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.


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


Zbl 1162.82031
Full Text: DOI Euclid