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Asymptotic Analysis

Time evolution of quantum resonance states

Markus Klein, J. Rama, Prof. Dr. rer. nat. Rainer Wüst

Published Jan 1, 2007• 1 citation
Publisher

Abstract

Let H 0 be self-adjoint with E 0 a possibly degenerate eigenvalue embedded in the continuous spectrum σ c (H 0 ) and ψ 0 a normalized eigenfunction. For a certain class of perturbations W and H=H 0 +W we investigate the asymptotics of the (naive) resonance state e −itH ψ 0 in the limit W→0. This amplifies previous results of Merkli and Sigal.

DOI
10.3233/asy-2007-798

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Let H 0 be self-adjoint with E 0 a possibly degenerate eigenvalue embedded in the continuous spectrum σ c (H 0 ) and ψ 0 a normalized eigenfunction. For a certain class of perturbations W and H=H 0 +W we investigate the asymptotics of the (naive) resonance state e −itH ψ 0 in the limit W→0. This amplifies previous results of Merkli and Sigal.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.
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In a previous paper we began our analysis on the role of non self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are {\em brute-force normalized}. Our main interest is on conserved quantities, and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.
It remains a difficult task to establish contact between the rigorous quantum theory built on sound foundations by Birkhoff and von Neumann and others and the models usually used to describe resonances. A theorem is proved which gives a set of necessary and sufficient conditions for the existence of solutions of formal perturbation and kindred equations in atomic physics and establishes certain uniqueness properties of solutions which exist. In another theorem it is proved that whenever a state vector belonging to a point eigenvalue of the Hamiltonian is perturbed, one has the phenomenon of spectral concentration and it is possible that in cases where the resulting spectral concentration does not constitute a point eigenvalue of the perturbed Hamiltonian, one has the phenomenon which probably corresponds to a resonance. It is shown that in the autoionising 2s2p 1.3 P states of helium, a solution of the formal perturbation equations exists in the underlying Hilbert space only in the zero order, though a formal solution which is outside the underlying Hilbert space exists and belongs to the space of distributions on a subset of the Hilbert space. It is also shown that in cases where a complex eigenvalue is obtained by interfering with a self-adjoint operator, the phenomenon is not related to deficiency indices. The mathematical background of the work of Sharma and Bowtell (1975) has been examined and some progress has been made towards finding the rigorous mathematical framework in which the theory should find a proper formulation.