On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete Riemannian manifolds
type: Item
instance of: scholarly article
title: On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete Riemannian manifolds
zbMATH Open document ID: 0848.35031
author: Yanyan Li
published in: Mathematical Notes
publication date: 1995-07-10T00:00:00Z
review text: The author studies on a complete Riemannian \(C^\infty\)-manifold \(M\) of dimension \(n\) the Schrödinger operator \(H= -\Delta+ q(x)\), \(x\in M\), in the space \(L^2(M)\), where \(\Delta\) is the Laplace-Beltrami operator, and the potential \(q\in L^\infty_{\text{loc}}(M)\) is a real-valued measurable function. The main result: let the potential \(q\) satisfy the inequality \(q(x)\geq - Q(x)\), where \(1\leq Q(x)\leq \infty\) and \(Q^{- 1/2}\) is a Lipschitz function on the manifold \(M\). Then, if for any arbitrary piecewise-smooth curve \(l\) outgoing to infinity \(\int_l Q^{- 1/2}(x) dl= \infty\), the operator \(H|_{C^\infty_0(M)}\) is essentially self-adjoint.
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 771971
zbMATH Keywords: essentially self-adjoint operator
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Lipschitz function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 771971