Comparison of probability and classical methods for the Schrödinger equation (Q1091682)

This is a brief expository paper in which a probabilistic proof [of the first author and \textit{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.