Spectral gap lower bound for the one-dimensional fractional Schr\"odinger operator in the interval

We prove the uniform lower bound for the difference $\lambda_2 - \lambda_1$ between first two eigenvalues of the fractional Schr\"odinger operator, which is related to the Feynman-Kac semigroup of the symmetric $\alpha$-stable process killed upon leaving open interval $(a,b) \in \R$ with symmetric differentiable single-well potential $V$ in the interval $(a,b)$, $\alpha \in (1,2)$. "Uniform" means that the positive constant appearing in our estimate $\lambda_2 - \lambda_1 \geq C_{\alpha} (b-a)^{-\alpha}$ is independent of the potential $V$. In general case of $\alpha \in (0,2)$, we also find uniform lower bound for the difference $\lambda_{*} - \lambda_1$, where $\lambda_{*}$ denotes the smallest eigenvalue related to the antisymmetric eigenfunction $\phi_{*}$. We discuss some properties of the corresponding ground state eigenfunction $\phi_1$. In particular, we show that it is symmetric and unimodal in the interval $(a,b)$. One of our key argument used in proving the spectral gap lower bound is some integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey-Lemma.Comment: 23 pages, 2 figure

Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes

We introduce a class of L\'{e}vy processes subject to specific regularity conditions, and consider their Feynman-Kac semigroups given under a Kato-class potential. Using new techniques, first we analyze the rate of decay of eigenfunctions at infinity. We prove bounds on $\lambda$-subaveraging functions, from which we derive two-sided sharp pointwise estimates on the ground state, and obtain upper bounds on all other eigenfunctions. Next, by using these results, we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions. We establish the relationships of these properties, derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the L\'{e}vy density and the potential at infinity, define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations. These results are amply illustrated by key examples.Comment: Published at http://dx.doi.org/10.1214/13-AOP897 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Transition in the decay rates of stationary distributions of L\'evy motion in an energy landscape

The time evolution of random variables with L\'evy statistics has the ability to develop jumps, displaying very different behaviors from continuously fluctuating cases. Such patterns appear in an ever broadening range of examples including random lasers, non-Gaussian kinetics or foraging strategies. The penalizing or reinforcing effect of the environment, however, has been little explored so far. We report a new phenomenon which manifests as a qualitative transition in the spatial decay behavior of the stationary measure of a jump process under an external potential, occurring on a combined change in the characteristics of the process and the lowest eigenvalue resulting from the effect of the potential. This also provides insight into the fundamental question of what is the mechanism of the spatial decay of a ground state

Fall-off of eigenfunctions for non-local Schr\"odinger operators with decaying potentials

We study the spatial decay of eigenfunctions of non-local Schr\"odinger operators whose kinetic terms are generators of symmetric jump-paring L\'evy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the L\'evy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the L\'evy intensity. We furthermore prove that under reasonable conditions the L\'evy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the L\'evy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition L\'evy intensity-driven decay becomes slower than the rate of decay of L\'evy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.Comment: to appear in Potential Analysis, 40 page

Upper estimates of transition densities for stable-dominated semigroups

We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Levy processComment: arXiv admin note: text overlap with arXiv:0903.529