Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture
type: Item
instance of: scholarly article
title: Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture
zbMATH Open document ID: 0879.34078
author: V. P. Maslov
author: Alexander Y. Gordon
author: S. Molchanov
published in: Russian Journal of Mathematical Physics
publication date: 1997-08-11T00:00:00Z
review text: The authors consider the stationary Schrödinger equation \(H\psi =E\psi\), where \(H\) is the Hamiltonian. It was postulated when the quantum mechanics theory was in its infancy that only square integrable solutions, or solutions bounded at infinity have physical meaning. Schrödinger conjectured that solutions can become unbounded only in the neighborhood of a singularity of \(H\), that is in the neighborhood of a particle. Variants of this statement can be found in several textbooks on quantum mechanics. Steklov's conjecture concerns a specific system of polynomials orthogonal on an interval \(\Delta\) with respect to the weight function \(\rho(\lambda)\), \(\lambda\in \Delta\), i.e. \(\int_\Delta [P_n(\lambda) P_m (\lambda) \rho(\lambda)] d\lambda =\delta (n-m)\). Steklov's conjecture states that \(|P_n (\lambda) |\leq C\Delta'\), with \(\lambda\in \Delta' \subset \text{int} \Delta\). The recurrence relation for these polynomials is \(\alpha_{n+1} P_{n+1} (\lambda) +\beta_n P_n(\lambda) +\alpha_{n-1} P_{n-1} (\lambda)=\lambda P_n(\lambda)\). With this relation the Steklov's conjecture becomes a discrete version of the Schrödinger conjecture. In a suitable Hilbert space, such as \(L^2 (\mathbb{R}^\nu)\) the operator \(H\) is essentially self-adjoint and allows spectral decomposition. The projector of this decomposition is an integral operator whose kernel is given in terms of solutions of the problem \(H\psi= E'\psi\), \(E'\leq E\). These solutions are known as generalized eigenfunctions. Of course the spectral decomposition of \(H\) is defined modulo sets of zero \(\sigma\)-measure. The authors outline the complicated history of Schrödingers and Steklov's conjectures. For a while Steklov's conjecture was regarded as a problem in algebra. George Szegö proved Steklov's conjecture under additional assumptions. K. Maurin and A. G. Kostiushenko published independently a theorem which implied that Schrödinger conjecture is true (in fact it implied more!) But their argum…
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47B80
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47N50
zbMATH DE Number: 1047127
zbMATH Keywords: stationary Schrödinger equation
zbMATH Keywords: Steklov's conjecture
reviewed by: Vadim Komkov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1047127