## 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