Abstract
The eigenvalues and eigenfunctions of the one-dimensional quasi-relativistic Hamiltonian (-ℏ2c2d2/dx2 + m2c4)1/2 + V well (x) (the Klein–Gordon square-root operator with electrostatic potential) with the infinite square well potential V well (x) are studied. Eigenvalues represent energies of a "massive particle in the box" quasi-relativistic model. Approximations to eigenvalues λn are given, uniformly in n, ℏ, m, c and a, with error less than C1ℏca-1 exp (-C2ℏ-1mca)n-1. Here 2a is the width of the potential well. As a consequence, the spectrum is simple and the nth eigenvalue is equal to (nπ/2 - π/8)ℏc/a + O(1/n) as n → ∞. Non-relativistic, zero mass and semi-classical asymptotic expansions are included as special cases. In the final part, some L2 and L∞ properties of eigenfunctions are studied.
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