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Comparison of probability and eigenvalue methods for the Schrödinger equation. (English) Zbl 0608.60061

Adv. Math., Suppl. Stud. 9, 25-34 (1986).
Let D be a bounded domain in \({\mathbb{R}}^ d\), \(d\geq 1\) and a function q be bounded in D satisfying a Hölder condition on D. We say that \(\phi\) is a solution of the Schrödinger boundary value problem (D,q,f) if \(\phi \in C^{(2)}(D)\cap C^{(0)}(\bar D)\), \((\Delta /2+q)\phi =0\) in D, and \(\phi =f\) on \(\partial D\). Such a solution is called ”positive” if \(\phi >0\) in \(\bar D.\) If f is not specified, such a problem will be denoted by (D,q).
Denote by \(\lambda_ 1\) the maximum eigenvalue of the operator \(L=\Delta /2+q\). Consider the following propositions: (i) \(\phi\) (D,q,1,\(\cdot)\not\equiv \infty\) in D; (ii) there exists a positive solution of (D,q); (iii) \(\lambda_ 1<0.\)
The main result of the paper is the following Theorem: The three propositions (i), (ii), and (iii) are equivalent. Comparison of probability and eigenvalue methods for the Schrödinger boundary value problem has been carried out.
Reviewer: G.Derfel

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
35J10 Schrödinger operator, Schrödinger equation
35J25 Boundary value problems for second-order elliptic equations