On the almost unperturbed Schrödinger pair of operators
type: Item
instance of: scholarly article
title: On the almost unperturbed Schrödinger pair of operators
zbMATH Open document ID: 0678.47032
DOI: 10.1007/BF01195116
published in: Integral Equations and Operator Theory
publication date: 1989-01-01T00:00:00Z
review text: A pair (U,V) of (unbounded) self-adjoint operators on a Hilbert space is called an almost unperturbed Schrödinger pair with parameter \(a\in {\mathbb{R}}\setminus \{0\}\) if there is a trace class operator D such that \[ i[U,V]\subset aI+D \] (where the domain of the commutator [U,V] is assumed to be large enough). This notion is related to the classical Schrödinger pair (satisfying the Heisenberg commutation relation, i.e. \(D=0)\) and to the case of almost commuting operators (i.e., \(a=0)\). The theory of pairs with these properties is well established, and in this paper the author starts to develop the theory of almost unperturbed Schrödinger pairs. It is shown, e.g., that under certain conditions an almost unperturbed Schrödinger pair has a singular integro-differential model, i.e., U is multiplication by the independent variable on \(L_ 2({\mathbb{R}})\) whereas V is differentiation plus a certain perturbation. Moreover, results on symbols, the principal distribution, a trace formula, and the determining function are established. In the last section perturbations of n-parameter groups of unitary operators are investigated.
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4111387
zbMATH Keywords: almost unperturbed Schrödinger pair
zbMATH Keywords: trace class operator
zbMATH Keywords: classical Schrödinger pair
zbMATH Keywords: Heisenberg commutation relation
zbMATH Keywords: singular integro- differential model
zbMATH Keywords: symbols
zbMATH Keywords: principal distribution
zbMATH Keywords: trace formula
zbMATH Keywords: perturbations of n-parameter groups of unitary operators
author: Daoxing Xiao
reviewed by: Jürgen Voigt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4111387