On the absolutely continuous spectrum of Stark HamiltoniansJaouad Sahbani Citation: Journal of Mathematical Physics 41, 8006 (2000); doi: 10.1063/1.1287922 View online: http://dx.doi.org/10.1063/1.1287922 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On geometric perturbations of critical Schrödinger operators with a surface interaction J. Math. Phys. 50, 112101 (2009); 10.1063/1.3243826 The classical dynamics of Rydberg Stark atoms in momentum space Am. J. Phys. 76, 1007 (2008); 10.1119/1.2961081 Stark and field-born resonances of an open square well in a static external electric field J. Chem. Phys. 122, 194101 (2005); 10.1063/1.1897370 Short-lived two-soliton bound states in weakly perturbed nonlinear Schrödinger equation Chaos 12, 324 (2002); 10.1063/1.1476951 Efficient adiabatic population transfer by two-photon excitation assisted by a laser-induced Stark shift J. Chem. Phys. 113, 534 (2000); 10.1063/1.481829

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On the absolutely continuous spectrum of StarkHamiltonians

Jaouad Sahbania)

Departments of Mathematics and Physics, Princeton University,Princeton, New Jersey 08544

~Received 11 May 2000; accepted for publication 25 May 2000!

We study the spectral properties of the Schro¨dinger operator with a constant elec-tric field perturbed by a bounded potential. It is shown that if the derivative of thepotential in the direction of the electric field is smaller at infinity than the electricfield, then the spectrum of the corresponding Stark operator is purely absolutelycontinuous. In one dimension, the absolute continuity of the spectrum is implied byjust the boundedness of the derivative of the potential. The sharpness of our crite-rion for higher dimensions is illustrated by constructing smooth potentials withbounded partial derivatives for which the corresponding Stark operators have adense point spectrum. ©2000 American Institute of Physics.@S0022-2488~00!03809-3#

I. INTRODUCTION

A. Overview

The Hamiltonian of an electron moving in a constant electric fieldFW and subject to an externalforce given by a potentialV is the Stark operator

H52D2x11V~x!

acting inH5L2(Rd), whered>1 is the dimension, andx1 is the multiplication operator by thecoordinatex1 . It is convenient to regard the potentialV as a perturbation of the Stark operator

H052D2x1 ,

where the electric field is taken asFW 5(21,0,...,0).The Stark operator is essentially self-adjoint inH and its spectrum is purely absolutely

continuous:

sac~H0!5sac~H0!5R,

ssc~H0!5spp~H0!5B.

Our purpose here is to study the stability of that spectral structure under a perturbation by a~bounded! potentialV. The stability here means that the perturbed HamiltonianH has a purelyabsolutely continuous spectrum up to a discrete set as a singular spectrum.

Perturbations of interest include the Anderson random potentials

V~x!5 (i PZd

l i~w!u~x2 i !, ~1.1!

a!This work was partially supported by the NSF Grant No. PHY-9971149. Permanent address: Universite´ Paris 7, U.F.R.de Mathematiques, 2, place Jussieu, 75251 Paris Cedex 05, Electronic mail: sahbani@math.jussieu.fr; jsahbani@princeton.edu

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 12 DECEMBER 2000

80060022-2488/2000/41(12)/8006/10/$17.00 © 2000 American Institute of Physics

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whereu is the single site potential and$l i(w)% is a collection of i.i.d. random variables. It is wellknown that when the electric field is zero the operatorH has only localized states~almost surelywith respect to the disorder! in dimension one. The same is expected in dimension 2. But indimensiond>3 there is a region in which the spectrum is pure point associated to the localizedstates, and a region of absolutely continuous spectrum which corresponds to the extended states,separated by some energies called the mobility edges~see Refs. 1–4 for a precise discussion ofsuch considerations!.

But in the presence of a nonzero electric field the spectral properties ofH change drastically.Indeed~see Refs. 5–7! in d51 if V is sufficiently regular~e.g., twice continuously differentiable,bounded with its derivatives! then the spectrum ofH is purely absolutely continuous. In particular,this is true for the Anderson potential~see however Ref. 8!.

This can be explained by the naive physical intuition: because the electron is uniformlyaccelerated by the electric field, a bounded potential could not affect its propagation and thespectral properties of its Hamiltonian. However, this is not always true, and therefore the absolutecontinuity of the spectrum of the Stark operator can be partially or completely destroyed by‘‘small’’ potential. This has been pointed out by Naboko–Pushnitskii in Ref. 9, where a smoothpotentialV has been construced, that satisfies

uV~x!u<C~x!

~11uxu!1/2, C~x!→` at infinity arbitrarily slowly

and the corresponding HamiltonianH has a dense set of eigenvalues. Notice that the derivative ofNaboko–Pushnitskii’s potential diverges at infinity exactly at the same rate of the functionC(x).This illustrates the important role of the derivative in the preservation of the absolutely continuousspectrum in the sense explained above.

We also mention the result of Ref. 10 where the one-dimensional Kro˝nig–Penny model withan electric field is studied. The authors establish a transition from purely continuous spectrum toa pure point spectrum as the electric field decreases. There are also several results dealing withstrongly singular potentialsV for which the corresponding operatorH has no absolutely continu-ous spectrum~see Refs. 8 and 11!.

Our first purpose in this paper is to describe a class of bounded potentialsV for which thecorresponding HamiltonianH has an absolutely continuous spectrum with at most a discrete set assingular spectrum. Basically, our class consists of sufficiently regular potential having a small~atinfinity! derivative in the direction of the electric field. In the one-dimensional case the derivativedoes not have to be small but only bounded.

The second goal of our paper is to study the role that the tail of the partial derivative of thepotential can play in the occurrence of a~dense! singular spectrum in higher dimension. Indeed, inthe case where the dimensiond>2 we construct a smooth potentialV, bounded with its firstpartial derivatives and leading to a dense point spectrum forH. Again, we notice here that indimension one this cannot happen, which illustrates the dimension effect in the problem.

B. The one-dimensional case

As we already mentioned above, ind51 if V is sufficiently regular then the absolute conti-nuity of the spectrum ofH @~Refs. 5–7!# is preserved, while it is partially or completely destroyedif the potential is singular~see Refs. 9–11!. What is then the minimal regularity ofV preservingthe absolute continuity ofH? A partial answer is the following.

Theorem 1: (1) Assume that V is of class BC1 with uniformly continuous first derivative.Then the setsp(H) of eigenvalues of H is discrete.

(2) Suppose that V is smooth in the Zygmund’s sense, i.e.,

E0

1

supxPR

uV~x1«!22V~x!1V~x2«!ud«

«2,`, ~1.2!

8007J. Math. Phys., Vol. 41, No. 12, December 2000 On the absolutely continuous spectrum . . .

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then H has no singular continuous spectrum.Remarks:

~1! We proved Theorem 1 in Ref. 12 under a slightly more restrictive assumption onV. This wasreobtained partially by Kiselev in Ref. 13 by using the Gilbert–Pearson method. The maindifference is that the result of Ref. 13 does not rule out the possibility to have a singularspectrum also filling the real axis. The slowly decaying potentials are also studied in Ref. 13,which combined with Ref. 9 provides an example of a Stark operator for which dense pointspectrum coexist with the absolutely continuous spectrum.

~2! Theorem 1 includes a large class of periodic, quasiperiodic or random potentials, because thereis no restriction on the bound of the potentialV nor on V8. This is not the case in themultidimensional case~cf. Theorems 3 and 4!.

Notice that Theorem 1 only ensures the discreteness of the set of eigenvalues ofH but doesnot exclude the possibility to have embedded eigenvalues. In fact, we have

Theorem 2: For each real numberl there exist a real-valued potential VPC`(R), such thatV and V8 are bounded andl is an eigenvalue of H.

Theorem 2 is an essential piece in our construction of a smooth and bounded potential suchthat H has a dense point spectrum in dimension 2 and more~cf. Theorem 4!. To prove it we usethe ideas of Ref. 9, and the potentialV will be sparse.

C. The multidimensional case

In d.1 there is also important literature on the spectral theory of Stark Hamiltonians~Refs.14–16 and references therein!. Most of these works assume thatV or “V tend to zero at infinity.This can be explained partially as follows. Whend51 the derivative of a bounded regularpotential isH0 compact. Which can be lost ind>2 if V or ]1V does not tend to zero at infinity~see, however, Ref. 16!.

For a functionW we setW1(x)5sup(0,W(x)). Putx5(x1 ,x8) with x85(x2 ,...,xd).Theorem 3: Assume that V and]1V are bounded and thati(]1V)1i`,1. Then H has no

eigenvalues. If moreover,

E0

1

supxPRd

uV~x11«,x8!22V~x!1V~x12«,x8!ud«

«2,`, ~1.3!

then H has no singular continuous spectrum.Remarks:

~1! Actually V does not have to be bounded, we only need thatH defines a self-adjoint operatorin H. This is not true in Theorem 1.

~2! It is not difficult to show that the assumptioni(]1V)1i`,1 can be replaced by only thesmallness at infinity, i.e.,

lim supuxu→`

~]1V!1~x!,1.

In which case, the possible singular spectrum thatH can have is discrete, and consists of finitelydegenerate eigenvalues.

Examples:~1! Theorem 3 obviously includes potentialsV such that]1V tends to zero at infinity, but without

asking thatV itself tends to zero at infinity~see discussion above!.~2! Let q : Rd→R be a bounded and sufficiently regular function. Then

H5H01q~ax1 ,x8!, aPR

8008 J. Math. Phys., Vol. 41, No. 12, December 2000 J. Sahbani

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has a purely absolutely continuous spectrum provided that the scaling constanta is sufficientlysmall. This includes a large class of periodic or quasiperiodic potentials. An analogous statementholds for the Anderson random potential~1.1!.

Let us rephrase the last theorem as follows. If the force induced by the electric field is strongerthan the force induced by the potential then the spectrum ofH is purely absolutely continuous.This bring us to ask the natural question: What happens when the electric field is weak? We havethe following theorem.

Theorem 4: Let l be a fixed real number. Then there exist a real-valued potential VPC`(Rd) such that V and]1V are bounded and@l,`),spp(H).

According to this theorem, it is not clear whether or not the Anderson model with a weakelectric field or a strong disorder in higher dimension has a purely absolutely continuous spectrum.This should be an interesting question to study.

The paper is organized as follows. In Sec. II we shall describe what we need in our proofs.Sections III and IV will be devoted to the proofs of Theorems 1 and 3, respectively. In Sec. V weprove Theorems 2 and 4.

II. BASIC NOTIONS

Our proofs are based on the conjugate operator method. It is an abstract theory which provesthat the HamiltonianH has an absolutely continuous spectrum if it has a conjugate operatorA,i.e., a self-adjoint operator such that the commutator@H,iA# is strictly positive in an adequatesense. In this section we give a short description of the main points of this theory. For more detailsconcerning the results of this section we refer the reader to Refs. 17–19. LetH,A be two self-adjoint operators in a Hilbert spaceH. The C0 group associated toA will be denoted bye2 iAt,tPR. We shall denote byR(z)5(H2z)21 the resolvent ofH for a complex numberzPC\s(H).

Definition 1: (1) We say that H is of class C1(A) if the map

R{t°e2 iAtR~z!eiAtPB~H!

is strongly of class C1, for some (and so for any) complex zPC\R.(2) We say that H is A-regular if

E0

1ie2 iA«R~z!eiA«22R~z!1eiA«R~z!e2 iA«i

d«

«2 ,`.

Remark that ifH is A-regular thenH is of classC1(A). Assume thatH is of classC1(A).Then the intersectionD(A)ùD(H) is dense inD(H) equipped with the graph topology associ-ated to the norm

i f iH5i f i1iH f i .

Moreover, the sesquilinear form defined onD(A)ùD(H) by

^ f ,@H,A#&5^A f ,Hg&2^H f ,Ag&

extends continuously toD(H). Then one can define the open setmA(H) of the real pointsl forwhich there exist a constanta.0, a compact operatorK in H such that

E~D!@H,iA#E~D!.aE~D!1K, ~2.1!

for some open intervalD containingl. HereE denotes the spectral measure ofH.Proposition 1: The eigenvalues of H contained inmA(H) are all finitely degenerate and

cannot accumulate inmA(H).In particular, the spectrum ofH is purely continuous in

8009J. Math. Phys., Vol. 41, No. 12, December 2000 On the absolutely continuous spectrum . . .

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mA~H !ªmA~H !\sp~H !,

wheresp(H) denotes the set of eigenvalues ofH. One can prove easily thatmA(H) is in fact theset of all real pointsl for which there exist a constanta.0 and an open intervalD{l such that

E~D!@H,iA#E~D!.aE~D!. ~2.2!

We have the following.Theorem 5: Assume that eiAt leaves invariant the domain D(H) and that H is A-regular.

Then H is purely absolutely continuous inmA(H).Remarks:

~1! This theorem is proved in Ref. 17, where one can find another version of this theorem in whichthe invariance of the domain under the action ofeiAt is replaced by the fact thatH has a gapin its spectrum~which is clearly inadequate for the model considered here!. In Ref. 19 we haveeliminated this condition on the domain without asking the existence of a spectral gap forH,but we askH to be ~locally! slightly more thanA-regular.

~2! Let us mention that ifH is of classC1(A) and @H,iA# is a bounded operator inH theneiAt

leaves invariant the domain ofH ~see Ref. 20!.

III. PROOF OF THEOREM 1

Using Mourre’s theory described in the last section the proof of Theorem 1 is reduced toprove thatH has a conjugate operator. This section is devoted to that.

~i! By straightforward computations we get

@H0 ,iA#51,

where A52 id/dx is the translation generator. ThenH0 is A-regular @in fact is even of classC`(A) in the sense that the map of Definition 1 is of classC`], and

mA~H0!5R.

On the other hand,

@V,iA#52V8~x!,

which is obviously bounded inH if and only if V8 is bounded as a function.To conclude the first part of Theorem 1 we have to prove that the Mourre estimate holds locallyon R, i.e., mA(H)5R. For this, and according to

E~D!@H,iA#E~D!5E~D!2E~D!V8~x!E~D!, ~3.1!

it is sufficient to prove thatE(D)V8(x)E(D) is a compact operator inH. But this property is asimple consequence of the following assertion which is proved by Bentoselaet al. in Ref. 7 ~seealso Ref. 21!. If a functionG is uniformly continuous and

limr→`

supxPR

U1r Ex2r

x1r

G~y!dyU50 ~3.2!

then E(D)G(x)E(D) is a compact operator inH. Clearly if V8 is bounded and uniformly con-tinuous, andV is bounded~which are ensured by our assumptions! then G5V8 satisfies~3.2!.Then mA(H)5R follows from ~3.1! and ~3.2!, and Proposition 1 finishes the proof of part 1 ofTheorem 1.

~ii ! Let us prove the second part of Theorem 1. According to Theorem 5 and the last part ofour proof we must only show thatH is A-regular and thatD(H) is invariant under the action ofeiAt. According to the obvious property

8010 J. Math. Phys., Vol. 41, No. 12, December 2000 J. Sahbani

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e2 iA«VeiA«5V~x2«!.

the operatorV is A-regular if and only if the functionV is smooth in Zygmund’s sense, which isexactly our assumption~1.2!. Then the operatorH is A-regular.

The invariance of the domainD(H) under the action ofeiAt follows from the fact that thecommutator@H,iA#512V8 is a bounded operator inH. This finishes the proof of Theorem 1.

IV. PROOF OF THEOREM 3

We shall prove that the self-adjoint operator

A52 i ]x1

is strictly conjugate toH on R. Indeed, we have

@H0 ,iA#51.

In particularH0 is of classC`(A) andA is strictly conjugate toH0 onR, i.e.,mA(H0)5R. On theother hand, we have

e2 i«AVei«A5V~x12«,x8!.

Then it is clear that the regularity assumptions on the functionV of Theorem 3 ensure thatV isA-regular, and soH is too ~sinceH0 is!. Moreover, the commutator

@H,iA#512]x1V

is obviously a bounded operator inH. And so the domain ofH is invariant under the action ofeiAt. It remains to show thatmA(H)5R. But this property follows easily as

@H,iA#512]x1V>12~]x1

V!1~x!>12i~]x1V!1i`.0

which is a global and strict Mourre estimate. This finishes the proof of the second part of Theorem3.

Remarks:~1! Let us remark that when we only assume that

lim supuxu→`

~]x1V!1),1

we only get a local Mourre estimate:E~D!@H,iA#E~D!>aE~D!1K,

whereK is a compact operator,a512 lim supuxu→`(]x1V)1.0 andD is any compact interval.

Which means thatmA(H)5R. Thus according to Proposition 1 there is only a discrete set ofpossible eigenvalues ofH and all these eigenvalues are finitely degenerate.

~2! We also mention the fact that our argument ignores completely if the particle is relativistic ornonrelativistic. More precisely, our proof is still valid for any operator of the form

H5h~2i¹!2x11V~x!,whereh is a divergent continuous function andV are as in Theorem 3. A physical interestingsituation is whenh(x)5A11uxu2. In such a case,H becomes

H5A2D112x11V~x!,which describes the motion of a charged relativistic particle in a constant electric field. How-ever, it is not clear whether or not Theorem 1 is still valid. To do this, we must describe therelative compact operatorV with respect toH05h(P)2x1 .

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V. PROOF OF THEOREM 2 AND 4

A. Proof of Theorem 4

Let us start by proving that Theorem 4 follows easily from Theorem 2 combined with Simon’sresult.22 Assume thatd52 and that

V~x1 ,x2!5V~x1!1V~x2!.

Then by a separation of variables we decomposeH as follows:

H5H1^ 111^ H2 ~5.1!

acting inL2(R) ^ L2(R), where

H152d2

dx12 2x11V1~x1!, ~5.2!

H252d2

dx22 1V2~x2!. ~5.3!

Let l be a fixed real number. Theorem 2 tells us that there exists a real-valued potentialV1 of classC` such thatV1 andV18 are bounded and thatl is an eigenvalue ofH1 . On the other hand, let$l i% i be a sequence of positive numbers. Then there exist~cf. Simon22! a potentialV2 of classC`

such that eachl i is an eigenvalue ofH2 . In particular, if the sequence$l i% i is dense in@0, )~which we assume from now! then we get that@0, ),spp(H1). But the decomposition~5.1! of Himplies that the numbersl1l i belong to the setsp(H) of the eigenvalues ofH. It follows then@l,`),spp(H). On the other hand, the potentialV5V11V2 is of class C` and ]1V(x)5V18(x1) which is clearly bounded. Finally, an obvious induction allows us to do the sameconstruction for any dimensiond.

B. Proof of Theorem 2

Reduction of the problem. Without loss of generality one can assume thatl50. We recall thatour goal is to construct a real-valued potentialV of classC`(R) such that

~i! V andV8 are bounded and~ii ! there existuPL2(R) solution of

2u92xu1Vu50.

Let us apply the Liouville transformation by setting forx>1

j5 23 x3/2, w~j!5x~j!1/4u~x~j!!.

One can show that~cf. Ref. 9! the problem~ii ! is equivalent to the following.~ii 8! There exist a functionwPL2((1, ),j22/3dj) solution of

2w91q~j!w5w. ~5.4!

The relation betweenq andV is

q~j!5V~x~j!!

x~j!1

5

36j2

or equivalently

8012 J. Math. Phys., Vol. 41, No. 12, December 2000 J. Sahbani

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V~x!5xS q~j~x!!25

36j~x!2D .

Moreover, it is not difficult to see that ifq of classC`(1, ) such that for each integerm>0

q(m)~j!5O~j21!, j→`

then V will be of classC`(R), suppV,@1, ), and is bounded with its derivative@i.e. ~i! issatisfied#. Let us mention however thatV9(x)5O(Ax) asx→`.

Remark:Notice that the Wigner–Von Neuwmann example~cf. Ref. 23! provides a potential satisfying

all of the requirements of the problem~ii 8!. We give, however, another construction which can beextended to construct a potentialV ~bounded with its derivative! for which the one-dimensionalStark operator has a discrete sequence of eigenvalues~cf. Theorem 3!. In that case, ifd>2 thenV can be chosen bounded with its partial derivatives and leading to a dense point spectrum on allR.

Construction of q. To constructq we apply the Pru¨fer transformation by setting

w5R cosf,

w85R sinf.

Straightforward computations give the equations

R8

R5

1

2q sin 2f,

~5.5!

f85211q cos2 f,

or equivalently

R25C expE1

j

q sin 2f dt,

~5.6!

f85211q cos2 f.

The potentialq will have the form

q~x!5 (k>1

qkj S j2jk

D D ,

wherejk and qk are two adequate sequences we have to construct, whileD.0 will be chosensufficiently small andj PC`((0,1)),j (x)>0,* j (x)51. Assume that

q sin 2f<2 12 q,

which is ensured iff(x)'3p/4 on the interval @jk ,jk1D#. ~Remark thatq50 on @jk

1D,jk11#.) Then

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E1

`

R2dj

j2/35CE1

`S expE1

j

q sin 2f dtD dj

j2/3

<C11C2(k.0

Ejk

jk11expS E

1

j

q sin 2f dtD dj

j2/3

<C11C2(k.0

Ejk

jk11expS 2

1

2 E1

j

q dtD dj

j2/3

<C11C2(k.0

@jk112jk#expS 21

2 E1

jkq dtD

<C11C2(k.0

@jk112jk#exp2S D

2 (j 51

j 5k

qj D .

Then if

jk112jk5O~1!,~5.7!

qk5C

jk,

whereC is sufficiently large constant, then the right-hand member of the last inequality is con-vergent, which means thatw lies in L2((1, ),j22/3dj). Moreover, by construction the potentialq satisfies

q(m)5O~1/j!, as x→`, ;m>0.

Thus, it is sufficient to construct by induction a sequencejk such thatf(jk)53p/4(modp) andq(j)5qkj @(j2jk)/D# on the interval@jk ,jk1D#; andq(j)50 on the interval@jk1D,jk11#.

Assume thatjk is constructed, and let us setq(j)50 for j.jk1D. Integrating Eq.~5.6!betweenjk1D,j, we obtain

f~j!52j1jk1D1f~jk1D!.

Let us choosejk11 as the nearest point on the right-hand side ofjk such thatf(jk11)53p/4(modp). We also have,

jk112jk5f~jk1D!2f~jk11!2D5O~1!.

Let us integrate~5.6! betweenjk andj<jk1D to obtain

uf~j!2f~jk!u<D1Ejk

j

q dt<D1Dqk .

For D sufficiently small we ensure the fact thatf(j)'3p/4(modp) on @jk ,jk1D#. The con-struction is now completed.

ACKNOWLEDGMENTS

The author is grateful to A. Klein and S. Jitomirskaya for their hospitality at the department ofMathematics at UC, Irvine, in which this work has been partially done. The author also thanks M.Aizenman for helpful discussions.

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