The Darboux transformation operator technique is applied to construct exactly solvable anharmonic singular oscillator potentials and to study their coherent states. Classical system corresponding to a transformed quantum system is constructed with the help of the coherent states technique. It is shown that at classical level the Darboux transformation may be treated as a transformation of Kähler potential which leads to a distortion of the initial phase space.
N-order Darboux transformation operator is defined on the basis of a general notion of transformation operators. Factorisation properties of this operator are studied. The Darboux transformation operator technique is applied to construct and investigate potentials with bound states at arbitrary energies for the spectral problem on semiaxis.
Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schrödinger ones with an additional functional dependence h(r) in the right-hand side of equations are constructed. The suggested generalized transformations turn into the Bargmann and Darboux transformations for both fixed and variable values of energy and an angular momentum.
A discrete version of the two-dimensional inverse scattering problem is considered. On this basis, algebraic transformations for the two-dimensional finite-difference Schredinger equation are elaborated.
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.
Chains of Darboux transformations for the matrix Schroedinger equation are considered. Matrix generalization of the well-known for the scalar equation Crum-Krein formulas for the resulting action of such chains is given.
We present the recent works \cite{trisetyarso2011} on the application of Darboux transformation on one-dimensional Dirac equation related to the field of Quantum Information and Computation (QIC). The representation of physical system in one-dimensional equation and its transformation due to the Bagrov, Baldiotti, Gitman, and Shamshutdinova (BBGS)-Darboux transformation showing the possibility admitting the concept of relativity and the trade-off of concurrent condition of quantum and classical physics play into the area of QIC. The applications in cavity quantum electrodynamics and on the pro…
A new strategy, using Darboux transformations, of finding self-switching solutions of $i\dotρ = [H, f(ρ)]$ is introduced. Unlike the previous ones, working for any f but for Hamiltonians whose spectrum contains at least three equally spaced eigenvalues, the strategy does not impose any restriction on the discrete part of the spectrum of H. The strategy is applied to the Bloch-Maxwell system.
The stationary Schroedinger equation of the harmonic oscillator is deformed by a Darboux transformation to construct time-dependent potentials with the oscillator profile. The Darboux (supersymmetric or factorization) method is usually developed in the spatial variables of the Schroedinger equation. Here we follow a variation introduced by Bagrov, Samsonov and Shekoyan to include the time-variable as a parameter of the transformation.
New types of irreducible second order Darboux transformations for the one dimensional Schroedinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the ground state energy of the initial (or reference) Hamiltonian. When such a transformation is presented as a chain of two first order transformations, an intermediate potential is singular and therefore intermediate Hamiltonian can not be Hermitian while the final potential is regular and the final Hamiltonian is Hermitian. Second derivative supersymmetric quan…
Darboux transformation of both Barut-Girardello and Perelomov coherent states for the time-dependent singular oscillator is studied. In both cases the measure that realizes the resolution of the identity operator in terms of coherent states is found and corresponding holomorphic representation is constructed. For the particular case of a free particle moving with a fixed value of the angular momentum equal to two it is shown that Barut-Giriardello coherent states are more localized at the initial time moment while the Perelomov coherent states are more stable with respect to time evolution. It…
A detailed analysis of matrix Darboux transformations under the condition that the derivative of the superpotential be self-adjoint is given. As a onsequence, a class of the symmetries associated to Schrödinger matrix Hamiltonians is characterized. The applications are oriented towards the Jaynes-Cummings eigenvalue problem, so that exactly solvable $2\times 2$ matrix Hamiltonians of the Jaynes-Cummings type are obtained. It is also established that the Jaynes-Cummings Hamiltonian is a quadratic function of a Dirac-type Hamiltonian.
A selective chronological survey of Darboux transformations as related to supersymmetric quantum mechanics, intertwining operators and inverse scattering techniques is presented. Short comments are appended to each quotation and basic concepts are explained in order to provide a useful primer
The nonlocal Darboux transformation of the two - dimensional stationary Schrödinger equation is considered and its relation to the Moutard transformation is established. It is shown that a special case of the nonlocal Darboux transformation provides the Moutard transformation. New examples of solvable two - dimensional stationary Schrödinger operators with smooth potentials are obtained as an application of the nonlocal Darboux transformation.
In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the "invariance" of the objects of the "Darboux theory of integrability". In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied.
It is proved that the Darboux transformation of the system of coherent states of a free particle leads to the states that may be treated as coherent states of soliton-like potentials.
We construct arbitrary-order Darboux transformations for Schroedinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates our equations with the standard Schroedinger case. We apply our method to generate several solvable Dunkl-Schroedinger equations.
Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple systems of Schrodinger equations and provide explicit representation for three classes of such systems. We show also that there is an elegant relationship between these transformations and analytic complex matrix functions.
Recently, Quesne on the one side and on the other side Bougie, Gangopadhyaya and Mallow, have discovered new translational shape invariant potentials not present in previous classifications. By using ordinary Bäcklund--Darboux transformations we propose a new framework in which such shape invariant potentials can be accommodated.
We construct a Darboux transformation for a class of two-dimensional Dirac systems at zero energy. Our starting equation features a position-dependent mass, a matrix potential, and an additional degree of freedom that can be interpreted either as a magnetic field perpendicular to the plane or a generalized Dirac oscillator interaction. We obtain a number of Darbouxtransformed Dirac equations for which the zero energy solutions are exactly known.
We develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two-level system. Thus, we find three classes of new solutions for the two-level system and the corresponding new potentials that allow such solutions.
We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct non-trivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one d…
Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain. A determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation is given.
The Fokker-Planck equation associated with the two - dimensional stationary Schrödinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schrödinger equations. The examples of exactly solvable two - dimensional…
Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by $4\times4$ Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scatterin…
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-Pöschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner …
An alternative derivation is presented of the infinitely many exceptional Wilson and Askey-Wilson polynomials, which were introduced by the present authors in 2009. Darboux-Crum transformations intertwining the discrete quantum mechanical systems of the original and the exceptional polynomials play an important role. Infinitely many continuous Hahn polynomials are derived in the same manner. The present method provides a simple proof of the shape invariance of these systems as in the corresponding cases of the exceptional Laguerre and Jacobi polynomials.
We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for the one-dimensional Dirac equation. The properties of the associated intertwining operators are discussed and the explicit form for higher-order transformations is presented. We utilize the results to derive a multi-parametric class of exactly solvable systems where the balanced gain and loss represented by the $PT$-symmetric refractive index can imply loca…
