Abstract Generated abstract
This note studies the negative spectrum of the multidimensional Schrödinger operator Mh u = minus h Delta u minus P(x)u in L2(Rn), for n at least 3 and nonnegative locally bounded potential P. Using capacity estimates for condensers and closed sets, it formulates criteria for positivity inequalities, compactness of Sobolev embeddings into weighted L2 spaces, and Friedrichs’ inequality. The main results give necessary and sufficient capacity conditions for the negative spectrum of Mh to be discrete, finite, or infinite for every h greater than 0, extending earlier sufficient conditions and relating the spectral problem to embedding compactness.
Full Text
MATHEMATICS
V. G. MAZ’YA
ON THE NEGATIVE SPECTRUM OF A MULTIDIMENSIONAL SCHRÖDINGER OPERATOR
(Presented by Academician V. I. Smirnov, January 20, 1962)
In the present note we formulate necessary and sufficient conditions for the discreteness, finiteness, and infiniteness, for all \(h>0\), of the negative spectrum of the operator
\[ M_h u=-h\Delta u-P(x)u \qquad (P(x)\geqslant 0) \]
in \(L_2(R_n)\) \((n\geqslant 3)\).* Here \(R_n\) is \(n\)-dimensional Euclidean space.
Let \(\Omega\) be an open subset of \(R_n\). By \(\overset{\circ}{C}{}^{(1)}(\Omega)\) we denote the set of functions continuously differentiable in \(\Omega\) and equal to zero outside some ball and near the boundary of \(\Omega\). By the space \(\overset{\circ}{L}{}^{(1)}_2(\Omega)\) \(\bigl(\overset{\circ}{W}{}^{(1)}_2(\Omega)\bigr)\) we shall mean the closure of \(\overset{\circ}{C}{}^{(1)}(\Omega)\) in the metric
\[ \int_{\Omega}(\operatorname{grad}u)^2\,dx \qquad \left( \int_{\Omega}(\operatorname{grad}u)^2\,dx+\int_{\Omega}u^2\,dx \right). \]
We also introduce the space \(L_2(P(x);\Omega)\) of functions for which
\[ \|u\|^2_{L_2(P(x);\Omega)}=\int_{\Omega}P(x)u^2\,dx<\infty . \]
Let \(E\) be an arbitrary bounded closed subset of \(\Omega\) with boundary \(\Gamma E\). By \(c(\Omega\setminus E)\) we shall mean the capacity of the condenser \(\Omega\setminus E\), i.e. the number
\[ \inf \frac{1}{(n-2)\omega_n}\int_{\Omega\setminus E}(\operatorname{grad}u)^2\,dx, \]
where \(\omega_n\) is the surface area of the unit \((n-1)\)-dimensional sphere and the infimum is taken over all functions
\(u(x)\in C(\Omega\setminus E)\cap C^{(1)}(\Omega\setminus E)\) equal to one on \(\Gamma E\) and to zero near \(\Gamma\Omega\) and outside some ball. By \(\operatorname{cap} E\) we denote the capacity of the set \(E\), i.e. \(c(R_n\setminus E)\).
In Theorem 1 conditions are given for the positivity of the operator \(M_1\).
Theorem 1. For the validity of the inequality
\[ \int_{\Omega}P(x)u^2\,dx\leqslant \int_{\Omega}(\operatorname{grad}u)^2\,dx, \tag{1} \]
where \(u(x)\in \overset{\circ}{L}{}^{(1)}_2(\Omega)\), it is sufficient that for every bounded closed set \(E\subseteq\Omega\) the inequality
\[ \int_E P(x)\,dx\leqslant \frac{n-2}{4}\,\omega_n c(\Omega\setminus E). \tag{2} \]
hold. The inequality
\[ \int_E P(x)\,dx\leqslant (n-2)\omega_n c(\Omega\setminus E) \tag{3} \]
is necessary for the validity of (1).
* An exhaustive bibliography on the questions considered can be found in \((^1)\).
Corollary. For Friedrichs’ inequality
\[ \int_{\Omega} u^2\,dx \leq \operatorname{const}\int_{\Omega}(\operatorname{grad}u)^2\,dx \quad \left(u\in \overset{\circ}{L}{}^{(1)}_{2}(\Omega)\right) \tag{4} \]
to hold, it is necessary and sufficient that the condition
\[ \sup_{E\subset\Omega}\frac{\operatorname{mes}_{n}E}{\operatorname{cap}(\Omega\setminus E)}<\infty \tag{5} \]
be satisfied.
In what follows we shall assume that the function \(P(x)\) is bounded in every ball of the space \(R_n\).
Theorem 2. For complete continuity of the embedding operator \(\overset{\circ}{W}{}^{(1)}_{2}(R_n)\) into \(L_2(P(x);R_n)\), it is necessary and sufficient that the condition
\[ \lim_{R\to\infty}\ \sup_{\substack{E\subset C S_R,\\ d(E)\leq 1}} \frac{\displaystyle\int_E P(x)\,dx}{\operatorname{cap}E}=0, \tag{6} \]
be satisfied, where the \(\sup\) is taken over closed sets \(E\) of diameter \(d(E)\leq 1\), situated outside the ball \(S_R\) of radius \(R\) with center at the origin.
Theorem 3. For complete continuity of the embedding operator \(\overset{\circ}{L}{}^{(1)}_{2}(R_n)\) into \(L_2(P(x);R_n)\), it is necessary and sufficient that the condition
\[ \lim_{R\to\infty}\ \sup_{E\subset C S_R} \frac{\displaystyle\int_E P(x)\,dx}{\operatorname{cap}E}=0, \tag{7} \]
be satisfied, where the \(\sup\) is taken over bounded closed sets \(E\), situated outside the ball \(S_R\) of radius \(R\) with center at the origin.
From the general theorem of M. Sh. Birman \((^{1})\) it follows that Theorems 2 and 3 admit the following equivalent formulations.
Theorem \(2'\). In order that the negative spectrum of the operator \(M_h\) be discrete for all \(h>0\), it is necessary and sufficient that condition (6) be satisfied.
Theorem \(3'\). In order that the negative spectrum of the operator \(M_h\) be finite for all \(h>0\), it is necessary and sufficient that condition (7) be satisfied.
The following theorem gives a criterion for infinitude of the negative spectrum of the operator \(M_h\) for all \(h>0\).
Theorem 4. The condition
\[ \sup_{E\subset R_n} \frac{\displaystyle\int_E P(x)\,dx}{\operatorname{cap}E}=\infty \tag{8} \]
is necessary and sufficient for the infinitude of the negative spectrum of the operator \(M_h\) for all \(h>0\).
We note that some sufficient conditions for discreteness, finiteness, and infinitude of the negative spectrum of the operator \(M_h\) for all \(h>0\) were given by M. Sh. Birman \((^{1})\).
Related in subject to the present note is the paper of A. M. Molchanov \((^{2})\), in which a necessary and sufficient condition was found for discreteness of the spectrum of the Schrödinger operator \(-\Delta u+Q(x)u\) with potential \(Q(x)\) bounded below.
Leningrad State University
named after A. A. Zhdanov
Received
18 I 1962
REFERENCES
\(^{1}\) M. Sh. Birman, Mat. sborn., 55, No. 22 (1961).
\(^{2}\) A. M. Molchanov, Tr. Mosk. matem. obshch., 2, 169 (1953).