We clarify a few conceptual problems of quantum field theory on the level of exactly solvable models with fermions. The ultimate goal of our study is to gain a deeper understanding of differences between the usual ("spacelike") and light-front forms of relativistic dynamics. We show that by incorporating solutions of the operator field equations to the canonical formalism the spacelike and light front Hamiltonians of the derivative-coupling model acquire an equivalent structure. The same is true for the massive solvable theory, the Federbush model. In the conventional approach, the physical predictions in the two schemes disagree. Moreover, the derivative-coupling model is found to be almost identical to a free theory, in contrast to the conventional canonical treatment. Physical vacuum state of the Thirring model is then obtained by a Bogoliubov transformation as a coherent state quadratic in composite boson operators. To perform the same task in the Federbush model, we derive a massive version of Klaiber's bosonization and show that its light-front form is much simpler.
First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed. In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now, our models admit (but still incomplete) sets of exact (algebraic) solutions.
The exactly and quasi-exactly solvable problems for spin one-half in one dimension on the basis of a hidden dynamical symmetry algebra of Hamiltonian are discussed. We take the supergroup, $OSP(2|1)$, as such a symmetry. A number of exactly solvable examples are considered and their spectrum are evaluated explicitly. Also, a class of quasi-exactly solvable problems on the basis of such a symmetry has been obtained.
We introduce a new concept of infinite quasi-exactly solvable models which are constructable through multi-parameter deformations of known exactly solvable ones. The spectral problem for these models admits exact solutions for infinitely many eigenstates but not for the whole spectrum. The hermiticity of their hamiltonians is guaranteed by construction. The proposed models have quasi-exactly solvable classical conterparts.
We reconsider the quasi exactly solvable matrix models constructed recently by R. Zhdanov. The 2$\times$2 matrix operators representing the algebra sl(2) are generalized to matrices of arbitrary dimension and a similar construction is achieved for the algebra sl(n).
Exactly solvable quantum theory of a singular at the origin scalar field with the self-interaction of Liouville type is proposed. The mean value of the scale factor in the FLRW metric as a function of conformal time is evaluated explicitly.
We construct exactly solvable models for four particles moving on a real line or on a circle with translation invariant two- and four-particle interactions.
An efficient procedure for constructing quasi-exactly solvable matrix models is suggested. It is based on the fact that the representation spaces of representations of the algebra sl(2,R) within the class of first-order matrix differential operators contain finite dimensional invariant subspaces.
We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the $sl(2)$ algebra.
A new exact analytically solvable Eckart-type potential is presented, a generalisation of the Hulthen potential. The study through Supersymmetric Quantum Mechanics is presented together with the hierarchy of Hamiltonians and the shape invariance property.
We construct a two-parametric family of exactly solvable Dirac Hamiltonians by the Darboux transformation method. We obtain intertwining relations between different members of the Hamiltonian family. We investigate the spectral properties of the obtained Hamiltonians and the explicit forms of their eigenfunctions.
We present a simple and effective method for constructing exactly solvable cosmological models containing inflation with exit. This method does not involve any parameter fitting. We discuss the problems arising with solutions that violate the weak energy condition.
In this paper we demonstrate that there exists a close relationship between quasi-exactly solvable quantum models and two special classes of classical dynamical systems. One of these systems can be considered a natural generalization of the multi-particle Calogero-Moser model and the second one is a classical matrix model.
A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra.
We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$.
We construct a quantum mechanical model of the Calogero type for the icosahedral group as the structural group. Exact solvability is proved and the spectrum is derived explicitly.
We suggest a generalization of the Lie algebraic approach for constructing quasi-exactly solvable one-dimensional Schroedinger equations which is due to Shifman and Turbiner in order to include into consideration matrix models. This generalization is based on representations of Lie algebras by first-order matrix differential operators. We have classified inequivalent representations of the Lie algebras of the dimension up to three by first-order matrix differential operators in one variable. Next we describe invariant finite-dimensional subspaces of the representation spaces of the one-, two-d…
By taking a product of two sl(2) representations, we obtain the differential operators preserving some space of polynomials in two variables. This allows us to construct the representations of osp(2,2) in terms of matrix differential operators in two variables. The corresponding operators provide the building blocks for the construction of quasi exactly solvable systems of two and four equations in two variables. Some generalisations are also sketched. The peculiar labelling used for the generators allows us to elaborate a nice deformation of osp(2,2). This gives an appropriate basis for analy…
We connect Quantum Hamilton-Jacobi Theory with supersymmetric quantum mechanics (SUSYQM). We show that the shape invariance, which is an integrability condition of SUSYQM, translates into fractional linear relations among the quantum momentum functions.
New examples of matrix quasi exactly solvable Schroedinger operators are constructed. One of them constitutes a matrix generalization of the quasi exactly solvable anharmonic oscillator, the corresponding invariant vector space is constructed explicitely. Also investigated are matrix generalizations of the Lame equation.
By exploiting the hidden algebraic structure of the Schrodinger Hamiltonian, namely the sl(2), we propose a unified approach of generating both exactly solvable and quasi-exactly solvable quantum potentials. We obtain, in this way, two new classes of quasi-exactly solvable systems one of which is of periodic type while the other hyperbolic.
Matrix quasi exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses one or several finite dimensional invariant vector spaces. New examples of $2\times 2$-matrix quasi exactly solvable operators are constructed with the emphasis set on PT-symmetric Hamiltonians.
In this paper we present a novel quasi-exactly solvable model with symmetric inverted potentials which are unbounded from below. The quasi-exactly solvable states are shown to be total transmission (or reflectionless) modes. From these modes even and odd wavefunctions can be constructed which are normalizable and flux-zero. Under the procedure of self-adjoint extension, a discrete spectrum of bound states can be obtained for these inverted potentials and the solvable part of the spectrum is the quasi-exactly solvable states we have discovered.
We extend the exactly solvable Hamiltonian describing $f$ quantum oscillators considered recently by J. Dorignac et al. by means of a new interaction which we choose as quasi exactly solvable. The properties of the spectrum of this new Hamiltonian are studied as function of the new coupling constant. This Hamiltonian as well as the original one are also related to adequate Lie structures.
We discuss a universal algebraic approach to quasi-exactly solvable models which allows us to interpret them as constrained Hamiltonian systems with a finite number of physical states. Using this approach we reproduce well-known two-dimensional Lie-algebraic quasi-exactly solvable system based on Lie algebra su(3).
We construct a family of triatomic models for heteronuclear and homonuclear molecular Bose-Einstein condensates. We show that these new generalized models are exactly solvable through the algebraic Bethe ansatz method and derive their corresponding Bethe ansatz equations and energies.
Two families of quasi exactly solvable 2*2 matrix Schroedinger operators are constructed. The first one is based on a polynomial matrix potential and depends on three parameters. The second is a one-parameter generalisation of the scalar Lame equation. The relationship between these operators and QES Hamiltonians already considered in the literature is pointed out.
We review three examples of quasi exactly solvable (QES) Hamitonians which possess multiple algebraisations. This includes the most prominent example, the Lame equation, as well as recently studied many-body Hamiltonians with Weierstrass interaction potential and finally, a 2x2 coupled channel Hamiltonian.
A new exactly solvable relativistic periodic potential is obtained by the periodic extension of a well-known transparent scalar potential. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Dirac Kronig-Penney model. The solutions of the Dirac equation are expressed in terms of elementary functions.
Several explicit examples of multi-particle quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multi-particle Hamiltonians, the Ruijsenaars-Schneider-van Diejen systems. These are difference analogues of the quasi exactly solvable multi-particle systems, the quantum Inozemtsev systems obtained by deforming the well-known exactly solvable Calogero-Sutherland systems. They have a finite number of exactly calculable eigenvalues and eigenfunctions. This paper is a multi-particle extension of the recent paper by one of the auth…
We have generated, using an sl(2,R) formalism, several new classes of quasi-solvable elliptic potentials, which in the appropriate limit go over to the exactly solvable forms. We have obtained exact solutions of the corresponding spectral problems for some real values of the potential parameters. We have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable.
The approach of multi-dimensional SUSY Quantum Mechanics is used in an explicit construction of exactly solvable 3-body (and quasi-exactly-solvable $N$-body) matrix problems on a line. From intertwining relations with time-dependent operators, we build exactly solvable non-stationary scalar and $2\times 2$ matrix 3-body models which are time-dependent extensions of the Calogero model. Finally, we investigate the invariant operators associated to these systems.
Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.
Using a new approach to the analysis of false vacuum decay based on the so-called tunneling potential, we develop a general method to find scalar potentials with a false vacuum with exactly solvable decay at the semi-classical level, including gravitational corrections. We examine in particular the decays of de Sitter vacua providing concrete examples that allow to explore analytically the transition between the Coleman-De Luccia and Hawking-Moss regimes.
We consider two quasi-exactly solvable problems in one dimension for which the Schrödinger equation can be converted to Heun's equation. We show that in neither case the Bender-Dunne polynomials form an orthogonal set. Using the anti-isopectral transformation we also discover a new quasi-exactly solvable problem and show that even in this case the polynomials do not form an orthogonal set.
We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Bardeen-Cooper-Schrieffer, Suhl-Matthias-Walker, the Lipkin-Meshkov-Glick, generalized Dicke, the Nuclear Interacting Boson Model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature ye…
Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lame potentials ma(a+1)sn^2(x,m) are computed for integer values a=1,2,3,.... For all cases (except a=1), we show that the partner potential is distinctly different from the original Lame potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lame potentials pm sn^2(x,m)+qm cn^2(x,m)/ dn^…
Brief introduction to the discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators, dynamical symmetry algebras including the $q$-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics is presented. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynom…
We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the $sl(2)$ algebra.
I introduce two family of exactly solvable models for multiatomic hetero-nuclear and homo-nuclear molecular Bose-Einstein condensates through the algebraic Bethe ansatz method. The conserved quantities of the respective models are also showed.
Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors. These equations are deformation of the well-known exactly solvable difference equations of the Meixner-Pollaczek, continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials. Up to an overall factor of the so-called pseudo ground state wavefunction, the eigenfunctions within the exactly solvable subspace are given by polynomials whose roots are solutions of the associated Bethe ans…
In this paper, we study the Schrödinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method.
Explicit examples of quasi-exactly-solvable $N$-body problems on the line are presented. These are related to the hidden algebra $sl_N$, and they are of two types -- containing up to $N$ (infinitely-many eigenstates are known, but not all) and up to 6 body interactions only (a finite number of eigenstates is known). Both types degenerate to the Calogero model.
We establish that by parameterizing the configuration space of a one-dimensional quantum system by polynomial invariants of q-deformed Coxeter groups it is possible to construct exactly solvable models of Calogero type. We adopt the previously introduced notion of solvability which consists of relating the Hamiltonian to finite dimensional representation spaces of a Lie algebra. We present explicitly the $G_2^q $-case for which we construct the potentials by means of suitable gauge transformations.
A systematic procedure to derive exact solutions of the associated Lame equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.
The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this curious property. These models have the notable feature that they do not preserve U(1) symmetry, which is typically associated to a non-conservation of particle number. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtained via the algebraic Bethe ansatz formalism.
Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable `matrix' quantum mechanics, which is recently proposed by Odake and the author. The ($q$-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of $q^x$ ($x$ being the population) corresponding to the $q$-Racah polynomi…
We propose the new exactly solvable model for bosons corresponding to the attractive pairing interaction. Using the electrostatic analogy, the solution of this model in thermodynamic limit is found. The transition from the superfluid phase with the Bose condensate and the Bogoliubov - type spectrum of excitations in the weak coupling regime to the incompressible phase with the gap in the excitation spectrum in the strong coupling regime is observed.
We construct a new class of quasi-exactly solvable many-body Hamiltonians in arbitrary dimensions, whose ground states can have any correlations we choose. Some of the known correlations in one dimension and some recent novel correlations in two and higher dimensions are reproduced as special cases. As specific interesting examples, we also write down some new models in two and higher dimensions with novel correlations.
