Bikashkali Midya

Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonians

Abstract Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type X 1 exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.

Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells

We report the existence and properties of localized modes described by nonlinear Schroedinger equation with complex PT-symmetric Rosen-Morse potential well. Exact analytical expressions of the localized modes are found in both one dimensional and two-dimensional geometry with self-focusing and self-defocusing Kerr nonlinearity. Linear stability analysis reveals that these localized modes are unstable for all real values of the potential parameters although corresponding linear Schroedinger eigenvalue problem possesses unbroken PT-symmetry. This result has been verified by the direct numerical simulation of the governing equation. The transverse power flow density associated with these localized modes has also been examined.

A generalized quantum nonlinear oscillator

We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the new exactly solvable potentials possess shape invariance symmetry. The solutions are obtained in terms of classical orthogonal polynomials.

A note on the PT invariant periodic potential V(x)=4cos2x+4iV0sin2x

Abstract It is shown that the PT symmetric Hamiltonian with the periodic potential V ( x ) = 4 cos 2 x + 4 i V 0 sin 2 x can be mapped into a Hermitian Hamiltonian for V 0 0.5 , by a similarity transformation. It is also shown that there exist a second critical point of the potential V ( x ) , apart from the known critical point V 0 = 0.5 , for V 0 c ∼ 0.888437 after which no part of the eigenvalues and the band structure remains real. Relevant physical consequence of this finding has been pointed out.

Position dependent mass Schrödinger equation and isospectral potentials: Intertwining operator approach

Here, we have studied first- and second-order intertwining approaches to generate isospectral partner potentials of position dependent (effective) mass Schrodinger equation. The second-order intertwiner is constructed directly by taking it as second-order linear differential operator with position dependent coefficients, and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained, which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s), and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. ...

Nonlinear localized modes in PT-symmetric optical media with competing gain and loss

Abstract The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expression of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effects of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. The spatial localized modes in two-dimensional geometry as well as the transverse power-flow density associated with these localized modes are also examined.

Supersymmetry-generated one-way-invisible PT -symmetric optical crystals

We use supersymmetry transformations to design transparent and one-way reflectionless (thus unidirectionally invisible) complex optical crystals with balanced gain and loss profiles. The scattering coefficients are investigated using a transfer matrix approach. It is shown that the amount of reflection from the left can be made arbitrarily close to zero whereas the reflection from the right is enhanced arbitrarily (or vice versa).

Effect of position-dependent mass on dynamical breaking of type B and type X2

We investigate effect of position-dependent mass profiles on dynamical breaking of -fold supersymmetry in several type B and type X2 models. We find that -fold supersymmetry in rational potentials in the constant-mass background is steady against the variation of mass profiles. On the other hand, some physically relevant mass profiles can change the pattern of dynamical -fold supersymmetry breaking in trigonometric, hyperbolic and exponential potentials of both type B and type X2. The latter results open the possibility of detecting experimentally the phase transition of -fold as well as ordinary supersymmetry at a realistic energy scale.

\mathcal {N}

The coherent state of a nonlinear oscillator having a nonlinear spectrum is constructed using Gazeau–Klauder formalism. The weighting distribution and the Mandel parameter are studied. Details of the revival structure arising from different time scales underlying the quadratic energy spectrum are investigated by the phase analysis of the autocorrelation function.

-fold supersymmetry

The modified factorization technique of a quantum system characterized by a position-dependent mass Hamiltonian is presented. It has been shown that the singular superpotential defined in terms of a mass function and an excited state wavefunction of a given position-dependent mass Hamiltonian can be used to construct non-singular isospectral Hamiltonians. The method has been illustrated with the help of a few examples.