Sturmian technique for the discrete spectrum of the Schrodinger equation

1980 ◽  
Vol 13 (12) ◽  
pp. 3605-3617 ◽  
Author(s):  
J P Gazeau
Keyword(s):  
Schrödinger Equation ◽  
Discrete Spectrum ◽  
Schrodinger Equation

scholarly journals Discrete Spectrum of 2 + 1-Dimensional Nonlinear Schrödinger Equation and Dynamics of Lumps

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Javier Villarroel ◽  
Julia Prada ◽  
Pilar G. Estévez
Keyword(s):  
Schrödinger Equation ◽  
Discrete Spectrum ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Body Problem ◽  
Dynamical Evolution ◽  
Nonlinear Schrodinger Equation ◽  
Spectral Operator ◽  
Nonlinear Schrödinger ◽  
Integrable Generalization

We consider a natural integrable generalization of nonlinear Schrödinger equation to2+1dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dynamical evolution of the associated pulses. For all cases under study we find that the relevant dynamics corresponds to acentral configurationof a certainN-body problem.

scholarly journals Comparison Theorems for the Position-Dependent Mass Schrödinger Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
D. A. Kulikov
Keyword(s):  
Schrödinger Equation ◽  
Discrete Spectrum ◽  
Schrodinger Equation ◽  
Comparison Theorems ◽  
Constant Mass ◽  
Dependent Mass ◽  
Feynman Theorem ◽  
Position Dependent Mass

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schrödinger equation are established. (i) If a constant mass m0 and a PDM m(x) are ordered everywhere, that is either, m0≤m(x) or m0≥m(x), then the corresponding eigenvalues of the constant-mass Hamiltonian and of the PDM Hamiltonian with the same potential and the BenDaniel-Duke ambiguity parameters are ordered. (ii) The corresponding eigenvalues of PDM Hamiltonians with the different sets of ambiguity parameters are ordered if ∇2(1/m(x)) has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.

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