Eigenvalues asymptotics of unbounded operators. Two-photon quantum Rabi model (2312.05646v1)

Abstract: In this work the general results about asymptotics of eigenvalues of unbounded operators are obtained. We consider here different cases of compact, relatively compact, selfadjoint or nonselfadjoint perturbations. In particular we prove a generalization of Janas-Naboko lemma about eigenvalues asymptotics of unbounded operators at compact perturbation. A generalization of our previous result about noncompact perturbation of oscillator spectrum is also given. As an example we consider two-photon quantum Rabi model. We obtain tree-term asymptotic formula for large eigenvalues of the energy operator of this model. The asymptotics of related to this model polynomials is found. We give also an original proof of the Perelomov factorization theorem for contraction operator of quantum optics.

• The paper generalizes the Janas-Naboko lemma for unbounded operators under compact perturbations and extends eigenvalue asymptotic analysis to non-compact perturbations.
• It derives a precise three-term asymptotic formula and expressions for eigenvalues of the two-photon quantum Rabi model's operator, showing oscillatory behavior for large n.
• Understanding eigenvalue shifts under perturbations is crucial for accurately predicting quantum system behavior in realistic applications like quantum optics, atomic physics, and condensed matter.

Eigenvalues Asymptotics of Unbounded Operators in the Quantum Rabi Model

The paper in focus advances the scientific discourse on the eigenvalues asymptotics of unbounded operators, particularly within the framework of quantum mechanics and operator theory. Notably, it addresses the asymptotic behavior of eigenvalues under various perturbations including compact, relatively compact, selfadjoint, and non-selfadjoint perturbations. Perturbation analysis is crucial for understanding physical phenomena in quantum systems, given that real-world models often deviate from idealized versions.

Core Contributions

A pivotal aspect of the paper is the generalization of the Janas-Naboko lemma concerning eigenvalue asymptotics of unbounded operators upon the imposition of compact perturbations. The results extend past studies by introducing methodologies to ascertain asymptotic behaviors under non-compact perturbations. The primary case paper for these theoretical advancements is the two-photon quantum Rabi model, a prominent representation for exploring interactions between light and matter without the rotating wave approximation. The authors derive a three-term asymptotic formula for the Hamiltonian's eigenvalues, highlighting the model's distinctive energy spectrum characteristics.

Key Numerical Findings

An impressive achievement in this work is the derivation of precise asymptotic expressions for the eigenvalues of the two-photon Rabi model's operator. For large $n$, the eigenvalues $E_n$ are determined to asymptotically approach a function of $n$ with an oscillatory behavior. Such results require meticulous boundary conditions, specifically for the operator transformations introduced.

Implications of Research

The expansion of eigenvalue asymptotic analysis in quantum models has far-reaching implications in both theoretical and applied physics. For instance, by understanding how eigenvalues shift under perturbation, physicists can predict the behavior of quantum systems far more accurately under various physical constraints or influences. The extension from ideal solutions to understandings that incorporate perturbations is critical for realistic applications involving quantum optics, atomic physics, and condensed matter.

Future Directions

The formulated hypotheses in the conclusions invite further investigation, specifically regarding the potential refinement of remainder terms in the asymptotic expansions. If proven, these refinements would enhance the applicability of operator theory in more complex systems, potentially allowing for more precise computational models in simulating quantum phenomena. There is also ample scope for exploration into non-compact phenomena within quantum Rabi models or related structures, leveraging the generalized perturbation approaches developed in this paper.

Conclusion

This paper significantly contributes to the field of mathematical physics, providing advanced tools for the eigenvalue analysis of unbounded operators subjected to complex perturbations. Its methodological advances align well with ongoing theoretical challenges in quantum mechanics, while its results facilitate a richer understanding of dynamic interactions in quantum systems. These accomplishments set foundational insights for future research initiatives in both the examination of quantum models and the development of advanced computational techniques in operator theory.