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Quasi-exactly solvable quartic Bose Hamiltonians

Dolya, S N, О. Б. Заславский

Published Jun 24, 2001• 4 citations
OpenAlex

Abstract

We consider Hamiltonians, which are even polynomials of the forth order with the respect to Bose operators. We find subspaces, preserved by the action of Hamiltonian These subspaces, being finite-dimensional, include, nonetheless, states with an \\QTR{it}{infinite} number of quasi-particles, corresponding to the original Bose operators. The basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labeling the representation. In some particular degenerate cases they turn (up to the power factor) into the trigonometric or hyperbolic functions, Bessel functions or combinations of the exponent and Hermit polynomials. We find explicitly the relationship between coefficients at different powers of Bose operators that ensure quasi-exact solvability of Hamiltonian.

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We consider Hamiltonians, which are even polynomials of the forth order with the respect to Bose operators. We find subspaces, preserved by the action of Hamiltonian These subspaces, being finite-dimensional, include, nonetheless, states with an \QTR{it}{infinite} number of quasi-particles, corresponding to the original Bose operators. The basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labeling the representation. In some particular degenerate cases they turn (up to the power factor) into the trigonometric or hyperbolic functions, Bessel functions or combinations of the exponent and Hermit polynomials. We find explicitly the relationship between coefficients at different powers of Bose operators that ensure quasi-exact solvability of Hamiltonian.
We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases n th harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigenspaces of the integrals of motion and show that the action of the Hamiltonian on these common eigenspaces can be represented by a quasi-exactly solvable reduced Hamiltonian, whose expression in terms of the usual generators of 𝔰𝔩 2 is computed explicitly.
We have constructed the quasi-exactly-solvable two-mode bosonic realizations of su(2) and su(1, 1) algebra. We derive the relations leading to the conditions for quasi-exact solvability of two-boson Hamiltonians by determining a general procedure which maps the Schwinger representations of the su(2) and su(1, 1) algebras to the Gelfand-Dyson representations, respectively. This mapping allows us to study nonlinear quantum-optical systems in the framework of quasi-exact solvability. Our approach also leads to a simple construction of special functions of two variables which are the most appropriate functions to study quasi-probabilities in quantum optics.
Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.
The non PT−P T-PT− symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field that couples to a harmonic oscillator via some pseudo-hermitian interaction is considered. We highlight all of the properties of both the original Mandal and Jaynes-Cummings Hamiltonians. We show that the Mandal Hamiltonian HMH_{M}HM​ is non hermitian and non invariant under the combined action of the parity operator P and the time-reversal operator T Even if the previous properties are not satisfied, it has been proved that the Mandal Hamiltonian HMH_{M}HM​ is pseudo-hermitian with respect to P and with respect to σ3\sigma_{3}σ3​ also [1,2]. Thus, we show that the original Jaynes-Cummings Hamiltonian is hermitian. We show the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator, similar to the direct approach to invariant vector spaces used in Refs [3,4].
We propose an explicit factorized formula for the ground state many-body wave function of a model quartic potential problem in two and three dimensions, using our recently developed method for solving Schrödinger equations. Our novel formula has a clear real-space structure: it is exactly factorized into three parts, including a non-integer pre-exponential power factor, dominant decaying terms on the exponent, and a modulator function, which is of minor importance. This result is a generalization of our previously obtained exact formula for the 1-body wave function. As in the 1-body case, here we show that for many-body ground wave functions that cannot achieve variable separation, our method is still far more efficient than the conventional basis expansion method both in representing the wave function and in calculating the energy. The exact factorization ansatz proposed in the present case study shall provide important insight into the general structure of many-body wave functions.
The single mode q-Bose field for -1<or=q<1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +or-(2/(1-q)) 1 2/, thus becoming unbounded in the Bose limit (q to 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatorics. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q to 1. Specifically it is shown that for O<or=q<1 the asymptotic growth is linear in time in contrast to the quadratic growth for q=1.