Higher order eigenvalues for non-local Schrödinger operators

Jacob, Niels; Wang, Feng‐Yu

doi:10.3934/cpaa.2018012

Cited by 14 publications

(9 citation statements)

References 31 publications

(32 reference statements)

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“…Typically, they play the role of profiles and majorants for Lévy kernels and related objects. The most recent progress includes equations involving convolution operators [17,18]), heat kernels of unbounded pseudodifferential operators [3,11,26,27,28,30,31] and the corresponding Schrödinger operators [2,22,25], structure of the spectrum and properties of eigenfunctions of non-local Schrödinger operators [14,23,24,35]. Of course, this list of references is far from being complete.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On directional convolution equivalent densities

Kaleta¹,

Daniel²

2021

Preprint

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We propose a definition of directional multivariate subexponential and convolution equivalent densities and find a useful characterization of these notions for a class of integrable and almost radial decreasing functions. We apply this result to show that the density of the absolutely continuous part of the compound Poisson measure built on a given density f is directionally convolution equivalent and inherits its asymptotic behaviour from f if and only if f is directionally convolution equivalent. We also extend this characterization to the densities of more general infinitely divisible distributions on R d , d 1, which are not pure compound Poisson.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On directional convolution equivalent densities

Kaleta¹,

Daniel²

2021

Preprint

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“…Over the past decade, there has been an increasing interest in non-local models involving Schrödinger operators associated with generators of Lévy processes with non-degenerate jump measures. Recent investigations include heat kernel and heat trace estimates [1,3,6,54], gradient estimates of harmonic functions [44], properties of radial solutions, ground states, eigenfunctions and eigenvalues, and spectral bounds [5,13,20,21,28,32,36,46,47,53], intrinsic hyper-and ultracontractivity properties of Schrödinger semigroups [14,15,33,35,37] as well as applications in quantum field theory [11,12,23,24].…”

Section: Introduction Assumptions and Statement Of Main Resultsmentioning

confidence: 99%

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

Kaleta¹,

Schilling²

2019

Preprint

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We study the long-time asymptotic behaviour of semigroups generated by nonlocal Schrödinger operators of the form H = −L + V ; the free operator L is the generator of a symmetric Lévy process in R d , d > 1 (with non-degenerate jump measure) and V is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp. decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.

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“…An estimate related to that of Theorem V is proved recently in [27]. Further results on spectral theory of non-local Schrödinger operators can be found, among others, in [6,17,20,28,29,30,33,34,36,39,40].…”

Section: 2mentioning

confidence: 86%

Extension technique for complete Bernstein functions of the Laplace operator

Kwaśnicki

Mucha

2018

J. Evol. Equ.

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We discuss representation of certain functions of the Laplace operator ∆ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies (−∆) 1/2 , the square root of the d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the (d + 1)-dimensional Laplace operator ∆ t,x in (0, ∞) × R d . Caffarelli and Silvestre extended this to fractional powers (−∆) α/2 , which correspond to operators ∇ t,x (t 1−α ∇ t,x ). We provide an analogous result for all complete Bernstein functions of −∆ using Krein's spectral theory of strings.Two sample applications are provided: a Courant-Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrödinger operators ψ(−∆) + V (x), as well as an upper bound for the eigenvalues of these operators. Here ψ is a complete Bernstein function and V is a confining potential.

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Higher order eigenvalues for non-local Schrödinger operators

Cited by 14 publications

References 31 publications

On directional convolution equivalent densities

On directional convolution equivalent densities

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

Extension technique for complete Bernstein functions of the Laplace operator

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