding initial coin state C can be sin- the Discrete Fourier Transform (DFT) coin [1, 2]. It has | (cid:105) | (cid:105) 0 | (cid:105) gle state ( L or R ) and superposed state (cos ω L + been shown that diﬀerent choices of coin ﬂip operator | (cid:105) | 0 (cid:105) Cˆ and initial state |ψ(0)(cid:105) may have diﬀerent dynamics. eiϕ0 sin ω0 R ). Consequently, the initial state ψ(0) can be summarized as follows, Here, we will sp|ecify (cid:105)the form for the coin operator Cˆ  and initial state ψ(0) . | (cid:105) ⊗ | (cid:105) | (cid:105) For the coin operator, we use the coin type of DTQWs | (cid:105) |1 (cid:105) ⊗R| (cid:105)= 1| ,R ,(cid:105) if i0 = 1 in Szegedy’s scheme [26], which can be generalized to ψ(0) = |N(cid:105) ⊗ L = |N,(cid:105)L , | (cid:105) if i∈0 = N (12) high dimensional case. If the particle is located at the i0 (cosω0 L + eiϕ0 sinω0 R ), i0 (1,N) two end points (i = 1, N), the particle will move to the Here, for the sake of simplicity, we set ω = π/4 neighboring node in the next step (do not need coin). If 0 and choose appropriate ϕ value to make the product the particle is located at the center nodes 1 < i < N, we |(cid:104) | (cid:105)| |(cid:104) 0 | (cid:105)| · |(cid:104) | (cid:105)| | (cid:105) | (cid:105) | (cid:105) Ψ ψ(0) 2 equals to Ψ i , L 2 i , L ψ(0) 2 + use an arbitrary coin state φ = cos 1ω L +eiϕ sin 1ω R |(cid:104) i,J | (cid:105)| · |(cid:104) | i(cid:105)0|,L 0 ∈ 0 to generate the coin operator, 2 2 Ψi0,R i0, R 2 i0, R ψ(0) 2 (i0 (1, N)). B. The moving and swapping shift operators | (cid:105)(cid:104) | − | (cid:105)(cid:104) | | (cid:105)(cid:104) | In this section, we deﬁne the moving and swapping Cˆ =2 φ −φ I =| c(cid:105)o(cid:104)s ω| −L L +| e(cid:105)iϕ(cid:104) si|n ω L R (8) shift operators. The moving shift operator Sˆm moves + e iϕ sin ω R L cos ω R R the particle to the neighboring position and keep the di- | (cid:105) | − (cid:105) rection J unchanged. This means Sˆm i, L = i 1, L ∈ | (cid:105) | (cid:105) ∈ − (i [3, N]), Sˆm i, R = i+1, R (i [1, N 2]). For the When ϕ = 0, the above coin operator becomes (cid:20) (cid:21) two boundary positions, we use the elastic reﬂection con- | (cid:105) | (cid:105) | − (cid:105) | (cid:105) dition: Sˆm 2, L = 1, R , Sˆm N 1, R = N, L . Thus, | (cid:105) | (cid:105)(cid:104) | | (cid:105)(cid:104) | the moving shift operator acting on i, J is summarized (cid:20) (cid:21) cos ω L L sin ω L R Cˆ = | (cid:105)(cid:104) | − | (cid:105)(cid:104) | as follows,  si|n ω(cid:105)(cid:104)R| L | (cid:105)c(cid:104)os|ω R R (9)  ≡ a L L b L R ≡ ≡ | (cid:105) (cid:54) (cid:54) − b|R(cid:105)(cid:104)L| −a|R(cid:105)(cid:104)R| , a cos ω, b sin ω. | (cid:105) |ii +− 11,, JJ(cid:105),, iiff JJ == RL aanndd 31 (cid:54) ii (cid:54) NN 2 Sˆm i, J = | (cid:105) − (13) N, L , if J = R and i = N 1 | (cid:105) The direct product of Iˆp and Cˆ can be written as, 1, R , if J = L and i = 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: order

text: to

text: calculate

text: the

text: analytical

text: expressions

text: for

text: Pst(ait,ets)

text: Cˆp

text: Iˆp

text: Cˆ

text: i

text: ,J

text: In

text: the

text: following

text: we

text: will

text: use

text: Eq.

text: to

text: derive

text: the

text: limit

text: N

text: probability

text: distribution

text: of

text: DTQWs

text: on

text: chain,

text: and

text: i

text: i

text: Cˆ.

text: put

text: emphasis

text: on

text: the

text: calculation

text: of

text: the

text: eigenvalues

text: and

text: eigenstates

text: of

text: Uˆ.

text: III.

label: section

text: DTQWS

text: ON

text: CHAIN

text: WITH

text: THE

text: In

text: the

text: Hilbert

text: space,

text: the

text: above

text: coin

text: operator

text: Cˆp

text: can

text: MOVING

text: AND

text: SWAPPING

text: SHIFT

text: OPERATORS

text: be

text: represented

text: as

text: the

text: following

text: matrix,

text: In

text: this

text: section,

text: we

text: will

text: continue

text: the

text: calculation

text: for

text: DTQWs

text: on

text: chain.

text: For

text: a

text: chain

text: system

text: composed

text: N

text: a

text: b

text: of

text: N

text: nodes,

text: which

text: are

text: labeled

text: as

text: i

text: i

text: N

text: each

text: node

text: i

text: N

text: is

text: connected

text: two

text: nearest

text: neighbors

text: and

text: except

text: the

text: leftmost

text: node

text: and

text: rightmost

text: node

text: a

text: b

text: N

text: ·h·i·ch,

text: |Nar−e

text: Cˆp

text: R

text: We

text: will

text: derive

text: the

text: matrix

text: form

text: for

text: the

text: coin

text: oper-

text: b

text: a

text: ator

text: and

text: shift

text: operator

text: in

text: the

text: Hilbert

text: space,

text: and

text: obtain

text: the

text: matrix

text: representation

text: for

text: the

text: evolution

text: operator.

text: We

text: also

text: determine

text: the

text: eigenvalues

text: and

text: eigenstates

text: of

text: Uˆ

text: based

text: b

text: a

text: on

text: the

text: matrix

text: representation.

text: A.

text: Coin

text: operator

text: Cˆ

text: and

text: initial

text: state

text: where

text: the

text: t|o

text: ized

text: basis

text: set

text: L

text: L

text: L

text: R

text: R

text: L

text: R

text: R

text: respectively.

text: In

text: this

text: paper,

text: we

text: will

text: use

text: this

text: coin

text: operator

text: to

text: do

text: ana-

text: The

text: probability

text: distribution

text: is

text: related

text: to

text: the

text: coin

text: op-

text: lytical

text: calculations.

text: erator

text: Cˆ

text: and

text: initial

text: state

text: In

text: the

text: scientiﬁc

text: com-

text: For

text: the

text: initial

text: state

text: i

text: C

text: the

text: initial

text: posi-

text: munity,

text: the

text: coin

text: ﬂip

text: operator

text: Cˆ

text: can

text: be

text: of

text: various

text: forms,

text: tion

text: i

text: can

text: be

text: at

text: the

text: two

text: end

text: points

text: and

text: the

text: center

text: nodes,

text: for

text: instance,

text: the

text: most

text: commonly

text: used

text: Grover

text: coin

text: and

text: and

text: the

text: corresponding

text: initial

text: coin

text: state

text: C

text: can

text: be

text: sin-

text: the

text: Discrete

text: Fourier

text: Transform

text: (DFT)

text: coin

text: It

text: has

text: gle

text: state

text: L

text: or

text: R

text: and

text: superposed

text: state

text: (cos

text: ω

text: L

text: been

text: shown

text: that

text: diﬀerent

text: choices

text: of

text: coin

text: ﬂip

text: operator

text: Cˆ

text: and

text: initial

text: state

text: may

text: have

text: diﬀerent

text: dynamics.

text: sin

text: ω0

text: R

text: Consequently,

text: the

text: initial

text: state

text: can

text: be

text: summarized

text: as

text: follows,

text: Here,

text: we

text: will

text: sp|ecify

text: form

text: for

text: the

text: coin

text: operator

text: Cˆ

text: and

text: initial

text: state

text: For

text: the

text: coin

text: operator,

text: we

text: use

text: the

text: coin

text: type

text: of

text: DTQWs

text: |1

text: ⊗R|

text: ,R

text: if

text: in

text: Szegedy’s

text: scheme

text: which

text: can

text: be

text: generalized

text: to

text: L

text: if

text: N

text: high

text: dimensional

text: case.

text: If

text: the

text: particle

text: is

text: located

text: at

text: the

text: L

text: R

text: two

text: end

text: points

text: (i

text: N),

text: the

text: particle

text: will

text: move

text: to

text: the

text: Here,

text: for

text: the

text: sake

text: of

text: simplicity,

text: we

text: set

text: ω

text: neighboring

text: node

text: in

text: the

text: next

text: step

text: (do

text: not

text: need

text: coin).

text: If

text: and

text: choose

text: appropriate

text: ϕ

text: value

text: to

text: make

text: the

text: product

text: the

text: particle

text: is

text: located

text: at

text: the

text: center

text: nodes

text: i

text: N,

text: we

text: Ψ

text: equals

text: to

text: Ψ

text: i

text: L

text: i

text: L

text: use

text: an

text: arbitrary

text: coin

text: state

text: φ

text: cos

text: 1ω

text: L

text: +eiϕ

text: sin

text: 1ω

text: R

text: i,J

text: to

text: generate

text: the

text: coin

text: operator,

text: R

text: R

text: N)).

text: B.

text: The

text: moving

text: and

text: swapping

text: shift

text: operators

text: In

text: this

text: section,

text: we

text: deﬁne

text: the

text: moving

text: and

text: swapping

text: Cˆ

text: φ

text: −φ

text: I

text: ω|

text: −L

text: L

text: si|n

text: ω

text: L

text: R

text: shift

text: operators.

text: The

text: moving

text: shift

text: operator

text: Sˆm

text: moves

text: e

text: iϕ

text: sin

text: ω

text: R

text: L

text: cos

text: ω

text: R

text: R

text: the

text: particle

text: to

text: the

text: neighboring

text: position

text: and

text: keep

text: the

text: di-

text: rection

text: J

text: unchanged.

text: This

text: means

text: Sˆm

text: i,

text: L

text: i

text: L

text: (i

text: N]),

text: Sˆm

text: i,

text: R

text: R

text: (i

text: N

text: For

text: the

text: When

text: ϕ

text: the

text: above

text: coin

text: operator

text: becomes

text: two

text: boundary

text: positions,

text: we

text: use

text: the

text: elastic

text: reﬂection

text: con-

text: dition:

text: Sˆm

text: L

text: R

text: Sˆm

text: N

text: R

text: N,

text: L

text: Thus,

text: the

text: moving

text: shift

text: operator

text: acting

text: on

text: i,

text: J

text: is

text: summarized

text: cos

text: ω

text: L

text: L

text: sin

text: ω

text: L

text: R

text: Cˆ

text: as

text: follows,

text: si|n

text: L

text: R

text: R

text: a

text: L

text: L

text: b

text: L

text: R

text: a

text: cos

text: ω,

text: b

text: sin

text: ω.

text: |ii

text: iiff

text: JJ

text: RL

text: aanndd

text: ii

text: NN

text: Sˆm

text: i,

text: J

text: N,

text: L

text: if

text: J

text: R

text: and

text: i

text: N

text: The

text: direct

text: product

text: of

text: Iˆp

text: and

text: Cˆ

text: can

text: be

text: written

text: as,

text: R

text: if

text: J

text: L

text: and

text: i

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: 5 | (cid:105) erator. Suppose the eigen equation of Uˆm is Uˆm Ψm = | (cid:105) | (cid:105) | (cid:105) | (cid:105) − u Ψm (Uˆm Ψm = um Ψm ), the eigenstates Ψm can Setting i = 2 in Eq. (27), we obtain bαm aαm = i,J i,J i,J 2,L 2,R be expre(cid:88)ssed as uαm . Substituting αm in Eq. (31) and αm in Eq. (22) 3,R 2,R 3,R (i = 2) into this equation, we get another equation for | (cid:105) | (cid:105) αm and αm , Ψm = αm i, J 3,L 2,L i,J i,J | (cid:105) | (cid:105) · · · | (cid:105) =α+2m,αLm1, L|1, R+(cid:105)α+3m,Lαm2, L|2,+R(cid:105) ++· · ·α+Nm,αLmN−, L |N − 1, R(cid:105) au(u2 − b)α3m,L − [(u2 − b)(u2 − b2) + a2b]α2m,L = 0 (32) 1,R 2,R N 1,R (21) Thus we have got two equations for αm and αm , Noting that the matrix form of Uˆm in Eq. (19), the 3,L 2,L | (cid:105) | (cid:105) Eq. (30) and Eq. (32). The two equations should have eigen equation Uˆm Ψm = u Ψm can be decomposed − nonzero solutions, hence the determinant of the four co- into the following 2N 2 linear equations, eﬃcients equals to 0. In the Appendix C, we show that the substraction of the coeﬃcient products can be sim- (cid:54) (cid:54) − pliﬁed to be a much simple form in Eq. (C10), thus we aαm + bαm = uαm , 2 i N 2 (22) i+1,L i+1,R i,L have obtained determinant equation for the eigenvalues, αm = uαm− , (23) N,L N 1,L − bαm− aαm− = uαm , (24) − N 1,L N 1,R N,L b 1 aα2m,L + bα2m,R = uα1m,R, (25) UN−1(x) + b + 1UN−3(x) = 0, (33) αm = uαm , (26) 1,R 2,R − (cid:54) (cid:54) − bαim,L aαim,R = uαim+1,R, 2 i N 2 (27) where Un(x) are Chebyshev polynomials of the second kind. There is no exact analytical solution for Eq. (33). How- Utilizing Eq. (−22) to eliminat−e αim,R and αim+1,R, Eq. (27) ever, when the system size N is large, the(cid:112)solution for becomes αim+1,L αim−1,L = u2au1αim,L. This is similar to Eq. (3−3) is close to the equation UN−1(x) = 0, wh·ic·h· lead−s the recursive relation of the variant Chebyshev polyno- to N 1 roots xk = cos θk, θk = kπ/N, k±= 1, 2, , N mials (see Appendix B). Noting the recursive relations 1. Us√ing the mapping relation u = ay 1 + a2y2 = − ≡ ± − − and the mapping relationship u2 1 2y in the deﬁnition axi 1 a2x2, the 2N 2 eigenvalues of Uˆm are given au of the variant Chebyshev polynomials, the variables αm by, ∈ − i,L (cid:113) (i [2, N 1]) can be expressed as a function of αm 3,L and αm , 2,L ± − kπ ∈ − u± = ax i 1 a2x2, x = cos , k [1, N 1]. (34) k k k k N ∈ − αm = V − (y)αm + V − (y)αm , i [2, N 1] (28) i,L i 3 3,L i 4 2,L | (cid:105) Now we analyze the eigenstates Ψm . For this pur- pose, we denote all the components αm in terms of αm . where V (y) = inU (x) (y = ix) is the variant Cheby- i,J 2,L n n First, according to Eq. (32), we can express αm as a shev polynomials (See Appendix B). Utilizing Eq. (22) 3,L function of αm , to eliminate αm− in Eqs. (23), (24), we get the rela- 2,L N 1,R tionship of αm− and αm− , N 1,L N 2,L − − (cid:104)(u2 b)(u2 b2) + a2(cid:105)b (1 − bu2)αm− − auαm− = 0. (29) α3m,L = au(u2 − b) α2m,L N 1,L N 2,L − (cid:104)u2 b2 ab (cid:105) = + − αm au u(u2 b) 2,L Combining Eqs. (28) and (29), we have, − − (cid:104)(u2 1) + (1 b2) (cid:105) ab = + − αm (35) − − au u(u2 b) 2,L [(1 bu2)V − (y) auV − (y)]αm − [auVN−6N(y)4− (1 − bu2N)VN5 −5(y)3],αL2m,L = 0 (30) =(cid:16)2iauaxu+ a2 +(cid:17)u(ua2b− b) α2m,L au In order to get another equation for α3m,L and α2m,L, we = 2ix + u2 − b α2m,L, combine Eqs. (25) and (26), − ≡ where a2 + b2 = 1 and mapping relation u2 1 2y = au aαm auαm 2ix are used. Substituting Eq. (35) into Eq. (28), we αm = −2,L , αm = −2,L . (31) 2,R u2 b 1,R u2 b have, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: erator.

text: Suppose

text: the

text: eigen

text: equation

text: of

text: Uˆm

text: is

text: Uˆm

text: Ψm

text: u

text: Ψm

text: (Uˆm

text: Ψm

text: um

text: Ψm

text: the

text: eigenstates

text: Ψm

text: can

text: Setting

text: i

text: in

text: Eq.

text: we

text: obtain

text: bαm

text: aαm

text: i,J

text: i,J

text: i,J

text: be

text: as

text: uαm

text: Substituting

text: αm

text: in

text: Eq.

text: and

text: αm

text: in

text: Eq.

text: (i

text: into

text: this

text: equation,

text: we

text: get

text: another

text: equation

text: for

text: αm

text: and

text: αm

text: Ψm

text: αm

text: i,

text: J

text: i,J

text: i,J

text: ·α+Nm,αLmN−,

text: L

text: |N

text: N

text: Thus

text: we

text: have

text: got

text: two

text: equations

text: for

text: αm

text: and

text: αm

text: Noting

text: that

text: the

text: matrix

text: form

text: of

text: Uˆm

text: in

text: Eq.

text: the

text: Eq.

text: and

text: Eq.

text: The

text: two

text: equations

text: should

text: have

text: eigen

text: equation

text: Uˆm

text: Ψm

text: u

text: Ψm

text: can

text: be

text: decomposed

text: nonzero

text: solutions,

text: hence

text: the

text: determinant

text: of

text: the

text: four

text: co-

text: into

text: the

text: following

text: linear

text: equations,

text: eﬃcients

text: equals

text: to

text: In

text: the

text: Appendix

text: C,

text: we

text: show

text: that

text: the

text: substraction

text: of

text: the

text: coeﬃcient

text: products

text: can

text: be

text: sim-

text: pliﬁed

text: to

text: be

text: a

text: much

text: simple

text: form

text: in

text: Eq.

text: thus

text: we

text: aαm

text: bαm

text: uαm

text: i

text: N

text: i,L

text: have

text: obtained

text: determinant

text: equation

text: for

text: the

text: eigenvalues,

text: αm

text: uαm−

text: N,L

text: N

text: bαm−

text: aαm−

text: uαm

text: N

text: N

text: N,L

text: b

text: b

text: αm

text: uαm

text: bαim,L

text: aαim,R

text: i

text: N

text: where

text: Un(x)

text: are

text: Chebyshev

text: polynomials

text: of

text: the

text: second

text: kind.

text: There

text: is

text: no

text: exact

text: analytical

text: solution

text: for

text: Eq.

text: How-

text: Utilizing

text: Eq.

text: to

text: eliminat−e

text: αim,R

text: and

text: Eq.

text: ever,

text: when

text: the

text: system

text: size

text: N

text: is

text: large,

text: for

text: becomes

text: This

text: is

text: similar

text: to

text: Eq.

text: is

text: close

text: to

text: the

text: equation

text: wh·ic·h·

text: lead−s

text: the

text: recursive

text: relation

text: of

text: the

text: variant

text: Chebyshev

text: polyno-

text: to

text: N

text: roots

text: xk

text: cos

text: θk,

text: θk

text: kπ/N,

text: k±=

text: N

text: mials

text: (see

text: Appendix

text: B).

text: Noting

text: the

text: recursive

text: relations

text: Us√ing

text: the

text: mapping

text: relation

text: u

text: ay

text: and

text: the

text: mapping

text: relationship

text: in

text: the

text: deﬁnition

text: axi

text: the

text: eigenvalues

text: of

text: Uˆm

text: are

text: given

text: au

text: of

text: the

text: variant

text: Chebyshev

text: polynomials,

text: the

text: variables

text: αm

text: by,

text: i,L

text: (i

text: N

text: can

text: be

text: expressed

text: as

text: a

text: function

text: of

text: αm

text: and

text: αm

text: kπ

text: u±

text: ax

text: i

text: x

text: cos

text: k

text: N

text: k

text: k

text: k

text: k

text: N

text: αm

text: V

text: (y)αm

text: V

text: (y)αm

text: i

text: N

text: i,L

text: i

text: i

text: Now

text: we

text: analyze

text: the

text: eigenstates

text: Ψm

text: For

text: this

text: pur-

text: pose,

text: we

text: denote

text: all

text: the

text: components

text: αm

text: in

text: terms

text: of

text: αm

text: where

text: V

text: (y)

text: inU

text: (x)

text: (y

text: ix)

text: is

text: the

text: variant

text: Cheby-

text: i,J

text: n

text: n

text: First,

text: according

text: to

text: Eq.

text: we

text: can

text: express

text: αm

text: as

text: a

text: shev

text: polynomials

text: (See

text: Appendix

text: B).

text: Utilizing

text: Eq.

text: function

text: of

text: αm

text: to

text: eliminate

text: αm−

text: in

text: Eqs.

text: we

text: get

text: the

text: rela-

text: N

text: tionship

text: of

text: αm−

text: and

text: αm−

text: N

text: N

text: auαm−

text: b)

text: N

text: N

text: ab

text: αm

text: au

text: b)

text: Combining

text: Eqs.

text: and

text: we

text: have,

text: ab

text: αm

text: au

text: b)

text: (y)

text: auV

text: (y)]αm

text: b)

text: au

text: In

text: order

text: to

text: get

text: another

text: equation

text: for

text: and

text: we

text: b

text: combine

text: Eqs.

text: and

text: where

text: and

text: mapping

text: relation

text: au

text: aαm

text: auαm

text: are

text: used.

text: Substituting

text: Eq.

text: into

text: Eq.

text: we

text: αm

text: αm

text: b

text: b

text: have,

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: Valleytronics on the Surface of Topological Crystalline Insulator: Elliptic Dichroism and Valley-Selective Optical Pumping Motohiko Ezawa Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan The low-energy theory of the surface of the topological crystalline insulator (TCI) is characterized by four Dirac cones anisotropic into the x and y directions. Recent experiments have shown that the band gap can be introduced in these Dirac cones by crystal distortion by applying strain to the crystal structure. The TCI surface provides us with a new way to valleytronics when gaps are given to Dirac cones. Indeed the system has the Chern number and three valley-Chern numbers. We investigate the optical absorption on the TCI surface. It shows a strong elliptic dichroism though the four Dirac cones have the same chiralities. Namely, it is found that the absorptions of the right- and left-polarized light are different, depending on the sign of mass and the location of the Dirac cones, owing to the anisotropy of the Dirac cone. By measuring this elliptic dichroism it is possible to determine the anisotropy of a Dirac cone experimentally. I. INTRODUCTION way of the valleytronics based on the TCI. The present paper is composed as follows. In Section II, we introduce the low-energy Hamiltonians H and H valid Valleytronics is a promising candidate of the next gener- X Y near the X and Y points for the [001] surface, which are ation electronics1–7. It is a technology of manipulating the related by the C discrete rotation symmetry. The Hamilto- degree of freedom to which inequivalent degenerate state an 4 nian contains the pseudospin degree of freedom representing electron belongs near the Fermi level. The main target of val- the cation and the anion. The X (Y ) point is separated into leytronics is the honeycomb lattice system such as graphene. (cid:48) (cid:48) a pair of the Λ and Λ (Λ and Λ ) points due to the Indeed, the honeycomb structure is an ideal play ground of X X Y Y spin-pseudospin mixing. We then derive the four low-energy valleytronics since it has two inequivalent Dirac cones or val- (cid:48) Hamiltonians describing four Dirac cones at the Λ , Λ , Λ leys. A key progress in valleytronics is valley-selective opti- (cid:48) X X Y and Λ points. They have in general Dirac electrons with dif- cal pumping4,5,8–12. By applying circular polarized light in a Y (cid:48) (cid:48) ferent masses m , m , m and m . In Section III, we study gapped Dirac system, we can selectively excite electrons in X X Y Y the spin and psuedospin structures around the X and Y points. one valley based on the property that two valleys have oppo- In Section IV, we analyze the Chern number for each Dirac site chiralities. It is known as the circular dichroism. Valley- ± cone. It is simply given by 1 depending on the sign of the selective pumping has been observed13–18 in the transition- 2 Dirac mass. Since there are four Dirac cones, there arise the metal dichalcogenides such as MoS , where there exists a di- 2 Chern number and three valley-Chern numbers. The Chern rect gap between the conduction and valence bands for Dirac number is a genuine topological number, while valley-Chern fermions. numbers are symmetry-protected topological numbers. When However, the valleytronics is not restricted to the hon- the mass is induced by the strain, the Chern number is zero eycomb system. Recently, the topological crystalline insu- because of the time-reversal symmetry. On the other hand, lator (TCI) attracts much attention due to its experimental when the mass is induced by the exchange effect, the Chern ± realizations19–21 in Pb1−xSnxTe. It is a topological insu- number is 2 per surface. In Section V, we investigate optical lator protected by the mirror symmetry22,23. The remark- absorption and elliptic dichroism by exciting massive Dirac able properties of the TCI is that there emerge four topo- electrons by the right or left elliptically polarized light. We logical protected surface Dirac cones, as has been observed show that the optical absorption is determined by the Chern in the angle-resolved photoelectron spectroscopy (ARPES) number of each Dirac cone and that the elliptic dichroism oc- experiment19–21. The appearance of several topologically pro- curs owing to the anisotropy of a Dirac cone. It is interesting tected Dirac cones enables us to use the TCI as the basic ma- that the elliptic dichroism is observable on the surface of the terial for the valleytronics. Recent experiments24 show that TCI with the Dirac mass being induced by the strain. the band gap can be introduced in the surface Dirac cones by II. HAMILTONIAN crystal distortion by applying strain to the crystal structure. In this paper, we investigate the optical absorption of the Recent ARPES experiments19–21 show that there are four (cid:48) (cid:48) TCI surface. The key properties of surface Dirac cones are Dirac cones at Λ , Λ , Λ and Λ points in the [001] sur- X X Y Y that all of them have the same chirality but that each of them face state of the TCI, whose band structure we show in has a particular anisotropy. Based on the anisotropy, we can Fig.1(a). They may be used as the valley degree of freedom. (cid:48) selectively excite electrons in different valley by the ellipti- Two Dirac cones are present at the Λ and Λ points near X X cally polarized light. This is a new type of dichroism differ- the X point but slightly away from the X point along the x ent from the circular dichroism. We call it an elliptic dichro- axis in the momentum space. The other two Dirac cones are (cid:48) ism. We propose an experimental method to determine the present at the Λ and Λ points near the Y point along the Y Y anisotropy of the velocities and the band gap of Dirac cones y axis. It is notable that the Dirac cones reside at the mirror with the use of elliptic dichroism. Our ﬁnding will open a new symmetry invariant points along the ΓX and ΓY lines rather ##LTLine##

text: Valleytronics

label: title

text: on

text: the

text: Surface

text: of

text: Topological

text: Crystalline

text: Insulator:

text: Elliptic

text: Dichroism

text: and

text: Valley-Selective

text: Optical

text: Pumping

text: Motohiko

label: author

text: Ezawa

text: Department

text: of

text: Applied

text: Physics,

text: University

text: of

text: Tokyo,

text: Hongo

text: Japan

text: The

label: abstract

text: low-energy

text: theory

text: of

text: the

text: surface

text: of

text: the

text: topological

text: crystalline

text: insulator

text: (TCI)

text: is

text: characterized

text: by

text: four

text: Dirac

text: cones

text: anisotropic

text: into

text: the

text: x

text: and

text: y

text: directions.

text: Recent

text: experiments

text: have

text: shown

text: that

text: the

text: band

text: gap

text: can

text: be

text: introduced

text: in

text: these

text: Dirac

text: cones

text: by

text: crystal

text: distortion

text: by

text: applying

text: strain

text: to

text: the

text: crystal

text: structure.

text: The

text: TCI

text: surface

text: provides

text: us

text: with

text: a

text: new

text: way

text: to

text: valleytronics

text: when

text: gaps

text: are

text: given

text: to

text: Dirac

text: cones.

text: Indeed

text: the

text: system

text: has

text: the

text: Chern

text: number

text: and

text: three

text: valley-Chern

text: numbers.

text: We

text: investigate

text: the

text: optical

text: absorption

text: on

text: the

text: TCI

text: surface.

text: It

text: shows

text: a

text: strong

text: elliptic

text: dichroism

text: though

text: the

text: four

text: Dirac

text: cones

text: have

text: the

text: same

text: chiralities.

text: Namely,

text: it

text: is

text: found

text: that

text: the

text: absorptions

text: of

text: the

text: right-

text: and

text: left-polarized

text: light

text: are

text: different,

text: depending

text: on

text: the

text: sign

text: of

text: mass

text: and

text: the

text: location

text: of

text: the

text: Dirac

text: cones,

text: owing

text: to

text: the

text: anisotropy

text: of

text: the

text: Dirac

text: cone.

text: By

text: measuring

text: this

text: elliptic

text: dichroism

text: it

text: is

text: possible

text: to

text: determine

text: the

text: anisotropy

text: of

text: a

text: Dirac

text: cone

text: experimentally.

text: I.

text: INTRODUCTION

text: way

text: of

text: the

text: valleytronics

text: based

text: on

text: the

text: TCI.

text: The

text: present

text: paper

text: is

text: composed

text: as

text: follows.

text: In

text: Section

text: II,

text: we

text: introduce

text: the

text: low-energy

text: Hamiltonians

text: H

text: and

text: H

text: valid

text: Valleytronics

text: is

text: a

text: promising

text: candidate

text: of

text: the

text: next

text: gener-

text: X

text: Y

text: near

text: the

text: X

text: and

text: Y

text: points

text: for

text: the

text: surface,

text: which

text: are

text: ation

text: It

text: is

text: a

text: technology

text: of

text: manipulating

text: the

text: related

text: by

text: the

text: C

text: discrete

text: rotation

text: symmetry.

text: The

text: Hamilto-

text: degree

text: of

text: freedom

text: to

text: which

text: inequivalent

text: degenerate

text: state

text: an

text: nian

text: contains

text: the

text: pseudospin

text: degree

text: of

text: freedom

text: representing

text: electron

text: belongs

text: near

text: the

text: Fermi

text: level.

text: The

text: main

text: target

text: of

text: val-

text: the

text: cation

text: and

text: the

text: anion.

text: The

text: X

text: (Y

text: point

text: is

text: separated

text: into

text: leytronics

text: is

text: the

text: honeycomb

text: lattice

text: system

text: such

text: as

text: graphene.

text: a

text: pair

text: of

text: the

text: Λ

text: and

text: Λ

text: (Λ

text: and

text: Λ

text: points

text: due

text: to

text: the

text: Indeed,

text: the

text: honeycomb

text: structure

text: is

text: an

text: ideal

text: play

text: ground

text: of

text: X

text: X

text: Y

text: Y

text: spin-pseudospin

text: mixing.

text: We

text: then

text: derive

text: the

text: four

text: low-energy

text: valleytronics

text: since

text: it

text: has

text: two

text: inequivalent

text: Dirac

text: cones

text: or

text: val-

text: Hamiltonians

text: describing

text: four

text: Dirac

text: cones

text: at

text: the

text: Λ

text: Λ

text: Λ

text: leys.

text: A

text: key

text: progress

text: in

text: valleytronics

text: is

text: valley-selective

text: opti-

text: X

text: X

text: Y

text: and

text: Λ

text: points.

text: They

text: have

text: in

text: general

text: Dirac

text: electrons

text: with

text: dif-

text: cal

text: By

text: applying

text: circular

text: polarized

text: light

text: in

text: a

text: Y

text: ferent

text: masses

text: m

text: m

text: m

text: and

text: m

text: In

text: Section

text: III,

text: we

text: study

text: gapped

text: Dirac

text: system,

text: we

text: can

text: selectively

text: excite

text: electrons

text: in

text: X

text: X

text: Y

text: Y

text: the

text: spin

text: and

text: psuedospin

text: structures

text: around

text: the

text: X

text: and

text: Y

text: points.

text: one

text: valley

text: based

text: on

text: the

text: property

text: that

text: two

text: valleys

text: have

text: oppo-

text: In

text: Section

text: IV,

text: we

text: analyze

text: the

text: Chern

text: number

text: for

text: each

text: Dirac

text: site

text: chiralities.

text: It

text: is

text: known

text: as

text: the

text: circular

text: dichroism.

text: Valley-

text: cone.

text: It

text: is

text: simply

text: given

text: by

text: depending

text: on

text: the

text: sign

text: of

text: the

text: selective

text: pumping

text: has

text: been

text: in

text: the

text: transition-

text: Dirac

text: mass.

text: Since

text: there

text: are

text: four

text: Dirac

text: cones,

text: there

text: arise

text: the

text: metal

text: dichalcogenides

text: such

text: as

text: MoS

text: where

text: there

text: exists

text: a

text: di-

text: Chern

text: number

text: and

text: three

text: valley-Chern

text: numbers.

text: The

text: Chern

text: rect

text: gap

text: between

text: the

text: conduction

text: and

text: valence

text: bands

text: for

text: Dirac

text: number

text: is

text: a

text: genuine

text: topological

text: number,

text: while

text: valley-Chern

text: fermions.

text: numbers

text: are

text: symmetry-protected

text: topological

text: numbers.

text: When

text: However,

text: the

text: valleytronics

text: is

text: not

text: restricted

text: to

text: the

text: hon-

text: the

text: mass

text: is

text: induced

text: by

text: the

text: strain,

text: the

text: Chern

text: number

text: is

text: zero

text: eycomb

text: system.

text: Recently,

text: the

text: topological

text: crystalline

text: insu-

text: because

text: of

text: the

text: time-reversal

text: symmetry.

text: On

text: the

text: other

text: hand,

text: lator

text: (TCI)

text: attracts

text: much

text: attention

text: due

text: to

text: its

text: experimental

text: when

text: the

text: mass

text: is

text: induced

text: by

text: the

text: exchange

text: effect,

text: the

text: Chern

text: in

text: It

text: is

text: a

text: topological

text: insu-

text: number

text: is

text: per

text: surface.

text: In

text: Section

text: V,

text: we

text: investigate

text: optical

text: lator

text: protected

text: by

text: the

text: mirror

text: The

text: remark-

text: absorption

text: and

text: elliptic

text: dichroism

text: by

text: exciting

text: massive

text: Dirac

text: able

text: properties

text: of

text: the

text: TCI

text: is

text: that

text: there

text: emerge

text: four

text: topo-

text: electrons

text: by

text: the

text: right

text: or

text: left

text: elliptically

text: polarized

text: light.

text: We

text: logical

text: protected

text: surface

text: Dirac

text: cones,

text: as

text: has

text: been

text: observed

text: show

text: that

text: the

text: optical

text: absorption

text: is

text: determined

text: by

text: the

text: Chern

text: in

text: the

text: angle-resolved

text: photoelectron

text: spectroscopy

text: (ARPES)

text: number

text: of

text: each

text: Dirac

text: cone

text: and

text: that

text: the

text: elliptic

text: dichroism

text: oc-

text: The

text: appearance

text: of

text: several

text: topologically

text: pro-

text: curs

text: owing

text: to

text: the

text: anisotropy

text: of

text: a

text: Dirac

text: cone.

text: It

text: is

text: interesting

text: tected

text: Dirac

text: cones

text: enables

text: us

text: to

text: use

text: the

text: TCI

text: as

text: the

text: basic

text: ma-

text: that

text: the

text: elliptic

text: dichroism

text: is

text: observable

text: on

text: the

text: surface

text: of

text: the

text: terial

text: for

text: the

text: valleytronics.

text: Recent

text: show

text: that

text: TCI

text: with

text: the

text: Dirac

text: mass

text: being

text: induced

text: by

text: the

text: strain.

text: the

text: band

text: gap

text: can

text: be

text: introduced

text: in

text: the

text: surface

text: Dirac

text: cones

text: by

text: II.

text: HAMILTONIAN

text: crystal

text: distortion

text: by

text: applying

text: strain

text: to

text: the

text: crystal

text: structure.

text: In

text: this

text: paper,

text: we

text: investigate

text: the

text: optical

text: absorption

text: of

text: the

text: Recent

text: ARPES

text: show

text: that

text: there

text: are

text: four

text: TCI

text: surface.

text: The

text: key

text: properties

text: of

text: surface

text: Dirac

text: cones

text: are

text: Dirac

text: cones

text: at

text: Λ

text: Λ

text: Λ

text: and

text: Λ

text: points

text: in

text: the

text: sur-

text: X

text: X

text: Y

text: Y

text: that

text: all

text: of

text: them

text: have

text: the

text: same

text: chirality

text: but

text: that

text: each

text: of

text: them

text: face

text: state

text: of

text: the

text: TCI,

text: whose

text: band

text: structure

text: we

text: show

text: in

text: has

text: a

text: particular

text: anisotropy.

text: Based

text: on

text: the

text: anisotropy,

text: we

text: can

text: They

text: may

text: be

text: used

text: as

text: the

text: valley

text: degree

text: of

text: freedom.

text: selectively

text: excite

text: electrons

text: in

text: different

text: valley

text: by

text: the

text: ellipti-

text: Two

text: Dirac

text: cones

text: are

text: present

text: at

text: the

text: Λ

text: and

text: Λ

text: points

text: near

text: X

text: X

text: cally

text: polarized

text: light.

text: This

text: is

text: a

text: new

text: type

text: of

text: dichroism

text: differ-

text: the

text: X

text: point

text: but

text: slightly

text: away

text: from

text: the

text: X

text: point

text: along

text: the

text: x

text: ent

text: from

text: the

text: circular

text: dichroism.

text: We

text: call

text: it

text: an

text: elliptic

text: dichro-

text: axis

text: in

text: the

text: momentum

text: space.

text: The

text: other

text: two

text: Dirac

text: cones

text: are

text: ism.

text: We

text: propose

text: an

text: experimental

text: method

text: to

text: determine

text: the

text: present

text: at

text: the

text: Λ

text: and

text: Λ

text: points

text: near

text: the

text: Y

text: point

text: along

text: the

text: Y

text: Y

text: anisotropy

text: of

text: the

text: velocities

text: and

text: the

text: band

text: gap

text: of

text: Dirac

text: cones

text: y

text: axis.

text: It

text: is

text: notable

text: that

text: the

text: Dirac

text: cones

text: reside

text: at

text: the

text: mirror

text: with

text: the

text: use

text: of

text: elliptic

text: dichroism.

text: Our

text: ﬁnding

text: will

text: open

text: a

text: new

text: symmetry

text: invariant

text: points

text: along

text: the

text: ΓX

text: and

text: ΓY

text: lines

text: rather

text: ##LTLine##


text: 14 L. ANDERSSON AND T.A. OLIYNYK is the Euclidean Laplacian. (cid:0) (cid:1) Lemma 3.3. The following estimates(cid:0)hold: (cid:1) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) q0 H − T b E T + Db E − T uλ T , k 0,s+1 (k+2)( n) s( n) s 1( n) Es+1,k( n) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) q1 H − T b E T + Db E − T uλ H − T k 0,s+1 (k+2)( n) s( n) s 1( n) k+1 1,s+1 (k+1)( n) and (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) q2 H − T b E T uλ H − T k 0,s+1 (k+2)( n) s( n) k+2 0,s+1 (k+2)( n) ≤ ≤ − for 0 k s 1. Proof. To begin, we observe that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) ≤ ≤ ≤ − J (bij− )∂ ∂ uλ H − T b − H − − T D2uλ H − − T 0 (cid:96) k s 1 λ k (cid:96) i j (cid:96) 0,s+1 (k+2)( n) k (cid:96) 0,s (k (cid:96))( n) (cid:96) 0,s 1 (cid:96)( n) follows directly from Proposition A.7 since s > n/2. Similar arguments show that the estimates (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) J (∂ bij− )∂ uλ H − T Db − H − − − T Duλ H − T λ i k (cid:96) j (cid:96) 0,s+1 (k+2)( n) k (cid:96) 0,s 1 (k (cid:96))( n) (cid:96) 0,s (cid:96)( n) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) J (b0j− )∂ uλ H − T b − H − − T Duλ H − T λ k (cid:96)+1 j (cid:96) 0,s+1 (k+2)( n) k (cid:96)+1 0,s (k (cid:96)+1)( n) (cid:96) 0,s (cid:96)( n) (cid:13) (cid:18) (cid:19) (cid:13) (cid:13)(cid:88)≤ ≤(cid:0) ≤ − (cid:1)(cid:13) hold fo(cid:13)r 0 (cid:96) k s 1. These inequalities give us the following es(cid:13)timate for the ﬁrst sum in q0: (cid:13) (cid:13) k k k J (bij− )∂ ∂ uλ + J (∂ bij− )∂ uλ+J ((cid:0)b0j− )∂ uλ (cid:1) (cid:96) λ k (cid:96) i j (cid:96) λ i k (cid:96) j (cid:96) λ k (cid:96)+1 j (cid:96) H − T (cid:96)=0 0,s+1 (k+2)( n) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) b E T + Db E − T uλ T s( n) s 1( n) Es+1,k( n) ≤ ≤ − (cid:3) for 0 k s 1. The required estimates for the rest of q0 and q1, q2 follow from similar arguments. k k k ≤ ≤ − We collect the equations (3.43) (0 k s 1) into a single system (cid:0) L (cid:1)(uλ− ) = K (cid:0)− , (cid:1) (3.44) (cid:15) s 1 s 1 where K − = f − + J χ h − + µ − + (cid:15) 0, . . . , 0, q2− , q1− + q2− Tr s 1 s 1 λ Ω1 s 1 s 1 s 2 s 1 s 1 and  −  L = L (cid:15)L (cid:15) 0 1   with  − − · · ·  ∆ ψ 0 1 0 0   − −   0 ∆ ψ 0 1 0     0 0 ∆ − ψ 0 ... ...    L0 =  0 0 0 ∆ − ψ ... −1  ... ... 0 · · · − 0 0 0 0 ∆ ψ and L (u − ) = q0− + (q1 + q2, . . . , q1− + q2− , q1− , 0)Tr. 1 s 1 s 1 0 0 s 3 s 3 s 1 Next, we observe that Lemma 3.3 shows that the operator norm of − −→ X − − L : Xs+1,s 1 s 1,s 1 1

text: L.

text: ANDERSSON

text: AND

text: T.A.

text: OLIYNYK

text: is

text: the

text: Euclidean

text: Laplacian.

text: Lemma

text: The

text: following

text: H

text: T

text: b

text: E

text: T

text: Db

text: E

text: T

text: uλ

text: T

text: k

text: n)

text: s(

text: n)

text: s

text: n)

text: n)

text: H

text: T

text: b

text: E

text: T

text: Db

text: E

text: T

text: uλ

text: H

text: T

text: k

text: n)

text: s(

text: n)

text: s

text: n)

text: n)

text: and

text: H

text: T

text: b

text: E

text: T

text: uλ

text: H

text: T

text: k

text: n)

text: s(

text: n)

text: n)

text: for

text: k

text: s

text: Proof.

text: To

text: begin,

text: we

text: observe

text: that

text: J

text: (bij−

text: uλ

text: H

text: T

text: b

text: H

text: T

text: H

text: T

text: k

text: s

text: λ

text: k

text: i

text: j

text: n)

text: k

text: (k

text: n)

text: n)

text: follows

text: directly

text: from

text: Proposition

text: since

text: s

text: Similar

text: arguments

text: show

text: that

text: the

text: estimates

text: J

text: bij−

text: uλ

text: H

text: T

text: Db

text: H

text: T

text: Duλ

text: H

text: T

text: λ

text: i

text: k

text: j

text: n)

text: k

text: (k

text: n)

text: n)

text: J

text: uλ

text: H

text: T

text: b

text: H

text: T

text: Duλ

text: H

text: T

text: λ

text: k

text: j

text: n)

text: k

text: (k

text: n)

text: n)

text: hold

text: k

text: s

text: These

text: inequalities

text: give

text: us

text: the

text: following

text: for

text: the

text: ﬁrst

text: sum

text: in

text: k

text: k

text: k

text: J

text: (bij−

text: uλ

text: J

text: bij−

text: uλ+J

text: uλ

text: λ

text: k

text: i

text: j

text: λ

text: i

text: k

text: j

text: λ

text: k

text: j

text: H

text: T

text: n)

text: b

text: E

text: T

text: Db

text: E

text: T

text: uλ

text: T

text: s(

text: n)

text: s

text: n)

text: n)

text: for

text: k

text: s

text: The

text: required

text: estimates

text: for

text: the

text: rest

text: of

text: and

text: follow

text: from

text: similar

text: arguments.

text: k

text: k

text: k

text: We

text: collect

text: the

text: equations

text: k

text: s

text: into

text: a

text: single

text: system

text: L

text: K

text: s

text: s

text: where

text: K

text: f

text: J

text: χ

text: h

text: µ

text: Tr

text: s

text: s

text: λ

text: Ω1

text: s

text: s

text: s

text: s

text: s

text: and

text: L

text: L

text: with

text: ψ

text: ψ

text: ψ

text: ψ

text: −1

text: ψ

text: and

text: L

text: (u

text: s

text: s

text: s

text: s

text: s

text: Next,

text: we

text: observe

text: that

text: Lemma

text: shows

text: that

text: the

text: operator

text: norm

text: of

text: X

text: L

text: s


text: 16 L. ANDERSSON AND T.A. OLIYNYK Hyperbolic es(cid:0)t(cid:0)imates: By ass(cid:1)umpti(cid:1)on ∂sµ = 0. T(cid:0)herefore, diﬀerent(cid:1)iating (3.38) s-times with respect t to t yields − − ∂ mµν + (cid:15)J (bµν) ∂ u(cid:18)λ (cid:19) ψuλ = (cid:15)∂ pµ sJ(cid:18)(b(cid:19)µ0)uλ + J f + J (χ h ), (3.50) µ λ (cid:88)ν s s µ (cid:88) λ 1 s λ s λ Ω1 s where − − − s 1 s − s 2 s pµ = J (bµ−j )∂ uλ J (bµ−0 )uλ . (cid:96) λ s (cid:96) j (cid:96) (cid:96) λ s (cid:96) (cid:96)+1 (cid:96)=0 (cid:96)=0 Recalling that s > n/2, it follows directly from Sobolev’s inequality, see Theorem A.2, and Corollary 2.2 that the inequalities (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) J bµν(t) ∞ T J bµν(t) H T b(t) E T , (3.51) λ L ( n) λ 0,s( n) s( n) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∂ J bµν(t) ∞ T J bµν(t) H T b(t) E T , (3.52) t λ L ( n) λ 1 0,s( n) s+1( n) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∂ J bµ0(t) T J bµ0(t) H − T ∂ b(t) E T (3.53) t λ 1 Ln( n) λ 2 0,s 1( n) t s( n) and (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) J bµ0(t) ∞ T J bµ0(t) H T ∂ b(t) E T (3.54) λ 1 L ( n) λ 1 0,s( n) t s( n) ≤ are satisﬁed for 0 t < T. Also, a slight adaptation of the arguments used to prove Lemma 3.3 show that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) pµ(t) T b(t) E T uλ(t) − T (3.55) L2( n) s( n) Es+1,s 1( n) (cid:0) (cid:1)(cid:0) (cid:1) and (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ∂ pµ(t) T b(t) E T + ∂ b(t) E T uλ(t) − T + uλ(t) T A(t) E R t L2( n) s( n) t s( n) Es+1,s 1( n) s H1( n) s+1( n) (3.56) ≤ hold for 0 t < T. We now are in a position to apply the energy estimates from Theorem C.2 to uλ since it solves the s wave equation (3.50). Doing so, w(cid:104)e see that these energy(cid:90)estimat(cid:16)es in conjunction with the inequalities (3.47), (3.51)-(3.56) and Corollary 2.2 show that uλ satisﬁes s (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:17) t (cid:107) (cid:107) (cid:105) uλ(t) T C(c ) β(0) uλ(0) T + β(τ) uλ(τ) − s E( n) L Es+1( n) Es+1,s 1 0 (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) + uλ(τ) T + f (τ) T + h (τ) dτ E( n) s L2( n) s L2(Ω1) ≤ for 0 t < T, where (cid:107) (cid:107) (cid:107) (cid:107) (cid:20) β(t) = 1 + b(t) E T + ∂ b(t) E T . s( n) t s( n) Combining this estimate with (3.48) gives (cid:107) (cid:90) (cid:107) (cid:16)≤ (cid:107) (cid:107) (cid:17) (cid:21) uλ(t) T C(c ) β(0) uλ(0) T s E( n) L Es+1( n) t (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) + β(τ) uλ(τ) T + f(τ) E T + h(τ) + µ(τ) E − T dτ (3.57) E( n) s( n) Es(Ω1) s 1( n) 0 ≤ for 0 t < T. (cid:104) Together, the estimates (3.49) and (3.57) show that uλ satisﬁes the uniform bound (cid:107) (cid:107) ≤ (cid:90) (cid:16) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:17) (cid:105) uλ(t) T C(c ) β(0) uλ(0) T + f(0) E − T + h(0) − Es+1( n) L Es+1( n) s 1( n) Es 1(Ω1) (cid:107) (cid:107) t (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) + µ(t) E − T + β(τ) uλ(τ) T + f(τ) E T + h(τ) + µ(τ) E − T dτ s 1( n) Es+1( n) s( n) Es(Ω1) s 1( n) 0

text: L.

text: ANDERSSON

text: AND

text: T.A.

text: OLIYNYK

text: Hyperbolic

text: By

text: ∂sµ

text: s-times

text: with

text: respect

text: t

text: to

text: t

text: yields

text: mµν

text: (bµν)

text: ψuλ

text: pµ

text: J

text: f

text: J

text: (χ

text: h

text: µ

text: λ

text: s

text: s

text: µ

text: λ

text: s

text: λ

text: s

text: λ

text: Ω1

text: s

text: where

text: s

text: s

text: s

text: s

text: pµ

text: J

text: (bµ−j

text: uλ

text: J

text: )uλ

text: λ

text: s

text: j

text: λ

text: s

text: Recalling

text: that

text: s

text: it

text: follows

text: directly

text: from

text: Sobolev’s

text: inequality,

text: see

text: Theorem

text: and

text: Corollary

text: that

text: the

text: inequalities

text: J

text: bµν(t)

text: T

text: J

text: bµν(t)

text: H

text: T

text: b(t)

text: E

text: T

text: λ

text: L

text: n)

text: λ

text: n)

text: s(

text: n)

text: J

text: bµν(t)

text: T

text: J

text: bµν(t)

text: H

text: T

text: b(t)

text: E

text: T

text: t

text: λ

text: L

text: n)

text: λ

text: n)

text: n)

text: J

text: T

text: J

text: H

text: T

text: b(t)

text: E

text: T

text: t

text: λ

text: Ln(

text: n)

text: λ

text: n)

text: t

text: s(

text: n)

text: and

text: J

text: T

text: J

text: H

text: T

text: b(t)

text: E

text: T

text: λ

text: L

text: n)

text: λ

text: n)

text: t

text: s(

text: n)

text: are

text: satisﬁed

text: for

text: t

text: T.

text: Also,

text: a

text: slight

text: adaptation

text: of

text: the

text: arguments

text: used

text: to

text: prove

text: Lemma

text: show

text: that

text: pµ(t)

text: T

text: b(t)

text: E

text: T

text: uλ(t)

text: T

text: n)

text: s(

text: n)

text: n)

text: and

text: pµ(t)

text: T

text: b(t)

text: E

text: T

text: b(t)

text: E

text: T

text: uλ(t)

text: T

text: uλ(t)

text: T

text: A(t)

text: E

text: R

text: t

text: n)

text: s(

text: n)

text: t

text: s(

text: n)

text: n)

text: s

text: n)

text: n)

text: hold

text: for

text: t

text: T.

text: We

text: now

text: are

text: in

text: a

text: position

text: to

text: apply

text: the

text: energy

text: estimates

text: from

text: Theorem

text: to

text: uλ

text: since

text: it

text: solves

text: the

text: s

text: wave

text: equation

text: Doing

text: so,

text: see

text: that

text: these

text: in

text: conjunction

text: with

text: the

text: inequalities

text: and

text: Corollary

text: show

text: that

text: uλ

text: satisﬁes

text: s

text: t

text: uλ(t)

text: T

text: C(c

text: T

text: β(τ)

text: uλ(τ)

text: s

text: E(

text: n)

text: L

text: n)

text: uλ(τ)

text: T

text: f

text: (τ)

text: T

text: h

text: (τ)

text: dτ

text: E(

text: n)

text: s

text: n)

text: s

text: for

text: t

text: T,

text: where

text: β(t)

text: b(t)

text: E

text: T

text: b(t)

text: E

text: T

text: s(

text: n)

text: t

text: s(

text: n)

text: Combining

text: this

text: estimate

text: with

text: gives

text: uλ(t)

text: T

text: C(c

text: T

text: s

text: E(

text: n)

text: L

text: n)

text: t

text: β(τ)

text: uλ(τ)

text: T

text: f(τ)

text: E

text: T

text: h(τ)

text: µ(τ)

text: E

text: T

text: dτ

text: E(

text: n)

text: s(

text: n)

text: s

text: n)

text: for

text: t

text: T.

text: Together,

text: the

text: estimates

text: and

text: show

text: that

text: uλ

text: satisﬁes

text: the

text: uniform

text: bound

text: uλ(t)

text: T

text: C(c

text: T

text: E

text: T

text: n)

text: L

text: n)

text: s

text: n)

text: Es

text: t

text: µ(t)

text: E

text: T

text: β(τ)

text: uλ(τ)

text: T

text: f(τ)

text: E

text: T

text: h(τ)

text: µ(τ)

text: E

text: T

text: dτ

text: s

text: n)

text: n)

text: s(

text: n)

text: s

text: n)


text: A TRANSMISSION PROBLEM FOR QUASI-LINEAR WAVE EQUATIONS 17 (cid:20) which, in turn, implies via Gronw(cid:82)all’s inequality that (cid:107)uλ(t)(cid:107)Es+1(Tn) ≤ C(cL)eC(cL) 0T(cid:90)β(τ)dτ β(cid:16)(0)(cid:107)uλ(0)(cid:107)Es+1(Tn) + (cid:107)f(0)(cid:107)Es−1(Tn) + (cid:107)h(0)(cid:107)E(cid:17)s−1((cid:21)Ω1) (cid:107) (cid:107) T (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) + µ(t) E − T + β(τ) f(τ) E T + h(τ) + µ(τ) E − T dτ (3.58) s 1( n) s( n) Es(Ω1) s 1( n) 0 ≤ for 0 t < T. This inequality implies that uλ is a bounded 1-parameter family of solutions in T Xs+1( n) to the IVP (3.41)-(3.42), and consequently, we know, by standard arguments, that there T (cid:38) exists a weakly convergent subsequence, again denoted uλ, that converges as λ 0 to the unique weak ∈ T solution u Xs+1( n) of (3.38)-(3.39). M(cid:20)oreover, by the uniqueness of weak limits, we see, by sending (cid:38) T λ 0 in the estimate (3.58), tha(cid:82)t u satisﬁes (cid:107)u(t)(cid:107)Es+1(Tn) ≤ C(cL)eC(cL) 0T(cid:90)β(τ)dτ β(cid:16)(0)(cid:107)u(0)(cid:107)Es+1(Tn) + (cid:107)f(0)(cid:107)Es−1(Tn) + (cid:107)h(0)(cid:107)E(cid:17)s−1(Ω(cid:21)1) (cid:107) (cid:107) T (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) + µ(t) E − T + β(τ) f(τ) E T + h(τ) + µ(τ) E − T dτ (3.59) s 1( n) s( n) Es(Ω1) s 1( n) 0 ≤ − for 0 t < T. Finally, a straightforward calculation shows that the diﬀerence u u satisﬁes a linear λ equation of the type (3.38)-(3.39), and hence, also an estimate of the type (3.59) from which it follows that (cid:107) − (cid:107) (cid:46) (cid:107) − (cid:107) u u T u(0) J u(0) T + c(λ) λ Xs+1( n) λ Es+1( n) T ∈ T for some constant c(λ) satisfying lim (cid:38) c(λ) = 0. Since u CXs+1( n), this estimate implies that λ 0 λ ∈T T (cid:3) u (t) converges uniformly on [0, T] to u(t), thereby showing that u CXs+1( n). λ T With the existence and uniqueness for the system (3.38)-(3.39) established, it is now straightforward, using the ﬁnite speed of propagation, to conclude an analogous uniqueness and existence result for the original system (3.1)-(3.2). ≥ ∈ Z ∈ X R ∈ X − R Theorem 3.4. Suppose n 3, s , T > 0, A = (Aµν), ∂ A s ( n), DA s 1( n), F ∈ Xs (Rn), H ∈(cid:0)Y s(Ω), (U˜ , U˜ ) ∈ H2>,sn+/12(Rn) ×(cid:1) H2,s(Rn) satisfy tthe comTpatibility conditioTns (3.4), T T 0 1 and let (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) α = sup A(t) E R + DA(t) E − R , β(t) = 1 + A(t) E R + ∂ A(t) E R ≤ s( n) s 1( n) s( n) t s( n) 0 t<T (cid:90) and T γ = β(τ) dτ. (cid:20) 0 ∈ R Then the IVP (3.1)-(3.2) has a unique solution U CXs+1( n) and satisﬁes the estimate T (cid:107) (cid:107) ≤ (cid:107) (cid:90) (cid:107) (cid:107) (cid:107) (cid:21) U(t) R c(α, β(0), γ) U(0) R(cid:0) + F(0) E − R (cid:1) Es+1( n) Es+1( n) s 1( n) (cid:107) (cid:107) T (cid:107) (cid:107) (cid:107) (cid:107) + H(0) − + β(τ) F(τ) E R + H(τ) dτ Es 1(Ω) s( n) Es(Ω) 0 ≤ for 0 t < T.

text: A

text: TRANSMISSION

text: PROBLEM

text: FOR

text: QUASI-LINEAR

text: WAVE

text: EQUATIONS

text: which,

text: in

text: turn,

text: implies

text: via

text: inequality

text: that

text: C(cL)eC(cL)

text: T

text: µ(t)

text: E

text: T

text: β(τ)

text: f(τ)

text: E

text: T

text: h(τ)

text: µ(τ)

text: E

text: T

text: dτ

text: s

text: n)

text: s(

text: n)

text: s

text: n)

text: for

text: t

text: T.

text: This

text: inequality

text: implies

text: that

text: uλ

text: is

text: a

text: bounded

text: family

text: of

text: solutions

text: in

text: T

text: n)

text: to

text: the

text: IVP

text: and

text: consequently,

text: we

text: know,

text: by

text: standard

text: arguments,

text: that

text: there

text: T

text: exists

text: a

text: weakly

text: convergent

text: subsequence,

text: again

text: denoted

text: uλ,

text: that

text: converges

text: as

text: λ

text: to

text: the

text: unique

text: weak

text: T

text: solution

text: u

text: n)

text: of

text: by

text: the

text: uniqueness

text: of

text: weak

text: limits,

text: we

text: see,

text: by

text: sending

text: T

text: λ

text: in

text: the

text: estimate

text: u

text: satisﬁes

text: C(cL)eC(cL)

text: T

text: µ(t)

text: E

text: T

text: β(τ)

text: f(τ)

text: E

text: T

text: h(τ)

text: µ(τ)

text: E

text: T

text: dτ

text: s

text: n)

text: s(

text: n)

text: s

text: n)

text: for

text: t

text: T.

text: Finally,

text: a

text: straightforward

text: calculation

text: shows

text: that

text: the

text: diﬀerence

text: u

text: u

text: satisﬁes

text: a

text: linear

text: λ

text: equation

text: of

text: the

text: type

text: and

text: hence,

text: also

text: an

text: estimate

text: of

text: the

text: type

text: from

text: which

text: it

text: follows

text: that

text: u

text: u

text: T

text: J

text: T

text: c(λ)

text: λ

text: n)

text: λ

text: n)

text: T

text: T

text: for

text: some

text: constant

text: c(λ)

text: satisfying

text: lim

text: c(λ)

text: Since

text: u

text: n),

text: this

text: estimate

text: implies

text: that

text: λ

text: λ

text: ∈T

text: T

text: u

text: (t)

text: converges

text: uniformly

text: on

text: T]

text: to

text: u(t),

text: thereby

text: showing

text: that

text: u

text: n).

text: λ

text: T

text: With

text: the

text: existence

text: and

text: uniqueness

text: for

text: the

text: system

text: established,

text: it

text: is

text: now

text: straightforward,

text: using

text: the

text: ﬁnite

text: speed

text: of

text: propagation,

text: to

text: conclude

text: an

text: analogous

text: uniqueness

text: and

text: existence

text: result

text: for

text: the

text: original

text: system

text: Z

text: X

text: R

text: X

text: R

text: Theorem

text: Suppose

text: n

text: s

text: T

text: A

text: (Aµν),

text: A

text: s

text: n),

text: DA

text: s

text: n),

text: F

text: Xs

text: (Rn),

text: H

text: s(Ω),

text: (U˜

text: U˜

text: satisfy

text: tthe

text: comTpatibility

text: conditioTns

text: T

text: T

text: and

text: let

text: α

text: sup

text: A(t)

text: E

text: R

text: DA(t)

text: E

text: R

text: β(t)

text: A(t)

text: E

text: R

text: A(t)

text: E

text: R

text: s(

text: n)

text: s

text: n)

text: s(

text: n)

text: t

text: s(

text: n)

text: t<T

text: and

text: T

text: γ

text: β(τ)

text: dτ.

text: R

text: Then

text: the

text: IVP

text: has

text: a

text: unique

text: solution

text: U

text: n)

text: and

text: satisﬁes

text: the

text: estimate

text: T

text: U(t)

text: R

text: c(α,

text: γ)

text: E

text: R

text: n)

text: n)

text: s

text: n)

text: T

text: β(τ)

text: F(τ)

text: E

text: R

text: H(τ)

text: dτ

text: Es

text: s(

text: n)

text: Es(Ω)

text: for

text: t

text: T.


text: 18 L. ANDERSSON AND T.A. OLIYNYK Proof. We begin by letting u ∈ CXs+1(Tn) be the u(cid:26)nique solu(cid:27)tion to the IVP (3.24)-(3.25) from Theorem ∈ T 3.2 with δ (0, 1] chosen small enough so that (cid:15) = δσ satisﬁes 1 1 (cid:15) = min , (3.60) (cid:0) (cid:1) 3c R 3 L with (cid:107) (cid:107) (cid:107) (cid:107) R = sup b(t) E T + Db(t) E − T . (3.61) ≤ s( n) s 1( n) 0 t<T ∈ | | We know from Proposition 3.1(cid:0) that there exists an η (cid:1) (0(cid:0), 1/4] such that(cid:1)the initial data (u , ∂ u ), 0 t=0 t t=0 the source term µ(t) and the coeﬃcient ψ satisfy | | | | u { }× , ∂ u { }× = u˜ , u˜ , 0 Qη0 t 0 Q| η0 0 Qη0 1 Qη0 ψ = 0 Qη0 and | µ × = 0, [0,T] Qη0 respectively. Appealing to the ﬁnite propagation speed for solutions of wave equations, we see, shrinking × T if necessary, that u solves the IVP(cid:34)(3.22)-(3.23) on the spacetime region [0, T) Q . Moreover, it η0/2 follows directly from Theorem 3.2 and Propositions 3.1, D.1 and D.2 that u satisﬁes the estimate (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:35) u(t) Es+1(Qη0/2) c(α¯, β¯(0), γ¯) U(0) Es+(cid:90)1(Q1) + δ(cid:0) F¯(0) Es−1(Q1) (cid:1) (cid:107) (cid:107) δT (cid:107) (cid:107) (cid:107) (cid:107) + δ H¯ (0) Es−1(Q(cid:0)+1 ) + 0 β¯(τ) F¯(τ) Es(Q1) + (cid:1)H¯ (τ) Es(Q+1 ) dτ (3.62) ≤ for 0 t < T, where (cid:107) (cid:107) (cid:107) (cid:107) α¯ = sup A¯(t) E + DA¯(t) E − , (3.63) ≤ s(Q1) s 1(Q1) 0 (cid:90)t<δT (cid:107) (cid:107) (cid:107) (cid:107) β¯(t) = 1 + A¯(t) E + ∂ A¯(t) E , s(Q1) t s(Q1) 1 δT γ¯ = β¯(τ) dτ δ 0 and U¯,A¯,F¯ and H¯ are as deﬁned previously by (3.17)-(3.21). Next, a simple change of variable argument shows that (cid:107) (cid:107) ≤ (cid:107) (cid:107) U¯(t) c(1/δ) u(t/δ) ≤ Es+1(Qδη0/2) Es+1(Qη0/2) for 0 t < δT, while the inequality 1 (cid:46) 1 + α¯1/σ δ (cid:20) follows from (3.60), (3.63) and Proposition D.1. Using these estimates together with (3.62), we see that (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:90) (cid:107) (cid:107) (cid:21) U¯(t) c(α¯, β¯(0), γ¯) U¯(0) + (cid:0)F¯(0) E − (cid:1) Es+1(Qδη0/2) Es+1(Q1) s 1(Q1) ∗ (cid:107) (cid:107) T (cid:107) (cid:107) (cid:107) (cid:107) + H¯ (0) Es−1(Q+1 ) + 0 β¯(τ) F¯(τ) Es(Q1) + H¯ (τ) Es(Q+1 ) dτ (3.64) ≤ ∗ ∗ for 0 t < T , where T = δT. × Since u solves the IVP (3.22)-(3.23) on the spacetime region [0, T) Q , U¯ must solve the ∗ × η0/2 IVP (3.14)-(3.15) on [0, T ) Q . Recalling the deﬁnitions (3.8)-(3.12), it is clear that U(t, x) = δη/2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L.

text: ANDERSSON

text: AND

text: T.A.

text: OLIYNYK

text: Proof.

text: We

text: begin

text: by

text: letting

text: u

text: be

text: the

text: to

text: the

text: IVP

text: from

text: Theorem

text: T

text: with

text: δ

text: chosen

text: small

text: enough

text: so

text: that

text: δσ

text: satisﬁes

text: min

text: R

text: L

text: with

text: R

text: sup

text: b(t)

text: E

text: T

text: Db(t)

text: E

text: T

text: s(

text: n)

text: s

text: n)

text: t<T

text: We

text: know

text: from

text: Proposition

text: that

text: there

text: exists

text: an

text: η

text: such

text: initial

text: data

text: (u

text: u

text: t

text: the

text: source

text: term

text: µ(t)

text: and

text: the

text: coeﬃcient

text: ψ

text: satisfy

text: u

text: u

text: u˜

text: u˜

text: t

text: Q|

text: η0

text: ψ

text: and

text: µ

text: respectively.

text: Appealing

text: to

text: the

text: ﬁnite

text: propagation

text: speed

text: for

text: solutions

text: of

text: wave

text: equations,

text: we

text: see,

text: shrinking

text: T

text: if

text: necessary,

text: that

text: u

text: solves

text: the

text: on

text: the

text: spacetime

text: region

text: T)

text: Q

text: Moreover,

text: it

text: follows

text: directly

text: from

text: Theorem

text: and

text: Propositions

text: and

text: that

text: u

text: satisﬁes

text: the

text: estimate

text: u(t)

text: c(α¯,

text: γ¯)

text: δT

text: δ

text: H¯

text: β¯(τ)

text: F¯(τ)

text: (τ)

text: dτ

text: for

text: t

text: T,

text: where

text: α¯

text: sup

text: A¯(t)

text: E

text: DA¯(t)

text: E

text: s

text: β¯(t)

text: A¯(t)

text: E

text: A¯(t)

text: E

text: t

text: δT

text: γ¯

text: β¯(τ)

text: dτ

text: δ

text: and

text: U¯,A¯,F¯

text: and

text: H¯

text: are

text: as

text: deﬁned

text: previously

text: by

text: Next,

text: a

text: simple

text: change

text: of

text: variable

text: argument

text: shows

text: that

text: U¯(t)

text: u(t/δ)

text: for

text: t

text: δT,

text: while

text: the

text: inequality

text: α¯1/σ

text: δ

text: follows

text: from

text: and

text: Proposition

text: Using

text: these

text: estimates

text: together

text: with

text: we

text: see

text: that

text: U¯(t)

text: c(α¯,

text: γ¯)

text: E

text: s

text: T

text: H¯

text: β¯(τ)

text: F¯(τ)

text: H¯

text: (τ)

text: dτ

text: for

text: t

text: T

text: where

text: T

text: δT.

text: Since

text: u

text: solves

text: the

text: IVP

text: on

text: the

text: spacetime

text: region

text: T)

text: Q

text: U¯

text: must

text: solve

text: the

text: IVP

text: on

text: T

text: Q

text: Recalling

text: the

text: deﬁnitions

text: it

text: is

text: clear

text: that

text: U(t,

text: x)

text: δη/2

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 L. ANDERSSON AND T.A. OLIYNYK ≤ (ii) 0 < δ 1, (iii) the initial data (uˆ , uˆ ) satisﬁes 0 1 (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) uˆ H T U(0) H R 0 s+1,2( n) s+1,2( n) and (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) uˆ H T ∂ U(0) H R , 0 s,2( n) t s,2( n) (cid:88) (iv) and − s 1 t(cid:96) µ = µ (cid:96)! (cid:96) (cid:96)=0 with the µ determined as in Proposition 3.1 so that (cid:96) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) u(0) U(0) R Es+1 Es+1( n) (cid:0) (cid:1) and (cid:107) (cid:107) ≤ − (cid:107) (cid:107) µ(t) E − T (1 + ts 1)C U(0) R . s 1( n) Es+1( n) In light of the above discussion, the following proposition completes the proof of Theorem 1.1. ∈ ∈ Proposition 4.1. Suppose uδ CXTs+δ(1Rn) is a family of solutions depending on δ (0, 1] to the IVP (4.26)-(4.27) satisfying the conditions (i)-(iv) above. If (cid:107) (cid:107) ≤ ∞ u ∞ ×T K < , δ W∗1, ((0,Tδ) n) ∈ then there exists a δ > 0 and a time T > T for each δ (0, δ ] such that the solution u can be 0 δ δ∗ × T 0 δ continued to a solution of (4.26)-(4.27) on [0, T ) n. δ Proof. Befo(cid:0)re proceeding with the(cid:1)proof, we will, in order to simplify calculations, suppress the explicit x-dependence of the functions bµν, f and h. Since u satisﬁes δ − × T ∂ (mµν + δbµν(u ))∂ u ψu = δ(f(u ∂u ) + χ h(u ∂u )) in [0, T ) n, (4.28) ν δ µ(cid:2) δ (cid:0) δ δ δ Ω1 (cid:1)δ δ (cid:0) δ ≤ ≤ − we see after diﬀerentiating k-times, where 0 k s 1, with respect to t that ∂ku satisﬁes (cid:1) t δ(cid:3) − − − (∆ ψ)∂ku = ∂k+2u + δ ∂k+1 b00(u )∂ u + b0i(u )∂ u ∂ ∂k bi0(u )∂ u t δ t δ t δ t δ δ i δ i t δ t δ + bij(u )∂ u + ∂kf(u , ∂u ) + χ ∂kh(u, ∂u) + ∂kµ. (4.29) δ j δ t δ δ Ω1 t t Since (cid:107) (cid:107) (cid:46) (cid:107) − (cid:107) ≤ ≤ − ∂tkuδ H2,(cid:0)s+1−k(Tn) (∆ ψ)∂tkuδ H0,s−k−1(Tn) 0 (cid:1)k s 1 by Proposition B.2, it follows from Prop(cid:0)osition A.10 and (4.29) that ∂tkuδ satis(cid:1)ﬁes (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) ∂ku(t) H − T c ∂k+2u H − − T + ∂kµ(t) H − − T t 2,s+1 k( n) t 0,s k 1( n) t 0,s k 1( n) (cid:107) (cid:107) (cid:107) (cid:107) ≤ + δC(K) 1 + u(t) H T + ∂ u(t) H T 0 t < T (4.30) 2,s+1( n) t 2,s( n) δ where (cid:107) (cid:107) ≤ ≤ u ∞ ×T K 0 < δ 1. (4.31) δ W1≤, ((0≤,Tδ) −n) We collect the estimates (4.30), for 0 k s 1, into the single matrix inequality | | (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) ≤ M X (t) ∂su (t) T + µ(t) E − T + δC(K) 0 t < T , (4.32) δ δ t δ E( n) s 1( n) δ ##LTLine##

text: L.

text: ANDERSSON

text: AND

text: T.A.

text: OLIYNYK

label: list

text: (ii)

text: δ

text: (iii)

text: the

text: initial

text: data

text: (uˆ

text: uˆ

text: satisﬁes

text: uˆ

text: H

text: T

text: H

text: R

text: n)

text: n)

text: and

text: uˆ

text: H

text: T

text: H

text: R

text: n)

text: t

text: n)

text: (iv)

text: and

text: s

text: µ

text: µ

text: with

text: the

text: µ

text: determined

text: as

text: in

text: Proposition

text: so

text: that

text: R

text: n)

text: and

text: µ(t)

text: E

text: T

text: ts

text: R

text: s

text: n)

text: n)

text: In

text: light

text: of

text: the

text: above

text: discussion,

text: the

text: following

text: proposition

text: completes

text: the

text: proof

text: of

text: Theorem

text: Proposition

text: Suppose

text: uδ

text: is

text: a

text: family

text: of

text: solutions

text: depending

text: on

text: δ

text: to

text: the

text: IVP

text: satisfying

text: the

text: conditions

text: (i)-(iv)

text: above.

text: If

text: u

text: ×T

text: K

text: δ

text: n)

text: then

text: there

text: exists

text: a

text: δ

text: and

text: a

text: time

text: T

text: T

text: for

text: each

text: δ

text: δ

text: such

text: that

text: the

text: solution

text: u

text: can

text: be

text: δ

text: δ∗

text: T

text: δ

text: continued

text: to

text: a

text: solution

text: of

text: on

text: T

text: n.

text: δ

text: Proof.

text: proceeding

text: with

text: we

text: will,

text: in

text: order

text: to

text: simplify

text: calculations,

text: suppress

text: the

text: explicit

text: x-dependence

text: of

text: the

text: functions

text: bµν,

text: f

text: and

text: h.

text: Since

text: u

text: satisﬁes

text: δ

text: T

text: (mµν

text: δbµν(u

text: u

text: ψu

text: δ(f(u

text: ∂u

text: χ

text: h(u

text: ∂u

text: in

text: T

text: n,

text: ν

text: δ

text: δ

text: δ

text: δ

text: δ

text: Ω1

text: δ

text: δ

text: we

text: see

text: after

text: diﬀerentiating

text: k-times,

text: where

text: k

text: s

text: with

text: respect

text: to

text: t

text: that

text: ∂ku

text: satisﬁes

text: t

text: ψ)∂ku

text: δ

text: u

text: u

text: ∂k

text: u

text: t

text: δ

text: t

text: δ

text: t

text: δ

text: t

text: δ

text: δ

text: i

text: δ

text: i

text: t

text: δ

text: t

text: δ

text: bij(u

text: u

text: ∂kf(u

text: ∂u

text: χ

text: ∂kh(u,

text: ∂u)

text: ∂kµ.

text: δ

text: j

text: δ

text: t

text: δ

text: δ

text: Ω1

text: t

text: t

text: Since

text: ∂tkuδ

text: ψ)∂tkuδ

text: s

text: by

text: Proposition

text: it

text: follows

text: from

text: and

text: that

text: ∂tkuδ

text: ∂ku(t)

text: H

text: T

text: c

text: H

text: T

text: ∂kµ(t)

text: H

text: T

text: t

text: k(

text: n)

text: t

text: k

text: n)

text: t

text: k

text: n)

text: δC(K)

text: u(t)

text: H

text: T

text: u(t)

text: H

text: T

text: t

text: T

text: n)

text: t

text: n)

text: δ

text: where

text: u

text: ×T

text: K

text: δ

text: δ

text: −n)

text: We

text: collect

text: the

text: estimates

text: for

text: k

text: s

text: into

text: the

text: single

text: matrix

text: inequality

text: M

text: X

text: (t)

text: ∂su

text: (t)

text: T

text: µ(t)

text: E

text: T

text: δC(K)

text: t

text: T

text: δ

text: δ

text: t

text: δ

text: E(

text: n)

text: s

text: n)

text: δ

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