Yu. Melnikov

Master equation for a quantum system driven by a strong periodic field in the quasienergy representation

The evolution of a quantum system weakly coupled with a thermal reservoir and influenced by an external periodic field is formulated in the quasienergy representation where the strong external periodic field disappears and the original model is transformed to a time-dependent system weakly coupled with thermal reservoir. Based on this time-dependent subdynamics, we derive a master equation of the reduced density operator for the driven quantum system. This equation is valid in the weak-coupling limit. Our method can be useful for obtaining the master equation of any system coupled with a thermal reservoir and driven by strong external periodic fields.

Resonances and the extension of dynamics to rigged Hilbert space

Abstract We propose a unified operator theoretic formulation of resonances and resonance states in rigged Hilbert space. Our approach shows how resonances are the basis for the extension of dynamics for unstable systems, and in a synthesis and a generalization of the work of the Brussels/Austin groups directed by Prigogine with the work of Bohm and Gadella on Gamow vectors and with the extension theory developed by the St. Petersburg group directed by Pavlov. We illustrate the approach for prototypes of resonances, namely resonances of the power spectrum, scattering resonances and resonances associated with the unstable Bloch states.

Gamow vectors for barrier wells

Abstract We study four examples of Gamow vectors in one-dimensional potential barriers, namely square barriers and delta barriers. We show that resonances appear when the potential has at least two relative maxima and investigate the emergence of double resonances given rise to Gamow–Jordan vectors as well.

Scattering on graphs and one-dimensional approximations to N-dimensional Schrödinger operators

In the present article we develop the spectral analysis of Schrodinger operators on lattice-type graphs. For the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star-like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices. Using an explicit expression for the resolvent of lattice-type operator we develop in the second section appropriate Lippmann–Schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix. It serves as a base for the approximation of the multi-dimensional Schrodinger operator by a one-dimensional operator on the graph: in the third section of the paper for given N-dimensional Schrodinger operators with rapidly decreasing potential we construct a lattice-type operator on a cubic graph embedded into RN and show that the original N-dimensional scattering problem can be approximated in a p...

On Spectral Analysis of an Integral-Difference Operator

A complete spectral analysis of an integral-difference operator arising as a collision operator in some nonequilibrium statistical physics models is presented. Eigenfunctions of both discrete and continuous spectrum are constructed.

Integral-difference collision operators: Analytical and numerical spectral analysis

We analyze spectral properties of a class of integral-difference collision operators arising in some nonequilibrium statistical physics models. We present analytical estimates and numerical results for the operators defined on finite intervals and corresponding to the truncated Gaussian equilibrium distribution function. Some conclusions are drawn about the spectrum of operators on whole axis. Physical limitations for these kinds of models are discussed.

On the concentration of the spectrum of integral-difference collision operator with Gaussian equilibrium distribution function in a vicinity of zero

We study integral-difference collision operators with a truncated Gaussian equilibrium distribution function. We prove that the number of eigenvalues in an arbitrary small vicinity of zero goes to infinity when the truncation parameter goes to infinity.

The connection between the rigged Hilbert space and the complex scaling approaches for resonances. The Friedrichs model

Abstract The rigged Hilbert space approach and the complex scaling method are compared for the Friedrichs model. It is shown that they define the same resonance pole. The choice of the test space for the rigged Hilbert space approach is discussed.

Extended spectral decompositions of the Renyi map

Abstract We present new general methods to obtain spectral decompositions of dynamical systems in rigged Hilbert spaces and investigate the existence of resonances and the completeness of the associated eigenfunctions. The results are illustrated explicitly for the simplest chaotic endomorphism, namely the Renyi map. This paper extends and completes our previous work.

Non-factorizable extensions of the Liouville operator: The Friedrichs model

Abstract We construct for the Friedrichs model extensions of the Liouville operator which do not reduce to any extension of the corresponding Hamilton operator. These extensions originate in the Brussels–Austin approach to irreversibility and acquire meaning in suitable rigged Hilbert spaces (RHS) associated with the Hilbert–Schmidt space, known as rigged Liouville space (RLS).