Y. Brihaye, Ancilla Nininahazwe, Bhabani Prasad Mandal
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.
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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.
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.
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.
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.
We present a unified approach for solving and classifying exactly solvable potentials. Our unified approach encompasses many well-known exactly solvable potentials. Moreover, the new approach can be used to search systematically for a new class of solvable potentials.
We obtained a new class of exactly-solvable potentials by means of the hypergeometric equation for Schrodinger equation, which different from the exactly-solvable potentials introduced by Bose and Natanzon. Using the new class of solvable potentials, we can obtain the corresponding complex PT-invariant potentials. This method can also apply to the other Fuchs equations.
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.
We provide an explanation to the behaviour of the spectra of two exactly-solvable one-dimensional Hamiltonians with PT symmetry proposed earlier. We calculate the branch points at which pairs of eigenvalues coalesce and discuss the perturbation series.
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…
Supersymmetric method of the constructing well-like quasi exactly solvable (QES) potentials with three known eigenstates has been extended to the case of periodic potentials. The explicit examples are presented. New QES potential with two known eigenstates has been obtained.
A general approach for constructing multidimensional quasi-exactly solvable (QES) potentials with explicitly known eigenfunctions for two energy levels is proposed. Examples of new QES potentials are presented.
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^…
Exactly solvable models of nonlinear extensions of the Schrödinger equation
本文提出一种构造Schrödinger方程精确可解非线性扩展的方法，通过建立精确可解的常微分Schrödinger方程与非线性理论间的对应关系，重新推导了孤子解并发现了新的精确可解非线性理论，可推广至相对论性孤子理论。
We propose a general method for constructing quasi-exactly solvable potentials with three analytic eigenstates. These potentials can be real or complex functions but the spectrum is real. A comparison with other methods is also performed.
We use a Lie algebraic technique to construct complex quasi exactly solvable potentials with real spectrum. In particular we obtain exact solutions of a complex sextic oscillator potential and also a complex potential belonging to the Morse family.
In this note we establish a relation between two exactly-solvable problems on circle, namely singular Coulomb and singular oscillator systems.
In this thesis, the quantum Hamilton Jacobi (QHJ) formalism is used to study various exactly solvable (ES) and quasi -exactly solvable (QES) models. Using this method, we obtain the bound state eigenvalues and the eigenfunctions for the models studied. The central entity of this formalism in the logarithmic derivative of the wave function, known as the quantum momentum function (QMF).It is assumed that the point at infinity is an isolated singular point.The kowledge of the singularity structure of the QMF is used to arrive at the required solutions. We show that there are marked differences be…
Using supersymmetric quantum mechanics we develop a new method for constructing quasi-exactly solvable (QES) potentials with two known eigenstates. This method is extended for constructing conditionally-exactly solvable potentials (CES). The considered QES potentials at certain values of parameters become exactly solvable and can be treated as CES ones.
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.
Quasi-exactly solvable rational potentials with known zero-energy solutions of the Schro\" odinger equation are constructed by starting from exactly solvable potentials for which the Schr\" odinger equation admits an so(2,1) potential algebra. For some of them, the zero-energy wave function is shown to be normalizable and to describe a bound state.
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.
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a quite general spherically symmetric form for these potentials and we analyse some exactly and quasi exactly solvable properties of the underlying matricial linear operators.
An extended notion of quasi-exactly solvable potential model is used here to treat quasi exactly solvable (QES) Bose systems. We report an analytic expression for the Ahoronov Anandan non-adiabatic geometric phase for the QES Bose system in general. The generalized expression is then used to study some particular cases of physical interest and we observe that the geometric phase can be tuned.
New exactly solvable quantum models are obtained with the help of the supersymmetric extencion of the nonstationary Schr/"odinger equation.
We deal with the exact solutions of Schrodinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, Poschl-Teller and Hulthen type potentials are considered respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.
We examine in detail the possibilty of applying Darboux transformation to non Hermitian hamiltonians. In particular we propose a simple method of constructing exactly solvable PT symmetric potentials by applying Darboux transformation to higher states of an exactly solvable PT symmetric potential. It is shown that the resulting hamiltonian and the original one are pseudo supersymmetric partners. We also discuss application of Darboux transformation to hamiltonians with spontaneously broken PT symmetry.
An exactly solvable model of a quantum spin interacting with a spin environment is considered. The interaction is chosen to be such that the state of the environment is conserved. The reduced density matrix of the spin is calculated for arbitrary coupling strength and arbitrary time. The stationary state of the spin is obtained explicitely in the $t \to \infty$ limit.
Recently, based on a supersymmetric approach, new classes of conditionally exactly solvable problems have been found, which exhibit a symmetry structure characterized by non-linear algebras. In this paper the associated ``non-linear'' coherent states are constructed and some of their properties are discussed in detail.
We develop a method to determine the eigenvalues and eigenfunctions of two-boson Hamiltonians include a wide class of quantum optical models. The quantum Hamiltonians have been transformed in the form of the one variable differential equation and the conditions for its solvability have been discussed. Applicability of the method is demonstrated on some simple physical systems.
We compare two recent approaches of quasi-exactly solvable Schr\" odinger equations, the first one being related to finite-dimensional representations of $sl(2,R)$ while the second one is based on supersymmetric developments. Our results are then illustrated on the Razavy potential, the sextic oscillator and a scalar field model.
A new supersymmetry method for the generation of the quasi-exactly solvable (QES) potentials with two known eigenstates is proposed. Using this method we obtained new QES potentials for which we found in explicit form the energy levels and wave functions of the ground state and first excited state.
The intertwining operator technique is applied to difference Schroedinger equations with operator-valued coefficients. It is shown that these equations appear naturally when a discrete basis is used for solving a multichannel Schroedinger equation. New families of exactly solvable multichannel Hamiltonians are found.
We construct a variety of new exactly-solvable quantum systems, the potentials of which are given in terms of Lambert-W functions. In particular, we generate Schrödinger models with energy-dependent potentials, conventional Schrödinger models using the supersymmetry formalism, and two-dimensional Dirac systems. In addition, we derive Wronskian integral formulas for Lambert-W functions.
A conditionally exactly solvable potential, the supersymmetric partner of the harmonic oscillator is investigated in the PT-symmetric setting. It is shown that a number of properties characterizing shape-invariant and Natanzon-class potentials generated by an imaginary coordinate shift $x-{\rm i}ε$ also hold for this potential outside the Natanzon class.
We comment that the conditionally exactly solvable potential of Dutt et al. and the exactly solvable potential from which it is derived form a dual system.
Using supersymmetric quantum mechanics we construct the quasi-exactly solvable (QES) potentials with arbitrary two known eigenstates. The QES potential and the wave functions of the two energy levels are expressed by some generating function the properties of which determine the state numbers of these levels. Choosing different generating functions we present a few explicit examples of the QES potentials.
A new class of quasi exactly solvable potentials with a variable mass in the Schroedinger equation is presented. We have derived a general expression for the potentials also including Natanzon confluent potentials. The general solution of the Schroedinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials.
One may obtain, using operator transformations, algebraic relations between the Fourier transforms of the causal propagators of different exactly solvable potentials. These relations are derived for the shape invariant potentials. Also, potentials related by real transformation functions are shown to have the same spectrum generating algebra with Hermitian generators related by this operator transformation.
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.
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.
We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of the first type give singular intermediate potentials and the ones of the second type give complex-valued intermediate potentials while final potentials are meaningful in all cases. These developments are illustrated on the so-called radial sextic oscillator.
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…
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.
We present in this paper a rather general method for the construction of so-called conditionally exactly solvable potentials. This method is based on algebraic tools known from supersymmetric quantum mechanics. Various families of one-dimensional potentials are constructed whose corresponding Schrödinger eigenvalue problem can be solved exactly under certain conditions of the potential parameters. Examples of quantum systems on the real line, the half line as well as on some finite interval are studied in detail.
In this paper we establish a relation between two exactly-solvable problems on one-dimensional hyperbolics space, namely singular Coulomb and singular oscillator systems.
A general procedure is presented to construct conditionally solvable (CES) potentials using the techniques of supersymmetric quantum mechanics.The method is illustrated with potentials related to the harmonic oscillator problem.Besides recovering known results,new CES potentials are also obtained within the framework of this general approach.The conditions under which this method leads to CES potentials are also discussed.
Recently developed supersymmetric perturbation theory has been successfully employed to make a complete mathematical analysis the reason behind exact solvability of some non-central potentials. This investigation clarifies once more the effectiveness of the present formalism.
We calculate the entanglement-assisted and unassisted channel capacities of an exactly solvable spin star system, which models the quantum dephasing channel. The capacities for this non-Markovian model exhibit a strong dependence on the coupling strengths of the bath spins with the system, the bath temperature, and the number of bath spins. For equal couplings and bath frequencies, the channel becomes periodically noiseless.
A series of exactly solvable non-trivial complex potentials (possessing real spectra) are generated by applying the Darboux transformation to the excited eigenstates of a non-Hermitian potential V(x). This method yields an infinite number of non-trivial partner potentials, defined over the whole real line, whose spectra are nearly exactly identical to the original potential.
In this paper, we write exactly solvable generalizations of 1-dimensional quantum XY and Ising-like models by using $2^d$-dimensional Gamma ($Γ$) matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner like transformations. We illustrate the techniques using a specific case of 4-dimensional $Γ$ matrices and explore the quantum phase transitions present in the model.
Exactly solvable rationally-extended radial oscillator potentials, whose wavefunctions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of $k$th-order supersymmetric quantum mechanics, with special emphasis on $k=2$. It is shown that for $μ=1$, 2, and 3, there exist exactly $μ$ distinct potentials of $μ$th type and associated families of exceptional orthogonal polynomials, where $μ$ denotes the degree of the polynomial $g_μ$ arising in the denominator of the potentials.
We consider the double sinh-Gordon potential which is a quasi-exactly solvable problem and show that in this case one has two sets of Bender-Dunne orthogonal polynomials . We study in some detail the various properties of these polynomials and the corresponding quotient polynomials. In particular, we show that the weight functions for these polynomials are not always positive. We also study the orthogonal polynomials of the double sine-Gordon potential which is related to the double sinh-Gordon case by an anti-isospectral transformation. Finally we discover a new quasi-exactly solvable problem…
In this work, we start from the well known Calogero-Wolfes type 3-body problems on a line and construct the corresponding exactly solvable rationally extended 3-body potentials. In particular, we obtain the corresponding energy eigenvalues and eigenfunctions which are in terms of the product of Xm Laguerre and Xp Jacobi exceptional orthogonal polynomials where both m,p = 1,2,3,....
We deform the real potential of Poeschl and Teller by a shift of its coordinate in imaginary direction. We show that the new model remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and proportional to Jacobi polynomials. Some of them diverge in the Hermitian limit. In contrast, all their energies prove real and shift-independent. In this sense the lost Hermiticity of our family of Hamiltonians seems replaced by their accidental PT symmetry.
We consider a particle with a position-dependent mass, moving in a three-dimensional semi-infinite parallelepipedal or cylindrical channel under the influence of some hyperbolic potential. We show that the lack of uniformity in the environment generates an infinite number of bound states.
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