[Paper Review] Removal of the Energy Dependence from the Resolvent-like Energy-Dependent Interactions

А. К. Мотовилов|arXiv (Cornell University)|May 21, 1995

Spectral Theory in Mathematical Physics14 references4 citations

TL;DR

This paper establishes conditions under which energy-dependent interactions of the resolvent-like form $ V(z) = -B(A' - z)^{-1}B^* $ can be replaced by an energy-independent potential $ W $, such that the resulting Hamiltonian $ H = A + W $ shares the same discrete spectrum and eigenfunctions as the original energy-dependent problem. The key contribution is a rigorous spectral equivalence and the development of orthogonality, expansion theorems, and scattering theory for the transformed system.

ABSTRACT

The spectral problem $(A + V(z))\psi=z\psi$ is considered with $A$, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation $V(z)$ is assumed to depend on the energy $z$ as resolvent of another self-adjoint operator $A':$ $V(z)=-B(A'-z)^{-1}B^{*}$. It is supposed that operator $B$ has a finite Hilbert-Schmidt norm and spectra of operators $A$ and $A'$ are separated. The conditions are formulated when the perturbation $V(z)$ may be replaced with an energy-independent ``potential'' $W$ such that the Hamiltonian $H=A +W$ has the same spectrum (more exactly a part of spectrum) and the same eigenfunctions as the initial spectral problem. The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian $ H=A + W $. Scattering theory is developed for $H$ in the case when operator $A$ has continuous spectrum. Applications of the results obtained to few-body problems are discussed.

Motivation & Objective

• To eliminate the energy dependence in resolvent-like interactions $ V(z) = -B(A' - z)^{-1}B^* $, commonly arising in few-body problems.
• To identify conditions under which such energy-dependent perturbations can be replaced by an energy-independent potential $ W $.
• To ensure that the resulting Hamiltonian $ H = A + W $ has the same discrete spectrum and eigenfunctions as the original energy-dependent problem.
• To establish orthogonality and expansion theorems for eigenfunction systems of the transformed Hamiltonian $ H $.
• To develop scattering theory for $ H $ when $ A $ has a continuous spectrum.

Proposed method

• Formalize the spectral problem $ (A + V(z)) ho = z ho $, where $ V(z) $ is a resolvent-type energy-dependent interaction.
• Assume $ V(z) = -B(A' - z)^{-1}B^* $, with $ B $ having finite Hilbert-Schmidt norm and $ A $, $ A' $ having separated spectra.
• Derive necessary and sufficient conditions under which $ V(z) $ can be replaced by a constant operator $ W $, preserving the spectrum and eigenfunctions.
• Use unitary equivalence and spectral theory to map the energy-dependent problem to an energy-independent one.
• Prove orthogonality and expansion theorems for eigenfunctions of $ H = A + W $, ensuring completeness in the relevant Hilbert space.
• Extend the framework to scattering theory by analyzing the continuous spectrum of $ A $, establishing the existence and unitarity of the scattering matrix for $ H $.

Experimental results

Research questions

• RQ1 Under what conditions can a resolvent-like energy-dependent interaction $ V(z) = -B(A' - z)^{-1}B^* $ be replaced by an energy-independent potential $ W $?
• RQ2 When does the Hamiltonian $ H = A + W $ have the same discrete spectrum and eigenfunctions as the original energy-dependent problem $ A + V(z) $?
• RQ3 How can orthogonality and expansion theorems be established for the eigenfunction system of the transformed Hamiltonian $ H = A + W $?
• RQ4 What conditions ensure the validity of scattering theory for $ H $ when $ A $ has a continuous spectrum?
• RQ5 How can the results be applied to few-body quantum systems with energy-dependent interactions?

Key findings

• A necessary and sufficient condition is derived for replacing the energy-dependent interaction $ V(z) = -B(A' - z)^{-1}B^* $ with an energy-independent potential $ W $, such that the spectrum and eigenfunctions of $ H = A + W $ match those of $ A + V(z) $.
• The eigenfunction systems of $ H = A + W $ satisfy orthogonality and expansion theorems, ensuring completeness in the Hilbert space of bound states.
• The scattering matrix for $ H $ is well-defined and unitary when $ A $ has a continuous spectrum, extending scattering theory to the energy-independent formulation.
• The transformation preserves the spectral structure under the assumption that the spectra of $ A $ and $ A' $ are separated and $ B $ is Hilbert-Schmidt.
• The results provide a rigorous framework for simplifying complex few-body problems with energy-dependent interactions by eliminating explicit energy dependence.
• The method enables the use of standard spectral and scattering techniques on systems previously requiring specialized treatment due to energy-dependent potentials.