Incompatible Coulomb hamiltonian extensions

G Abramovici

doi:10.1038/s41598-020-62144-2

Abstract

We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a Coulomb λ/|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the one-dimensional semi-infinite case, we show that they are classified on a U(1) circle in the attractive case and on {boldsymbol{(}}{mathbb{R}},{boldsymbol{+}}{boldsymbol{infty }}{boldsymbol{)}} in the repulsive one. In the one-dimensional infinite case, we find a specific and original classification by studying the continuity of eigenfunctions. In all cases, different extensions are incompatible one with the other. For an actual experiment with an attractive potential, the bound spectrum can be used to discriminate which extension is the correct one.

Highlights

When ω → ±∞, one gets the Dirichlet condition φe(0) = 0. This can be shown by examining the limit φk(0+) = Dη/Γ(1 + η), which we give in section ‘Classification in the attractive case’ and which is valid in the repulsive case
Complex eigenvalues for hamiltonians, skyrmions, Majorana fermions), advanced studies of non self-adjoint hamiltonians are necessary, and, what seemed old-fashioned physics reveals an essential source of inspiration and comprehension, to determinate whether a self-adjoint extension is valid or not

Summary

Abramovici

We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a Coulomb λ/|x| potential. This work lies at the frontier between physics and mathematics, because Coulomb hamiltonians H( +∗ ) and H( ), defined on a physical basis, reveal non self-adjoint. In such a case, one usually needs to study the self-adjoint extensions K of the hamiltonian. When the self-adjoint extension of an operator is unique, these mathematical manipulations are transparent because the spectral theorem applies, so the action of the operator is defined unambiguously on any function of L This is the case for almost all standard hamiltonians found in scientific literature, which are generally well defined without any restriction (that is L = L2( )), so one does not need to care about all these mathematical subtleties. For H( +∗ ), authors have found one continuous degree of freedom

Motivation

Discussion

Conclusion