New quasi-exactly solvable double-well potentials

A new three-parameter family of quasi-exactly solvable double-well potentials is introduced. We show that the solutions of this family of double-well potentials are expressed in terms of the Heun confluent functions. Under a certain parameter condition, some of the bound-state wavefunctions and asso...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2012-05, Vol.45 (17), p.175302-7
Main Author: Xie, Qiong-Tao
Format: Article
Language:English
Subjects:
Construction
Eigenvalues
Exact sciences and technology
Exact solutions
Functions (mathematics)
Mathematical analysis
Nonlinearity
Physics
Schroedinger equation
Wavefunctions
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Summary:A new three-parameter family of quasi-exactly solvable double-well potentials is introduced. We show that the solutions of this family of double-well potentials are expressed in terms of the Heun confluent functions. Under a certain parameter condition, some of the bound-state wavefunctions and associated energies can be found exactly in explicit form. In particular, we develop an analytical method to derive the conditions for the energy eigenvalues of the bound states. It is also shown that our analytical results can be applied to construct exact solutions of the nonlinear Schrödinger equation with double-well potentials and spatially localized nonlinearities.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/45/17/175302