One-dimensional quasi-relativistic particle in the box

Abstract

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x) (the Klein-Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) is given: the n-th eigenvalue is equal to (n pi/2 - pi/8) h c/a + O(1/n), where 2a is the width of the potential well. Simplicity of eigenvalues is proved. Some L^2 and L^infinity properties of eigenfunctions are also studied. Eigenvalues represent energies of a `massive particle in the box' quasi-relativistic model.

40 pages, 4 figures; minor corrections

References in corpus (6)

• Heat kernel estimates for the fractional Laplacian with Dirichlet conditions
• Spectral properties of the Cauchy process
• Two-sided Green function estimates for killed subordinate Brownian motions
• Sharp heat kernel estimates for relativistic stable processes in open sets
• Spectral analysis of subordinate Brownian motions in half-line
• First passage times for subordinate Brownian motions

Cited by in corpus (8)

• Comment on "Fractional quantum mechanics" and "Fractional Schroedinger equation"
• Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well
• Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials
• Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well
• Spectral theory for one-dimensional symmetric Levy processes killed upon hitting the origin
• The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach
• Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval
• Ultrarelativistic bound states in the spherical well