Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator

TL;DR

This paper explores methods for constructing matrix intertwining operators for non-Hermitian Hamiltonians, demonstrating how they can generate new Hamiltonians with added bound states, including cases with multiple eigenstates at the same energy.

Contribution

It introduces new methods for constructing first-order matrix intertwining operators and analyzes their relations, expanding the toolkit for matrix Hamiltonian transformations.

Findings

Constructed 2x2 matrix Hamiltonian intertwined with zero potential.

Showed how to add up to two bound states at different energies.

Demonstrated addition of bound states with same energy via eigenfunctions.

Abstract

We study intertwining relations for $n\times n$ matrix non-Hermitian, in general, one-dimensional Hamiltonians by $n\times n$ matrix linear differential operators with nondegenerate coefficients at $d/dx$ in the highest degree. Some methods of constructing of $n\times n$ matrix intertwining operator of the first order of general form are proposed and their interrelation is examined. As example we construct $2\times2$ matrix Hamiltonian of general form intertwined by operator of the first order with the Hamiltonian with zero matrix potential. It is shown that one can add for the final $2\times2$ matrix Hamiltonian with respect to the initial matrix Hamiltonian with the help of intertwining operator of the first order either up to two bound states for different energy values or up to two bound states described by vector-eigenfunctions for the same energy value or up to two bound states described by vector-eigenfunction and associated vector-function for the same energy value.