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

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 $2\times 2$ 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 $2\times 2$ 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.

arXiv comment39 pages

References in corpus (32)

• Making Sense of Non-Hermitian Hamiltonians
• PT-Symmetric Quantum Mechanics
• Self-Isospectral Periodic Potentials and Supersymmetric Quantum Mechanics
• Nonlinear supersymmetry in Quantum Mechanics: algebraic properties and differential representation
• Nonlinear Supersymmetry for Spectral Design in Quantum Mechanics
• New possibilities for supersymmetry breakdown in quantum mechanics and second order irreducible Darboux transformations
• New Solvable and Quasi Exactly Solvable Periodic Potentials
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics
• The confluent algorithm in second order supersymmetric quantum mechanics
• Matrix Hamiltonians: SUSY approach to hidden symmetries
• Second order SUSY transformations with `complex energies'
• Nonlocal supersymmetric deformations of periodic potentials
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• Chains of Darboux transformations for the matrix Schroedinger equation
• SUSY approach to Pauli Hamiltonians with an axial symmetry
• New exactly solvable relativistic models with anomalous interaction
• Pauli equation and the method of supersymmetric factorization
• Matrix superpotentials
• Single- and coupled-channel radial inverse scattering with supersymmetric transformations
• Irreducible second order SUSY transformations between real and complex potentials
• Supersymmetry of a spin 1/2 particle on the real line
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results
• Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics. I: general classification of reducibility and analysis of the third-order algebra
• Higher Order Matrix SUSY Transformations in Two-Dimensional Quantum Mechanics
• N-fold Supersymmetry in Quantum Mechanical Matrix Models
• Singular matrix Darboux transformations in the inverse scattering method
• Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics
• Matrix Superpotential Linear in Variable Parameter
• Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator
• Polynomial Supersymmetry for Matrix Hamiltonians
• Factorization of non-linear supersymmetry in one-dimensional Quantum Mechanics. II: proofs of theorems on reducibility

Cited by in corpus (4)

• Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator
• Minimal Realizations of Supersymmetry for Matrix Hamiltonians
• Polynomial supersymmetry for matrix Hamiltonians: proofs
• Symmetries and Supersymmetries of Generalized Schrödinger equations