LONG RANGE SCATTERING WITH STARK EFFECT AND ALMOST PERIODIC POTENTIALS

Spectral and scattering theory is discussed for the Stark effect Hamiltonians H0=−(1/2)Δ−x1 and H=H0


Introduction
This note concerns quantum mechanical scattering in the presence of a constant electric field.The corresponding Stark effect" Hamiltonians are, H0 -Azl where A (1.1) j--I U Ho+ V Ho+ V + V+ Vs where -z is the potential corresponding to the constant electric field in the positive z direction (x R) and H is a perturbation of H0 by another potential V V + V + Vs consisting of two long range terms and a short range term.More precisely assume Conditon LRI.V C(R)/s real valued and .forsome e > 0 IDVL()[ < C,() -Il/z-' for every multi-indez where (z,)' (1 + ).
Conditon LR2.VL 0 if the space dimension is n > bet i n then Vz, C(R) is real valued and bounded along with all its derivatives and Va x + -"r )d'r ezists as an improper Riemann integral for every z.
Condition SR.Vs is a symmetric operator which is Ho-compact and f, II'(x > )Vs(Uo +/)-all dr.< oo where F(.) is multiplication bt the characteristic function of the indicated set.
It should be clarified that the term VL2 which allows an almost periodic potential is only present in the one dimensional case.Much of the recent worlq has focused on the one dimen- sional case; see Hislop-Nakamura" [1], Jensen [2], Jensen-Yajima [3] and Ozawa [4] for example.
The methods here are primarily multi-dimensional and the role of the assumption n will be highlighted below.Almost periodic potentials and the existence and completeness of the ordinary wave operators (of equation (1.6) below) have been studied previously [2].Long range scattering in this setting is relatively new but there has been some interest recently because there is a discrepancy between quantum and classical mechanics; the ordinary wave operators exist in the classical setting but not the quantum one when n and VL2 0 as was noted recently by Jensen-Yajima [3] and Jensen-Ozawa [5].In Theorem 2 below an additional condition on the potential is introduced which assures that the two Hilbert space wave operators equal the ordinary wave operators.In general one expects that the two Hilbert space wave operators are equal, up to a phase function, to the Dollard's modified wave operators; the justification of this expectation will be the subject of a future investigation.That result and Theorem below will precisely delineate for which potentials of those considered here, the usual wave operators exist and are complete.For the potentials V1 the condition is e > 1/2 (see Jensen-Yajima [3] and Ozawa [4] when n=l) but for Vt; the condition is not obvious; see Theorem 2.
The main result is the existence and completeness of the two Hilbert space wave operators, W + W+(H, H0; d+) defined by W +/s-lim eitHJ+e-itH (1.3) where J+/are two bounded operators on L(R'), to be specified and "s-lim" is the limit in the strong operator topology.Note the reversal of sign in equation (1.3) which is for historical reasons (see Reed and Simon's third volume [6, p. 17]).The two operators J+ could equally well be replaced by a single operator as was previously noted by the author [7][equation (2.10)] and as is customary in most discussions of two Hilbert space scattering.(The operators J+ are not unique.)Isozaki and Kitada [8] were the first to introduce J+ ("time independent modifiers") in this context; they are defined by where is the Fourier transform of b, dx (2r)-n/dx and 0 + are functions to be chosen.The functions 0 + will be specified in 3 below but roughly they are chosen so that the commutator HJ + J+Ho is small. (Recall Cook's method.)The symbol of HJ + J+Ho is, at least in the case that Vs 0: ___00+ (,) " v0(,) + (,0- + 1/2v0(,).v0(, )+ v() + v() (.) and this should be small; that is roughly "short range" when q= > 0. The main results may now be stated.
Theorem 1 Assume the Conditions Litl, LHP. and Sit.Then for .I 4-as defined above (for appropriate 04-), the wave operators W4-of (1.3) ezist, are isometries and are complete.Moreover H has no singularly continuous spectrum, and its eigenvalues are discrete and offinite multiplicity.
Theorem 2 Under the hypotheses of Theorem 1 and in the special case n 1, VL1 0 and lim VL,(x 1/2' + It')dr 0 it is possible to choose J4to be the identity operator in Theorem I, that is W +/-s-lim eitue-*u.
(1.6) t Examples.The conditions of Thereto are verified if: for e > 0 and a, 1/2 and 7 > 1/2 and b, , b are real, VL(X) where U is real function which is bound along with all its derivmtiv (but V m 0 if n > 1.)The conditions of Threms and 2 are linear and m the t of potentials vered by thee results form linear spe.For exampl of the short range potentials Vs s Yajima [9].Roughly Vs should be O((x)-/-) for > 0 and rome > 0 and o((x)) for < 0.
One simple example indicates that there is indd a difference betwn one and more dimen- sions: for V(x,x) sinx +sinx and n 2 the wave operatom (1.6) do not exist by a tensor product argument where they do if VL(m) sin m when n 1.If V(x) sin(x/3) then Threm applies but not Thereto 2. Jenn, [2] using Mourre's [10] method obtains a rult similar to Thereto 2 for a cls of real valu almost periodic potentis, r i_ g() e " () prvia ( + -(() < (and ith e and g 0).Threm 2 ruir more differenfibility but ne les enti- derivative; mre predsely, for eh N Threm 1, hieh indudes both lly deeyi end m periodic ptenfial, i ne.
2 Two Hilbert Space Scattering.
The derivation of Theorem 1.1 is broken into two main steps.The first step (Theorem 2.1 below) is to derive necessary conditions on J4-or, in view of (1.4), conditions on 04-to conclude Theorem 1.1.The second step (in 3) is to construct 04-satisfying those conditions, from the long range potential VL.Portions of the proofs will be cited from White [7] which will be quoted frequently simply as [W] for brevity's sake.More general potentials Vta are allowed in the one dimensional case because F(ID, < r))(Ho + i)is compact if n 1. (2.1)  since it is unitarily equivalent via exp(-iD/6) to a Hilbert Schmidt operator (see Perry [11, Proposition 19.1]); here D =-iO/cz.
Before stating the conditions on 0 +/-it is convenient to recall the Calder6n-Vaillancourt the- orem which will be required to show that J: and related operators are bounded.Introduce therefore the pseudo-differential operator R for all fi $(R").Here "Os-" indicates that the integrals are oscillatory integrals (see Kumana- go [12]).The theorem can be stated in terms of certain norms on the symbol, p: for each integer k Ipl, ,p{IOO,(z,,)l, lD'O,(,,)l} where the sup is over all (z, y, ) /R 3" and all or, , 3' so that 0 < I1, I'rl _< 2, ,nd 0 _< I1 < 2([n/2] + k + 1) and where rn is the least integer m > 5n/4.The Calder6n-Vaillancourt theorem [13] says that there is a constant C not depending on p so that IIRll < Clpl0 where II" ao h operator norm on Anticipating the use of Enss's time dependent method ([14]) we introduce smooth versions of the incoming and outgoing operators, F(D < 0) and F(D > 0) respectively.Choose t/in C(R) so that if > 1, r/(,) and t/(,) + r/(-,) 1.
Theorem 1 Define Ho by (I.I) and let H Ho+ V. + Vs where VL V(x) is a real valued function which as an operator acts multiplicatively, is Ho-bounded, and has Ho-bound strictly less than 1 and where Vs satisfies Condition SR.If J+ and p are as defined in equations (1.) and (1.5) and if 0 4-and p+ satisfy the above hypotheses then the conclusions of Theorem I. are valid.
Outline of the Proof.The proof is by Enss's time dependent method which here is adapted to a two Hilbert space setting appropriate for studying long range scattering.The method can be described as follows.Let H be any self adjoint operator on L2(R") with spectral measure by E (but H0 is defined by (1.1)) and suppose J+ be bounded operators.Assume further Hypothesis H1.There is ao < -1 so that, for all a, a > ao ((H + i)-lJ + J+(Ho + i)-)y(q=Dl -a) are compact.Hypothesis H2.For every compact real interval I, there is an integer N so that /t IIE(I)(HJ + J+Ho)h(:FD/r)y(xl/r2)(Ho + i)-lldr < oo.Hypothesis H3.The following operator is compact (H + i)-[J+l(-D)(J+) + J-l(D)(J-)'-I](H +,i)-.Hypothesis H4. eitHo((J+)*J : 1)e -irate --, 0 weakly as Too.
Enss method arguments apply to derive the present theorem from these hypotheses; see [W, Theorems 2.1 and 2.2].The first two hypotheses correspond to the standard Enss assumptions for the short range case; H3 assures that the operators J are "almost" unitary and H4 assures that the wave operators W : are isometries, if they exist.
Check therefore Hypotheses H1 through H4.Suppose n since the case of n > was considered in [W].Only the outgoing "-" case is considered but see (2.11).
Let P (H0 + VL)J-J-Ho so that for pdefined by (1.5).Since Vs is H0-compact by Condition SR, it suffices to show that (H + i)-P(Ho + i)-?(D-a) is compact for all a to verify H1.Since 1-y(D/r) is H0-compact for any r > 0, by (2.1), this may be simplified to showing that (H+i)-' Prl,(IDI/r)(Ho+i)-'tt(D-a) can be made arbitrarily close to a compact operator by choosing r large enough.The proof of this latter statement is the same as the proof of H1 given in [W] but P there is replaced by P,,(IOl/").(It should be remarked that [W, Proposition 3.2 (a) and (b)], which is used in the proof cited above, assumes that 0 is real valued but the proofs apply without change to the complex valued case provided the factor exp(-0+/-) appearing in the Fourier integral operators is treated as part of the symbol (and not the phase) as in the proof of H3 below.
The proof of H2 is exactly as in [W]; it uses the critical short range assumption (2.8) and the Calder6n-Vaillancourt theorem.
The following theorem will be used for the proof of H4 and Theorem 1.2 both.