James S. Howland

The Livsic matrix in perturbation theory

In 1957, M. S. Livsic introduced into scattering theory a certain dissipative operator-valued function B(x) [22-251. His theorem, which relates the characteristic function of B(z) to a certain S-matrix, is based on a simple formula for the compression of a resolvent to a subspace. This formula, proved below as Proposition 1, and the matrix B(z) apparently have been rediscovered several times, and occur in the physics literature under a variety of names. The earliest reference known to the author‘is [22], and we shall therefore refer to B(z) as the Livsic matrix. The purpose of the present article is to indicate an application of this formula to the perturbation theory of embedded eigenvalues. A perturbation of an isolated eigenvalue may be called regular if the perturbed resolvent is analytic in the perturbation parameter K. The perturbed eigenvalues then have convergent Puiseux expansions in K. The Livsic matrix leads to a natural extension of the idea of a regular perturbation to eigenvalues which are not isolated, in which the Puiseux series describe resonances rather than eigenvalues [14]. Moreover, calculation of the coefficients of these series is reduced to the known procedures of the theory of isolated eigenvalues. A somewhat similar reduction, relying on dilatation analyticity, was obtained by Simon [28] for Balslev-Combes Hamiltonians. The general properties of B(z) are given in Section 1, and Livsic’s Theorem is proved in an abstract framework. A general theorem on resonnaces and spectral concentration is proved in Section 2. Regular perturbations of embedded eigenvalues occupy the next two sections, which include a remark on the breakdown of the ordinary perturbation method. The final section indicates how eigenvalues which are not regular can often be discussed, and gives an example in which spectral concentration occurs, but there is no analytic family of resonances. The method used here may be considered as a combination of the two approaches to the problem referred to by Simon in his introduction [28]. This article by no means exhausts the applications of the Livsic matrix, and the author recommends its use to other writers on perturbation theory.

Perturbation theory of dense point spectra

Abstract The property of a self-adjoint operator having pure point spectrum is stable under certain random compact perturbations. In particular, a compact perturbation of a self-adjoint operator whose spectrum has Lebesgue measure zero has pure point spectrum, almost surely.

On the Weinstein-Aronszajn formula

(with poles counted as zeros of negative multiplicity). This result was extended by KURODA [12] to perturbations V in the trace class cg 1. We shall generalize this result in several directions. In the first place, we shall consider possibly unbounded symmetric perturbations of the form V=B*A, where the operator Q(z)=AG(z)B is compact. (See [3, 5, 9, 11, 13] and the references given in these papers.) In this case, we show that in the neighborhood of any point 2o as above, a meromorphic function A (z) can be defined which serves the same purpose as the Weinstein determinant. The proof uses a construction of STEINBERG [15]. This is carried out in w167 1-2. In w 4, we extend Kurodas result by showing that if Q(z)e

Complex scaling of ac Stark Hamiltonians

For a two‐body atom in a temporally periodic, spatially uniform field, it is shown that in an appropriate gauge the essential spectrum of the Floquet Hamiltonian rotates about a certain set of thresholds when subjected to a complex scaling transformation.

Perturbation of embedded eigenvalues by operators of finite rank

In this classical paper of 1948, K. 0. Friedrichs considered an example of a self-adjoint operator T,, having a point eigenvalue embedded in its continuous spectrum which disappears under a perturbation EL’ of rank 2 [l, Sections 6-8 and Section IO]. Friedrichs analyzed the perturbed operator T, = TO

On a theorem of Ismagilov

EV, and showed that although there was no eigenvalue of T, the density function of the spectral measure of T, was peaked, for small positive E, near the solution h, of the formal perturbation equations-an example of the “spectral concentration” phenomenon, which has been discussed by several authors for isolated eigenvalues. (See [2], [3, Chapt. VIII] and [4].) In the present paper, as a first step toward a general analysis of the problem, we shall consider the case of a perturbation Y of arbitrary finite rank. First, we shall give sufficient conditions under which an embedded eigenvalue h, of T,, of simple multiplicity vanishes under perturbation by I’ (Theorem I). These conditions state essentially that (a) except for the point mass at /\a , the spectral measure of To is rather smooth near ha, and (b) there is, in a certain sense, no “degeneracy” of I’ at h, . Under these conditions, the part of T,, on the orthogonal complement of the eigenspace of A,, is absolutely continuous near ha and is unitarily equivalent near h, to the perturbed operator T = T,, + I? The main tool in the proof of this, and the subsequent theorems is the formula (1.6) for the inverse of the Weinstein-Aronszajn matrix W(z). We do not employ the contour integral techniques used by Friedrichs [I]. Secondly, we shall consider the asymptotic properties of the family of operators T, = T,, + EV as E -+ 0 + . We shall give sufficient conditions that Theorem 1 apply for all sufficiently small positive e (Theorem 1), and -__* This work was supported by NSF Institutional Grant GU-1978,

A note on spectral concentration for non-isolated eigenvalues☆

Abstract If A and B are bounded selfadjoint operators and AB is trace class, then the absolutely continuous part of A + B is unitarily equivalent to the direct sum of the absolutely continuous parts of A and B .

On a theorem of Carey and Pincus

Abstract A classical sufficient condition for solution of the perturbation equations to order p is extended to the case of non-isolated eigenvalues, and a new proof is given.

Imaginary part of a resonance in barrier penetration

Abstract If H is self-adjoint and spectrally singular with respect to Lebesgue measure, then there exists a self-adjoint V of arbitrarily small trace norm such that H + V has pure point spectrum. If the multiplicity of H does not exceed r , then V may be chosen of rank r .

Dilations and Mehler's kernel

Abstract A simple model of quantum mechanical barrier penetration is discussed in which a certain resonance converges to a true bound state energy of a limiting Hamiltonian as the width R of the barrier tends to infinity. An asymptotic formula for the imaginary part of this resonance for large R is obtained, and compared with the standard quasi-classical approximation.