Spin dependent point potentials in one and three dimensions

TL;DR

This paper characterizes all spin-dependent point potentials in one and three dimensions for a quantum particle interacting with an array of spins, providing explicit resolvent formulas and examples of solvable models.

Contribution

It introduces a comprehensive classification of spin-dependent point interactions via boundary conditions and supplies explicit resolvent formulas for these Hamiltonians.

Findings

Explicit formulas for resolvents of spin-dependent point potentials.

Characterization of all such Hamiltonians via boundary conditions.

Examples of solvable multi-component models.

Abstract

We consider a system realized with one spinless quantum particle and an array of $N$ spins 1/2 in dimension one and three. We characterize all the Hamiltonians obtained as point perturbations of an assigned free dynamics in terms of some ``generalized boundary conditions''. For every boundary condition we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin dependent point potentials that could be of interest as multi-component solvable models.