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Asymptotic expansion of resolvent kernels and behavior of spectral functions for symmetric stable processes. (English) Zbl 1381.60088

In this paper, the author studies the asymptotic behavior of spectral functions for a rotationally invariant \(\alpha\)-stable Lévy process \(\{X_t\}_{t\geq 0}\) on \(\mathbb R^d\) with \(0<\alpha<2\). Denote by \(H=(-\Delta)^{\alpha/2}\) the infinitesimal generator of \(X_t\) and let \((\mathcal E,\mathcal F)\) denote the associated Dirichlet form on \(L^2(\mathbb R^d)\). Let \(V\) be a nonnegative continuous function on \(\mathbb R^d\) with compact support and consider the Schrödinger-type operator \(H^\lambda=H-\lambda V\) with \(\lambda\geq 0\). The bottom of the spectrum of the operator \((1/V)H\) is defined by \[ \lambda_V=\inf\left\{\mathcal E(u,u):u\in\mathcal F,\int_{\mathbb R^d}u^2(x)V(x)dx=1\right\}. \] The spectral function \(C(\lambda)\) is defined by \[ C(\lambda)=-\inf\left\{\mathcal E(u,u)-\lambda\int_{\mathbb R^d}u^2(x)V(x)dx:u\in\mathcal F,\int_{\mathbb R^d}u^2(x)dx=1\right\}, \] i.e., \(-C(\lambda)\) is the bottom of the spectrum of the operator \(H^\lambda\). The author gives precise asymptotics of the function \(C(\lambda)\) as \(\lambda\downarrow 0\) or \(\lambda\downarrow\lambda_V\), depending on whether the process \(X_t\) is recurrent or transient. This extends [M. Cranston et al., J. Funct. Anal. 256, No. 8, 2656–2696 (2009; Zbl 1162.82031)], where the same problem was studied for the Brownian motion.

MSC:

60G52 Stable stochastic processes
60J45 Probabilistic potential theory
35C20 Asymptotic expansions of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation

Citations:

Zbl 1162.82031
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