A probabilistic approach to bounded/positive solutions for Schrödinger operators with certain classes of potentials

Pinsky, Ross

doi:10.1090/S0002-9947-08-04473-5

A probabilistic approach to bounded/positive solutions for Schrödinger operators with certain classes of potentials
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by Ross G. Pinsky PDF

Trans. Amer. Math. Soc. 360 (2008), 6545-6554 Request permission

Abstract:

Consider the equation \[ {(^*)\qquad \qquad \qquad \qquad \qquad \frac 12\Delta u-Vu=0 \text { in }R^d, \qquad \qquad \qquad \qquad \qquad \qquad }\] for $d\ge 3$. For certain classes of potentials $V$, we use probabilistic tools to study the bounded solutions and the positive solutions for (*). A primary motivation is to offer probabilistic intuition for the results.

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