Discreteness of spectrum and positivity criteria for Schrödinger operators. (English) Zbl 1106.35043

Maz’ya, Vladimir; Shubin, Mikhail

Ann. Math. (2) 162, No. 2, 919-942 (2005).

Consider the Schrödinger operator \(-\Delta+V\) in \(L^2(\mathbb{R}^n)\) with a locally integrable potential \(V\). As was proved by K. Friedrichs, the spectrum of this operator is discrete if \(V(x)\to+\infty\) as \(| x| \to \infty.\) As was observed by A. M. Molchanov [Tr. Mosk. Mat. Obshch. 2, 169–199 (1953; Zbl 0052.10201)], if \(V\) is semi-bounded below, then the discreteness of the spectrum easily implies that for every \(d>0\) \[ \int_{Q_d}V(x)\,dx\to+\infty, \tag{1} \] as \(Q_d\to\infty\), where \(Q_d\) is an open cube with edge length \(d\) and with edges parallel to the coordinate axes. Here \(Q_d\to\infty\) means that the cube goes to infinity with fixed \(d\).
Molchanov also showed that this condition is in fact necessary and sufficient in case \(n=1\) but not sufficient for \(n\geqslant 2\). He discovered a modification of condition (1) which is fully equivalent to the discreteness of the spectrum in the case \(n\geqslant 2\). It states that for every \(d>0\) \[ \inf_{F}\int_{Q_d\setminus F}V(x)\,dx\to+\infty \] as \(Q_d\to\infty\), where the infimum is taken over all compact subsets \(F\) of the closure \(\overline{Q}_d\), which are called negligible. The negligibility of \(F\) means that cap\((F)\leqslant \gamma \text{cap}(Q_d),\) where cap is the Wiener capacity and \(\gamma >0\) is a sufficiently small constant. More precisely, Molchanov proved that we can take \(\gamma=c_n\) where for \(n\geqslant 3\) \[ c_n=(4n)^{-4n}( \text{cap}(Q_1))^{-1}. \]
In 1953, I. M. Gel’fand raised the question about the best possible constant \(c_n\). In the paper under review the authors answer this question by proving that \(c_n\) can be replaced by an arbitrary constant \(\gamma\), \(0<\gamma<1.\) The authors also derive a stronger result, allowing negligibility conditions of the form \(\text{cap}(F)\leqslant \gamma(d)\text{cap}(Q_d)\) and completely describing all admissible functions \(\gamma\). The results of the authors are even proved in a more general context.

Reviewer: Maksim Sokolov (Tashkent)

MSC:

35P20	Asymptotic distributions of eigenvalues in context of PDEs
35Q40	PDEs in connection with quantum mechanics
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
35J10	Schrödinger operator, Schrödinger equation

Keywords:

Schrödinger operator; locally integrable potentials; discrete; Wiener capacity