Joachim Weidmann

Spectral properties of Hamiltonian operators

Spectra and essential spectra of selfadjoint and essentially selfadjoint operators.- Schrodinger operators.- Perturbations small at infinity.- Examples.- Operators acting only on part of the variables.- N-particle Hamiltonians.- Symmetries of the Hamiltonian.- The spectrum of the Hamiltonian of a free system.- A lower bound of the essential spectrum.- The essential spectrum of the Hamiltonian of an N-particle system with external forces.- The essential spectrum of the internal Hamiltonian of a free system.- Proof of theorem 11.16.

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {Sr{ of regular S-L problems with the properties(i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {Sr{(ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {Sr{ has to converge to an eigenvalue of S(iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.

Expansions in generalized eigenfunctions of selfadjoint operators

Let a quantum mechanical system be in the normalized state f; measuring the observable quantity which is represented by H, we have the probability I~.l 2 to find the value a(n) and the probability density to find a(2) in the continuous spectrum is Ic~(2)l 2. b) Let for example H = A + V be a one-body Hamiltonian (i.e. V(x) ~ 0 for [xJ--.oo). We expect the T~ to be bounded or at most slowly increasing (plane waves for V=0). On the other hand, for values E

Approximation of isolated eigenvalues of general singular ordinary differential operators

a(H) the solutions T of (--A + V) T = E 7 ~ should grow fast (exponentially) at infinity. These conjectures can be summarized to

Spectral Theory of Sturm-Liouville Operators Approximation by Regular Problems

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), − ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations An of τ on subintervals (an, bn) of (a, b) with an → a, bn → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators.In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ − λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ.

Asymptotic estimates of wave functions and the existence of wave operators

It is the aim of this article to present a brief overview of the theory of Sturm-Liouville operators, self-adjointness and spectral theory: minimal and maximal operators, Weyl’s alternative (limit point/limit circle case), deficiency indices, self-adjoint realizations, spectral representation.

Closed linear operators

Abstract The existence of wave operators is proved for the case, where the unperturbed operator is the operator of multiplication by a smooth function in momentum space and the perturbation is an arbitrary operator satisfying a fall off condition near infinity or a weighted L p -estimate in configuration space. Under somewhat more restrictive conditions the invariance principle is also proved.

Scattering Theory for a General Class of Differential Operators

In what follows H, H1 and H2 will always be Hilbert spaces. As long as no adjoint operators (in particular no symmetric or self-adjoint operators) are treated, we could also consider Banach spaces; the proofs may be somewhat harder, in this case. An operator T from H1 into H2 is said to be closed if its graph G(T) (cf. Section 4.4) in H1 × H2 is closed. An operator T is said to be closable if \(\overline {G(T)} \) is a graph. From the proof of Theorem 4.15 we know that there exists then a uniquely determined operator \(\overline T \) such that \(G(\overline T )\, = \,\overline {G(T)} ;\,\overline T \) is closed and is called the closure of T.

On the Spectra of Selfadjoint Extensions

Abstract In this paper we give an account of some recent results on the existence of wave operators for very general unperturbed operators and perturbations. In section 5 we give a completeness result under similar conditions for the case of dimension 1.

The spectral theory of self-adjoint and normal operators

For selfadjoint extensions of closed symmetric operators with a gap and infinite deficiency indices the paper summarizes results on spectra which may occur inside the gap. Generalizing a result of Krein it is found that an arbitrary point spectrum can be generated by extensions inside the gap. In particular, a dense point spectrum is possible. Moreover, modulo a discrete spectrum every purely singular or absolutely continuous spectrum can be obtained provided the operator is an significantly symmetric one and the spectrum is a regular set. Finally, certain combinations of all three kinds of spectra are realizable.