Completeness Of Wave Operators Research Articles

Schrödinger Operators with Multiple Aharonov–Bohm Fluxes

AbstractWe study the Schrödinger operator describing a two-dimensional quantum particle moving in the presence of $$ N \geqslant 1$$ N ⩾ 1 Aharonov–Bohm magnetic fluxes. We classify all the self-adjont realizations of such an operator, providing an explicit characterization of their domains and actions. Moreover, we examine their spectral and scattering properties, proving in particular the existence and completeness of wave operators in relation with the free dynamics.

Wave operators on Sobolev spaces

We provide a simple sufficient condition in an abstract framework to deduce the existence and completeness of wave operators (resp. modified wave operators) on Sobolev spaces from the existence and completeness of the usual wave operators (resp. modified wave operators). We then give some examples of Schr\"odinger operators to which our abstract result applies. An application to scattering theory for the nonlinear Schr\"odinger equation with a potential is also given.

Scattering Theory and Banach Space Valued Singular Integrals

We give a new sufficient condition for existence and completeness of wave operators in abstract scattering theory. This condition generalises both trace class and smooth approaches to scattering theory. Our construction is based on estimates for the Cauchy transforms of operator valued measures.

Scattering Theory for Klein–Gordon Equations with Non-Positive Energy

We study the scattering theory for charged Klein-Gordon equations: \[\{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x, D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)= f_{1}, {array}. \] where: \[\epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x),\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \[ h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. \] We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case.

Energy transfer in scattering by rotating potentials

Quantum mechanical scattering theory is studied for time-dependent Schrödinger operators, in particular for particles in a rotating potential. Under various assumptions about the decay rate at infinity we show uniform boundedness in time for the kinetic energy of scattering states, existence and completeness of wave operators, and existence of a conserved quantity under scattering. In a simple model we determine the energy transferred to a particle by collision with a rotating blade.

Spectral and scattering properties of three-dimensional anisotropic Schrödinger operators

A three-dimensional Schrödinger operator with a small, zero order, pseudodifferential potential is considered. Equivalence of H and in is shown. Eigenfunction expansion is given explicitly in terms of incoming and outgoing waves. Existence and completeness of wave operators are proven. Scattering operator and scattering matrix are studied.

On the Completeness of Wave Operators under Loss of Local Compactness

We present a general criterion for the existence and completeness of the wave operators in obstacle (or potential) scattering which does not require local compactness near the boundary of the obstacle. The scope of our basic theorem is illustrated by new classes of examples, namely, exterior "Rooms and Passages"-domains, where completeness holds, and exterior "Jelly Roll"-domains, where it fails.

Moving potentials and the completeness of wave operators. Part III: Existence and completeness

Moving potentials and the completeness of wave operators. Part III: Existence and completeness

Moving potentials and the completeness of wave operators. Part I: The propagator

Moving potentials and the completeness of wave operators. Part I: The propagator

Theory of potential scattering, taking into account spatial anisotropy

We get new tests for the existence and completeness of wave operators under perturbation of a pseudodifferential operator with constant symbol P(ξ) by a bounded potential v(x). The term anisotropic is understood in the sense that the growth of P(ξ) as ξ→∞ and the decrease of v(x) as x→∞ can depend essentially on the direction of the vectors ξ and x respectively. This permits us to include in the sphere of applications of the abstract scattering theory of a nonelliptic unperturbed operator the D'Alembert operator, an ultrahyperbolic operator, nonstationary Schrödinger operator, etc. In view of the anisotropic character of the assumptions on the potential, the results obtained are new even in the elliptic case. As an example we consider a Schrödinger operator with potential close to the energy of a pair of interacting systems of many particles.

Spectral and scattering theory for symmetric systems of nonconstant deficit

We study the spectral and scattering problems for symmetric systems of nonconstant deficit. We prove (i) discreteness of eigenvalues, (ii) absolute continuity of continuous spectrum. (iii) principle of limiting absorption. As an application to the scattering problems, the decay of local energy and the completeness of wave operators are obtained. The class of systems we consider here includes the linearized equation of magnetohydrodynamics as a typical example.

Completeness of wave operators in relativistic quantum mechanics

A combination of geometric and algebraic methods is used to prove asymptotic completeness for Schrodinger-type equations with potential not vanishing at infinity along hyperboloids (in spacetime), and with the free Hamiltonian given by the (not bounded below) relativistic (mass)2 operator. The proof is based on the use of a modified form of local compactness and additional geometric properties of asymptotic scattering states which are needed to distinguish them from states ‘trapped’ inside some hyperboloid for all times.

Scattering theory for pseudodifferential operators. II. The completeness of wave operators

Scattering theory for pseudodifferential operators. II. The completeness of wave operators

New criterion for existence and completeness of wave operators and applications to scattering by unbounded obstacles

(1981). New criterion for existence and completeness of wave operators and applications to scattering by unbounded obstacles. Communications in Partial Differential Equations: Vol. 6, No. 11, pp. 1179-1223.

Another look at scattering theory

We present a new approach to existence and completeness of wave operators. We do not use the subspace of absolute continuity, but rather the orthogonal complement of the eigenvectors. This is more natural from the physical point of view. We give sufficient conditions for existence and completeness of the wave operators. These results are both simpler and stronger than those obtained previously.

On the Long-Range Stationary Wave Operator

In the present paper we shall be concerned with the stationary theory of scattering associated with the Schrodinger operator with a long-range potential. In quantum theory of scattering, many authors have investigated the existence and completeness of wave operators W = s-lim ee~y where J-»±oo HI and H2 are self-adjoint operators acting on a Hilbert space M. Among them, Kato and Kuroda gave an abstract time-independent approach to scattering theory. They derived a stationary form of the wave operator in the following way:

On a limiting case of a theorem of Kato and Kuroda

Cook [5] and Jauch [6] introduced an abstract version of the two characteristic matrices of Mijller [2] and called them wave operators. At the same time Jauch [6] emphasized that wave operators corresponding to physically realistic systems have an additional property that he called completeness. Several different authors formulated criteria for the existence and completeness of wave operators for various abstract and concrete classes of operators [3, 4, 7, 8, 10, 12, 15-17, 20, 22-241. For the pair of operators (d, d + p) recently this problem was taken up again by Kato and Kuroda [25, 26, 33, 341. Here, as usual, il denotes the negative Laplacian acting in free space and p denotes a potential. They formulated general criteria on the potential for the existence and completeness of the wave operators. According to their abstract theory it also follows that the absolutely continuous parts of the perturbed and unperturbed operator are unitarily equivalent. In this paper, we consider a limiting case of their cnndition. Roughly speaking, we treat potentials with decay exponent equal to one and satisfying an additional smallness condition at infinity. This additional smallness condition is in accordance with a result of Dollard [14] which says that, for the Coulomb potential, the wave operators do not exist. At the same time in this paper, we illustrate that an estimate formulated elsewhere [27] in conncction with a different problem [18] can be applied to study wave operators. In Section 2, our condition is formulated and the corresponding Theorem 2.1 is stated. In Section 3, we state a version of an abstract theorem of Kato and Kuroda. This theorem is implicit in their work [25] and we state it for completeness only. In Section 4 we verify the assumptions of this abstract theorem for potentials satisfying the assumptions of Section 2.

On the existence and completeness of wave operators in nonrelativistic scattering

A new direct proof of the existence of the wave operatorsW+-= s-lim exp [iH2t] exp [-iH1 t] of nonrelativistic quantum mechanicst->+-<o is given. Also the results of Prosser and Hunziker for the two and threebody cases onL 2 completeness are extended and an error in the literature is corrected.

SCATTERING PROBLEMS FOR DIFFERENTIAL OPERATORS WITH PERTURBATION OF THE SPACE

For differential operators with constant coefficients in Rm we study the existence and completeness of wave operators corresponding to perturbation of the metric of the original Hilbert space. The results are applied, in particular, to scattering problems of Maxwell's system in an anisotropic medium and for the wave equation. The study is based on abstract criteria (of "nuclear type") of existence of complete wave operators.