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Ergodic-type limit theorem for fundamental solutions of critical Schrödinger operators. (English) Zbl 1442.60077

Summary: Let \(\{X_t\}_{t \geq 0}\) be the symmetric \(\alpha \)-stable process with generator \(H = (-\Delta )^{\alpha /2}\) for \(0 < \alpha \leq 2\). For a positive Radon measure \(\mu \), we define the Schrödinger operator \(H^\mu = H - \mu \) and consider the fundamental solution of the equation \(\partial u/\partial t = - H^{\mu } u\). If \(\mu \) is critical, the behavior of the fundamental solution is different from that of the transition density function of \(\{X_t\}_{t \geq 0}\). In this paper, we give a certain ergodic-type limit theorem for fundamental solutions of critical Schrödinger operators.

MSC:

60J45 Probabilistic potential theory
60J40 Right processes
35J10 Schrödinger operator, Schrödinger equation
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