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arXiv:math-ph/0312004 · Journal of Physics A Mathematical and General

The evolution operator of the Hartree-type equation with a quadratic potential

A L Lisok, Andrey Yu. Trifonov, Alexander V. Shapovalov

Published Apr 5, 2004• 18 citations
arXivHTMLPDF

Abstract

Based on the ideology of the Maslov's complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov-Anandan geometric phases are found in explicit form for the quasi-energy states.

DOI
10.1088/0305-4470/37/16/005

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Based on the ideology of the Maslov's complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov-Anandan geometric phases are found in explicit form for the quasi-energy states.
We study the semiclassical mechanics of systems with second-class constraints. We assume that the quantum mechanics of such a system is constructed by the Batalin–Fradkin–Tyutin method, where some additional coordinates and momenta are introduced and second-class constraints are converted into firstclass constraints. We also assume that the algebraic quantization method is used to quantize the extended system. To construct the semiclassical approximation, we use the Maslov complex germ theory. We study the semiclassical states of a system with second-class constraints, their scalar product, and the action of semiclassical observables in the first order of the semiclassical expansion. We consider the transformation of semiclassical states in the course of evolution.
Quantum systems with constraints are often considered in modern theoretical physcics. All realistic field models based on the idea of gauge symmetry are of this type. A partial case of constraints being linear in coordinate and momenta operators is very important. Namely, when one applies semiclassical methods to an arbitrary constrained system, the constraints in "general position case" become linear. In this paper, different mathematicals constructions for the Hilbert space space for the constraint system are discussed. Properties of Gaussian and quasi-Gaussian wave functions for these systems are investigated. An analog of the notion of Maslov complex germ is suggested. Properties of Hamiltonians being quadratic with respect to the coordinate and momenta operators are discussed. The Maslov theorem (it says that there exists a Gaussian eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian system is stable) is generalized to the constrained systems. The case of infinite number degrees of freedom (constrained Fock space) is also discussed.
Abstract Quantum models of interacting bosons have a wide range of applications, including the propagation of optical modes in nonlinear media, such as the k-photon down-conversion. Many of these models are related to nonlinear deformations of finite group algebras and, in this sense, are exactly solvable. While advanced group-theoretic methods were developed to study the eigenvalue spectrum, in quantum optics, the primary focus is not on the spectrum of the Hamiltonian but rather on the evolution of an initial state—such as the generation of optical signal modes by a strong pump mode propagating through a nonlinear medium. I propose a simple and general method to solve the state evolution problem, applicable to a broad class of quantum models of interacting bosons. For the k-photon down-conversion model and its generalizations, the solution to the state evolution problem is expressed as an infinite series expansion in powers of the propagation time, with coefficients determined by a recursion relation involving only a single polynomial function. This polynomial function is unique to each nonlinear model. As an application, I compare the exact solution of the parametric down-conversion process with the semiclassical parametric approximation.
This paper contains an evaluation of the Berry phases associated with the following class of nonlinear spin Hamiltonians : H (B, n ) = γS S · B + γQ((S · n) 2 - S2/3), with B · n = 0. Examples of these Hamiltonians are given in Atomic and Solid State Physics. We compute exactly the Berry phases for S = 1, 3/2 and for S > 3/2, to second order with respect to the ratio γQ/(γS B ). The results involve loop integrals of an Abelian gauge field which can be considered as a generalization of the magnetic monopole field, the projective complex plane P2(C) playing the role of the sphere S2. In the case of the spin-1 the relation between the Aharonov and Anandan cyclic quantum phase and the Berry adiabatic phase is made explicit. The connection between the semiclassical limit of the Berry phase and the classical adiabatic Hannay angle is analyzed using the classical spin model recently proposed by Nielsen and Rohrlich.
We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach.The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems.We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position.Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space.We present analytical solutions in L 2 (R) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.