[Paper Review] Scattering Theory for Open Quantum Systems

Jussi Behrndt, M. M. Malamud+1 more|ArXiv.org|Oct 31, 2006

Spectral Theory in Mathematical Physics40 references4 citations

TL;DR

This paper develops a scattering theory framework for open quantum systems modeled by maximal dissipative operators (pseudo-Hamiltonians), showing that their scattering matrices can be fully recovered from unitary scattering matrices of self-adjoint dilations. The key contribution is a rigorous construction of a semibounded Hamiltonian embedding for energy-dependent dissipative systems, proving that the scattering matrix of the closed system coincides with that of the original open system via characteristic function theory.

ABSTRACT

Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator $A_D$ in a Hilbert space $\sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation $\widetilde K$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system $\{A_D,\sH\}$, but since $\widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family $\{A(μ)\}$ of maximal dissipative operators depending on energy $μ$, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.

Motivation & Objective

• To develop a rigorous scattering theory for open quantum systems described by maximal dissipative operators (pseudo-Hamiltonians).
• To address the physical inconsistency of non-semibounded self-adjoint dilations in standard dilation theory.
• To construct a semibounded closed system embedding for open systems modeled by energy-dependent families of maximal dissipative operators.
• To establish a complete correspondence between the scattering matrix of the closed system and that of the original open system via characteristic functions.
• To apply the framework to Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.

Proposed method

• Use of boundary triplet theory and Weyl functions to characterize extensions of symmetric operators.
• Construction of a self-adjoint dilation $\widetilde{K}$ of a maximal dissipative operator $A_D$ via a singular finite-rank perturbation of $K_0 = A_0 \oplus G_0$.
• Employment of the Lax-Phillips scattering framework to define wave operators and the scattering operator $S(\widetilde{K}, K_0)$.
• Derivation of the scattering matrix $\{\widetilde{S}(\lambda)\}$ as a unitary multiplication operator in a spectral representation.
• Establishment of a link between the scattering matrix and the characteristic function $W_{A_{-\tau(\lambda)}}(\lambda - i0)^*$ of the dissipative extension $A_{-\tau(\lambda)}$.
• Application of the theory to Sturm-Liouville operators with point interactions, using the Buslaev-Fomin operator as a model.

Experimental results

Research questions

• RQ1 Can a physically consistent semibounded Hamiltonian be constructed to embed an open quantum system described by a single maximal dissipative operator?
• RQ2 How is the scattering matrix of a closed system constructed via self-adjoint dilation related to the scattering matrix of the original open system?
• RQ3 Can the scattering matrix of an energy-dependent family of maximal dissipative operators be recovered from the scattering matrices of individual pseudo-Hamiltonians?
• RQ4 What is the precise relationship between the characteristic function of a maximal dissipative operator and the scattering matrix of its self-adjoint dilation?
• RQ5 How can the framework be applied to physical systems such as dissipative Schrödinger-Poisson systems with point interactions?

Key findings

• The scattering matrix $\{\widetilde{S}(\lambda)\}$ of the self-adjoint dilation $\widetilde{K}$ decomposes into a $2\times2$ block matrix, with the upper-left block equal to the scattering matrix $\{S_D(\lambda)\}$ of the dissipative system $\{A_D, A_0\}$.
• The scattering matrix $\{\widetilde{S}(\lambda)\}$ is unitarily equivalent to a multiplication operator induced by $\{\widetilde{S}(\lambda)\}$ in the spectral representation of $K_0^{ac}$.
• The scattering matrix $\{\widetilde{S}(\lambda)\}$ is explicitly given by $\widetilde{S}(\lambda) = I - 2iP_{\tau(\lambda)}\sqrt{\Im(\tau(\lambda))}(M(\lambda) + \tau(\lambda))^{-1}\sqrt{\Im(\tau(\lambda))}$, where $M(\lambda)$ is the Weyl function and $\tau(\lambda)$ the Weyl function of the dissipative extension.
• The scattering matrix of the closed system $\{\widetilde{L}, L_0\}$ satisfies $\widetilde{S}(\lambda) = W_{A_{-\tau(\lambda)}}(\lambda - i0)^*$, establishing a direct link between characteristic functions and scattering matrices.
• For Sturm-Liouville systems with point interactions, the self-adjoint dilation $\widetilde{L}$ coincides with the Buslaev-Fomin operator, confirming the physical relevance of the construction.
• The framework allows for a semibounded embedding of open systems by using a family $\{A(\mu)\}$ of maximal dissipative operators depending on energy $\mu$, resolving the issue of non-semiboundedness in standard dilation theory.