Decay of resolvent kernels and Schrödinger eigenstates for Lévy operators

TL;DR

This work analyzes the spatial decay of resolvent kernels for a broad class of symmetric Lévy operators and the corresponding bound states of non-local Schrödinger operators. By casting the problem in the convolution-semigroup/Lévy-Khintchine framework and introducing a profile function for the Lévy density, the authors establish a sharp dichotomy: subexponential Levy tails yield resolvent decay g_alpha(x) that matches the Levy density tail f(|x|), while exponential tails lead to directionally dependent decay with a sharp alpha-dependent threshold omega_star determining when g_alpha is comparable to nu. The paper proves directional and radial bounds for the heat kernel p_t and the resolvent g_alpha, and derives two-sided decay estimates for bound states and the ground state in both regimes, including explicit results for relativistic-type operators. Methodologically, the analysis relies on operator semigroup techniques and meticulous control of exponential moments, enabling a unified, non-probabilistic treatment of decay phenomena that extends classical results of Carmona-Masters-Simon and related works. The findings clarify the mechanisms behind the subexponential-to-exponential transition and the alpha-parameter dependence of exponential decay rates, with implications for the qualitative behavior of quantum states in non-local media.

Abstract

We study the spatial decay behaviour of resolvent kernels for a large class of non-local Lévy operators and bound states of the corresponding Schrödinger operators. Our findings naturally lead us to proving results for Lévy measures, which have subexponential or exponential decay, respectively. This leads to sharp transitions in the the decay rates of the resolvent kernels. We obtain estimates that allow us to describe and understand the intricate decay behaviour of the resolvent kernels and the bound states in either regime, extending findings by Carmona, Masters and Simon for fractional Laplacians (the subexponential regime) and classical relativistic operators (the exponential regime). Our proofs are mainly based on methods from the theory of operator semigroups.

Figures (1)

• Figure 1: The left panel shows the function $s\mapsto \omega(s\theta)$ for any fixed $\theta\in \mathds{S}^{d-1}$ in the relativistic case, i.e. $\Psi(\xi) = \sqrt{|\xi|^2+m^2}-m$ (note that we are in the radial case). It is strictly convex and continuous in $[0,\kappa]$, with finite and positive derivatives of any order in the interval $[0,\kappa)$. In the open interval $(\kappa,\infty)$ it is infinite. The (generalized) inverse $\alpha\mapsto\gamma_\alpha$ is shown in the right panel. Notice that $\gamma_\alpha$ is continuous in $[0,\omega(\kappa\theta)]$, and it is continuously extended by $\kappa = \gamma_{\omega(\kappa\theta)}$ on the interval $[\omega(\kappa\theta),\infty)$. Recall that the Schrödinger operator $H=-L+V$ serves as the energy operator (Hamiltonian) in the mathematical model describing the motion of a (relativistic) quantum particle. The kinetic term $-L = \Psi(\frac{1}{i} \nabla)$ is a positive, homogeneous pseudo-differential operator. Here, $\frac{1}{i} \nabla$ is the momentum operator in a multidimensional setting, consistent with the quantization rules of quantum mechanics. The threshold $\omega(\kappa\theta) = - \Psi(\frac{1}{i} \kappa\theta)$ seems to relate to the particle's kinetic energy. It is critical in the sense that $\omega(\xi) = \infty$ whenever $|\xi| > \kappa$.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2: sharp transition in the exponential rate
• Theorem 1.3: sharp radial estimates above threshold for general Lévy measures
• Corollary 2.1: Lévy densities with subexponential profiles
• Corollary 2.2: Lévy densities with exponential profiles
• Example 2.3: transition from the subexponential to the exponential regime
• Example 2.4: relativistic $\beta$-stable Lévy operators, resp., semigroups
• Lemma 3.1
• proof
• Lemma 3.2