Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1101.3637 · Journal of Mathematical Physics

Band structure and accumulation point in the spectrum of quantum collision operator in a one-dimensional molecular chain

Buang Ann Tay, Kazuki Kanki, Satoshi Tanaka, Tomio Yamakoshi Petrosky

Published Feb 1, 2011• 4 citations
arXivHTMLPDF

Abstract

We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point, and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.

DOI
10.1063/1.3553201

Related papers

Accumulation Point in the Spectrum of the Collision Operator for a One-Dimensional Polaron System

Kazuki Kanki, Satoshi Tanaka, Buang Ann Tay +1 more

The eigenvalue problem of a collision operator in a quantum kinetic equation for a momentum relaxation process of a one-dimenstional polaron system is considered. There appear an accumulation point and band structures in the spectrum of the collision operator.

Progress of Theoretical Physics Supplement•Jan 1, 2010•6 citations

Complex Spectral Representation of the Liouvillian and Kinetic Theory in Nonequilibrium Physics

Tomio Yamakoshi Petrosky

Kinetic theory in nonequilibrium statistical physics is discussed in terms of the complex spectral representation of the Liouvillian for quantum systems. We show that the collision operator appearing in the kinetic equation is the “self-energy part” of the Liouvillian. Solving the dispersion equation associated to the “self-energy part”, one can construct the resonance states of the Liouvillian which have complex eigenvalues. The imaginary part of the eigenvalue gives a decay rate in irreversible process in nonequilibrium statistical physics. As illustrations of the resonance states of the Liouvillian we consider two examples; one is a one-dimensional system in which a polaron is weakly interacting with a thermal reservoir consisting of an acoustic phonon field, and the other is a molecular chain that is described by the Davydov Hamiltonian, which has been introduced as a simple model that describes a protein chain. We show that the imaginary part of the spectrum of the Liouvillian in these systems has rich structures showing band structure, an accumulation point, and a fractal structure reminiscent of Hofstadter's butterfly.

Progress of Theoretical Physics•Mar 1, 2010•19 citations

Dual Spaces Structure of Quantum Kinetic Equations

Buang Ann Tay, Tomio Yamakoshi Petrosky

The relation between the Hilbert space structure and the generalized spaces structure represented by dual states for dissipative kinetic equation is discussed for quantum systems. As working examples, we consider the systems of a harmonic oscillator or a particle interacting with a thermal reservoir and construct analytic solutions to the eigenvalue problem of the quantum collision operators of these systems. The generalized spaces structure of the eigenfunctions indicates that dissipation destroys the Hilbert space structure of the undamped system. In the Wigner representation where the quantum collision operators closely resemble the classical kinetic operators in phase space, the Hilbert space structure can be restored to certain extent by introducing a weighted norm or a similarity transformation on the operators. However, in the position space where the collision operators have no classical counterpart, generalized spaces description cannot be avoided.

Progress of Theoretical Physics Supplement•Jan 1, 2010•1 citation

The point spectrum of the three-particle Schrödinger operator for a system comprising two identical bosons and one fermion on Z

Zahriddin Ishkobilovich Muminov, Актамова Васила Уктамовна

We consider the Hamiltonian of a system of three quantum particles (two identical bosons and a fermion) on the one-dimensional lattice interacting by means of zero-range attractive or repulsive potentials. We investigate the point spectrum of the three-particle discrete Schrödinger operator H(K), K ∈ T which possesses infinitely many eigenvalues depending on repulsive or attractive interactions, under the assumption that the bosons in the system have infinite mass

Nanosystems Physics Chemistry Mathematics•Aug 30, 2024•3 citations

Simplified quantum oscillator solution

Thomas F. Jordan

To find the eigenvalues and eigenvectors of the quantum oscillator Hamiltonian, it is enough to construct eigenvectors with the raising operator and see that these eigenvectors are complete. The subspace they span must be the entire space of square-integrable functions because it is invariant for the raising and lowering operators and therefore invariant for the position and momentum operators. This shows that the Hamiltonian has no continuous spectrum.

American Journal of Physics•Oct 1, 2001•1 citation

Modelling light-induced kinetic effects in gases with eigenvalues of collision operators

S Kryszewski, Gerard Nienhuis

The authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.

Journal of Physics B Atomic Molecular and Optical Physics•Nov 14, 1989•10 citations
We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point, and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.
Kinetic theory in nonequilibrium statistical physics is discussed in terms of the complex spectral representation of the Liouvillian for quantum systems. We show that the collision operator appearing in the kinetic equation is the “self-energy part” of the Liouvillian. Solving the dispersion equation associated to the “self-energy part”, one can construct the resonance states of the Liouvillian which have complex eigenvalues. The imaginary part of the eigenvalue gives a decay rate in irreversible process in nonequilibrium statistical physics. As illustrations of the resonance states of the Liouvillian we consider two examples; one is a one-dimensional system in which a polaron is weakly interacting with a thermal reservoir consisting of an acoustic phonon field, and the other is a molecular chain that is described by the Davydov Hamiltonian, which has been introduced as a simple model that describes a protein chain. We show that the imaginary part of the spectrum of the Liouvillian in these systems has rich structures showing band structure, an accumulation point, and a fractal structure reminiscent of Hofstadter's butterfly.
The relation between the Hilbert space structure and the generalized spaces structure represented by dual states for dissipative kinetic equation is discussed for quantum systems. As working examples, we consider the systems of a harmonic oscillator or a particle interacting with a thermal reservoir and construct analytic solutions to the eigenvalue problem of the quantum collision operators of these systems. The generalized spaces structure of the eigenfunctions indicates that dissipation destroys the Hilbert space structure of the undamped system. In the Wigner representation where the quantum collision operators closely resemble the classical kinetic operators in phase space, the Hilbert space structure can be restored to certain extent by introducing a weighted norm or a similarity transformation on the operators. However, in the position space where the collision operators have no classical counterpart, generalized spaces description cannot be avoided.
The authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.