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arXiv:math-ph/0604058 · Journal of Mathematical Physics

Extended weak coupling limit for Friedrichs Hamiltonians

Jan Dereziński, Wojciech De Roeck

Published Jan 1, 2007• 6 citations
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Abstract

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one called a “reservoir.” The operator, which we call a “Friedrichs Hamiltonian,” has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before (Accardi et al., 1990) in more complicated models under the name of “stochastic limit.”

DOI
10.1063/1.2405402

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We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one called a “reservoir.” The operator, which we call a “Friedrichs Hamiltonian,” has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before (Accardi et al., 1990) in more complicated models under the name of “stochastic limit.”
We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one – a “reservoir”. The operator, which we call a “Friedrichs Hamiltonian”, has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before [AFL] in more complicated models and they are usually referred to as ”stochastic Limit”.
The weak coupling limit of a ‘‘system+reservoir’’ model with nonlinear (quadratic) interaction is discussed. In a previous paper [Accardi and Lu, Stoch. Process. Phys. Geom., 1–26], it has been shown that, if the matrix elements of the van Hove rescaled evolution are taken with respect to the same collective vectors used for linear interactions, then the limiting evolution is not unitary. In the present paper, we choose a new type of collective vectors and obtain, in the weak coupling limit, a unitary evolution satisfying a quantum stochastic differential equation. A remarkable feature of the limiting process is that the quadratic nature of the interaction shows up in a nontrivial choice of the Hilbert space where the limiting quantum noise takes its values, which is qualitatively new with respect to all the models considered up to now.
This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles in the dipole approximation. The free system Hamiltonian and the system part of the Hamiltonian describing interaction with the reservoir are considered as unbounded operators with continuous spectrum which are commuting in a weak sense. We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir which are non-zero in the WCL. Then we prove that the resulting reduced system dynamics converges to unitary dynamics (such behavior sometimes called as Quantum Cheshire Cat effect) with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian. We obtain exact form of the modified Hamiltonian and estimate the rate of convergence to the limiting dynamics. For Fermi reservoir, we prove the convergence of the full Dyson series. For Bose case the convergence is understood term by term.
The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C *-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C *-algebras are investigated and it is shown that pure states could evolve into mixed states.
When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian H as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian.In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs extension of the restriction of H to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from H itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with.As a practical demonstration of such results, we consider the quantum particle in the box, and we show that, for a wide class of bases (which include associated Legendre polynomials as a concrete example) the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed.