Comparison of probability and classical methods for the Schrödinger equation. (English) Zbl 0623.60097

This is a brief expository paper in which a probabilistic proof [of the first author and K. M. Rao, see C. R. Acad. Sci., Paris, Sér. A 290, 629-631 (1980; Zbl 0439.31005) and Stochastic processes, Semin. Evanston/Ill. 1981, Progr. Probab. Stat. 1, 1-29 (1981; Zbl 0492.60073)] of an elliptic boundary result is given side by side with a p.d.e. proof. For D a bounded domain in \(R^ d\), \(q: D\to R\), and \(f: \partial D\to R\), we say that \(\phi\) is a solution of the Schrödinger boundary value problem (D,q,f) if \(A(\Delta /2+q)\phi =0\) in D, \(\phi =f\) on \(\partial D\) (omitting all regularity assumptions). The authors are concerned with the equivalence of (i), (ii) and (iii):
(i) A function called the gauge is always finite on D;
(ii) There exists a positive solution of (D,q,1);
(iii) The top eigenvalue of \((\Delta /2+q)\) is strictly negative.
Reviewer: W.R.Darling


60J45 Probabilistic potential theory
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35J10 Schrödinger operator, Schrödinger equation
60H25 Random operators and equations (aspects of stochastic analysis)