Discreteness of spectrum and positivity criteria for Schrödinger operators.

*(English)*Zbl 1106.35043Consider 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.

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)