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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

MSC:

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)
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