Proceedings of a Workshop Held at Aarhus, Denmark 15 May - 1 August 1991
The objective of stochastic spectral analysis is explained. It is used to study regular perturbations for a general class of generators of Feller semigroups, also called generalized Schrödinger operators. Upon introducing the Kato-Feller norm, the asymptotic behaviour of several spectral data can be studied. In the present article mainly the convergence of scattering matrices is considered.
In this paper, we study some relations between the local geometry of a potential V on R n and the spectrum of the corresponding self-adjoint Hamiltonian H = −Δ + V on L 2(R n ). More specifically, for a fixed energy interval I,we want to examine the effects of strong local fluctuations of V near energies in I on the spectrum...
In this work we describe some results on the spectral theory of the Schrödinger operator H = −Δ + V acting on L 2(R n ), where V is a bounded real-valued function of class C l on R n \ {0} with n ≥ 2 that satisfies 1.1 ...
We sketch a proof of the asymptotic completeness of N-body quantum systems with potentials that decay like x −μ where ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\mu > \sqrt 3 - 1$$ .
We prove that N—particle Hamiltonians with long range pair potentials decaying at infinity like < x >−μ for μ > 1/2 are asymptotically clustering at all non-threshold energies.
Consider a molecule consisting of N quantized electrons at positions xi, and M nuclei of charges Z = (Z1,..., ZM) fixed at positions y = (yl,..., yM). The Schrödinger Hamiltonian of such a system is given by ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${H_{Z,H}}...
Stability of the motion of a discrete quantum system under the influence of a time-dependent perturbation is discussed. Various results are reviewed, and a few new ones obtained, notably the absence of absolutely continuous spectrum for the pulsed rotor.
In recent years, there has been a considerable progress in scattering theory for many body Schrödinger operators. Namely, the completeness of wave operators was proved for 3-body systems by Enss [4] and for the general N-body systems by Sigal-Soffer [13]. See also an elegant proof of Graf [7].
We describe some results of Klein-Martinez-Seiler-Wang [12] and Martinez-Messirdi [15], concerning the study of the discrete spectrum and the resonances of molecular systems in the Born-Oppenheimer approximation.
Polyatomic molecules are studied in the limit as the total charge Z becomes infinite with the number of nuclei and their charge ratios fixed. It is shown that, in the Born-Oppenheimer approximation, if such a system has a stable bound state then it is asymptotically neutral in the sense that it satisfies the inequality |Z − N| < C 1Z1−ε where N denotes the number of electrons,...
This paper is divided into two parts. In the first part, we give a modified definition of time-delay as the limit as r → ∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ‘fuzzy’ ball of radius r in ℝ. The potential W is assumed to be smooth and behave like | x |−α (α > 1) at infinity. For earlier studies on this...
In this note we give an account on some recent results on the smoothness of quantum mechanical N-body scattering amplitudes [S1]. We have results for the 2-cluster—2-cluster and 2-cluster—N-cluster amplitudes under a short range condition on the potential and in addition under a discreteness assumption on the 2-cluster channel energies. This gives a rather complete picture for N = 3 while a number...
Let H ћ = −ћ 2 ∂ φ 2 /2+U(φ) be a Schrödinger operator acting in L 2(T) with T an ℓ-dimensional torus and U an analytic periodic function on T. Approximate semiclassical expansions for the eigenfunctions and eigenvalues of H ћ are developed which are asymptotic...
In this work, we study the resonances of Schrödinger operator with homogeneous magnetic and electric fields: ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$P = {\left( {{D_x} - \vec A\left( x \right)} \right)^2} + \vec e.x + V\left( x \right)$$ in the case the strength of the magnetic field tends to ∞.
The correct form of radiation conditions is found in scattering problem for N-particle quantum systems. The estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations.
We begin with a brief review of the method of WKB-approximation for the Cauchy problem for time dependent Schrödinger equations 1.1 ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ih{\partial _t}u - {H^h}\left( t \right)u = 0,{H^h}\left( t \right) = \left( {1/2} \right){\left(...