Darboux Transformation Operator Research Articles

Darboux transformations of the modified BKP hierarchy by fermionic approach

The modified BKP (mBKP) hierarchy is an important integrable hierarchy related with BKP hierarchy. Darboux transformation is a powerful method to obtain various solutions of integrable systems. But for mBKP hierarchy, it is usually very difficult to construct the corresponding Darboux transformations, since it has a more complicated constraint on the Lax operator, compared with the BKP hierarchy. In this paper, we use the fermionic approach to obtain the explicit Darboux transformation operators of the mBKP hierarchy starting from the changes in tau functions. Also we investigate the relations among these new Darboux transformation operators. It is usually very difficult to convert the changes in tau functions to the ones in the Lax operator, while here we provide one successful example.

Obtaining multisoliton solutions of the (2+1)-dimensional Camassa–Holm system using Darboux transformations

We construct and study Darboux transformations for the $$(2{+}1)$$ -dimensional Camassa–Holm system. We apply a reciprocal transformation that relates the $$(2{+}1)$$ -dimensional Camassa–Holm system and the linear system associated with the modified Kadomtsev–Petviashvili hierarchy. Using three Darboux transformation operators, we obtain three types of solutions for the $$(2{+}1)$$ -dimensional Camassa–Holm system, of which one is a multisoliton solution. In addition, we briefly discuss rational solutions.

Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation

In this paper, analytically investigated is a generalized one-dimensional time-dependent Schrödinger equation. Using Darboux transformation operator technique, we construct the first-order Darboux transformation and the real-valued condition of transformed potential for the generalized Schrödinger equation. To prove the equivalence of the supersymmetry formalism and the Darboux transformation, we investigate the relationship among first-order Darboux transformation, supersymmetry and factorization of the corresponding effective mass Hamiltonian. Furthermore, the nth-order Darboux transformations are constructed by means of different method. Finally, by using Darboux transformation, some analytical solutions are generated in a recursive manner for some examples of the Schrödinger equation.

Mathematical modeling of quantum well potentials via generalized Darboux transformations

The Darboux transformation operator technique is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. It is shown how to construct the quantum well potentials in nanoelectronic with a given spectrum. The method is illustrated by several examples.

Darboux transformations in integral and differential forms for generalized Schrödinger equations

The Darboux transformation operator technique is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations. An interrelation is established between the differential and integral transformation operators. It is shown how to construct the quantum well potentials in nanoelectronic with a given spectrum.

Exactly solvable pseudoscalar periodic Dirac potentials from Darboux transformations and underlying nonlinear supersymmetry

Darboux transformation method is applied to constructing new exactly solvable periodic pseudoscalar potentials for the one-dimensional Dirac equation. Factorization properties of Darboux transformation operators are established. Underlying quadratic supersymmetry is revealed.

Intertwining technique for the one-dimensional stationary Dirac equation

The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples.

Coherent states for transparent potentials

The Darboux transformation operator method is applied to the investigation of coherent states of transparent multisoliton potentials. An isometric correspondence between Hilbert spaces of the states of a free particle and a particle moving in the soliton potential is established. It is shown that the Darboux transformation operator being unbounded and closed cannot realize an isometric mapping between Hilbert spaces. A quasispectral representation of transformation operators in terms of continuous basis sets is obtained. Different families of coherent states of the multisoliton potential are introduced. Measures that realize the resolution of the unity in terms of the projectors on the coherent states vectors are calculated. These measures are defined by ordinary smooth functions for the states obtained with the help of bounded transformation operators and by generalized ones otherwise.

The Darboux Transform for the One-Dimensional Stationary Dirac Equation

Based on the method oftransformation operators, the Darboux transformation operator has been constructed for the one-dimensional stationary Dirac equation. The properties of this operator have been studied. As an application, exactly solvable transparent potentials and potentials with the spectrum of a relativistic harmonic oscillator have been obtained.

Coherent states of potentials of soliton origin

The spectral properties of the most general time-dependent potentials of the soliton type described by a self-adjoint operator acting in Hilbert space are discussed. The spectral decomposition for these potentials and the quasispectral decomposition for the Darboux transformation operators are obtained. The coherent states of such systems are examined. Finally, the measure realizing the decomposition of the identity operator in the projectors on the coherent states is calculated.

Distortion of a phase space under the Darboux transformation

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.

Holomorphic representation of coherent states of a singular oscillator and their Darboux representation

We consider the coherent states of a singular oscillator; these states are defined to be the characteristic states of an operator that reduces the number of basis functions for a discrete spectrum to one. We use the Darboux transform to study the coherent states of a transformed Hamiltonian. We obtain an expression for measures that can be used to decompose unity. We construct a holomorphic representation for the state vectors in the space of functions holomorphic everywhere in the complex plane, including vectors for discrete spectra and coherent states. We obtain a holomorphic representation of the Darboux transformation operators.

Darboux transformation of coherent states for a Lobachevski plane

A Darboux transformation of coherent states is considered for a singular oscillator. A coordinate representation and a holomorphic representation are obtained for the Darboux transformation operator and the coherent states. The Hermitian metric and the Kaler potential of the transformed system are calculated. It is established that a Darboux transformation corresponds to curvature of the phase space of the classical system.

Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.