Abstract
A PT-symmetric Hamiltonian associated with a trigonometric Razhavi potential is analyzed. Along the same lines of the general quasi-exactly solvable analytic method considered in the [1] [2] [3], three necessary and sufficient algebraic conditions for this Hamiltonian to have a finite-dimensional invariant vector space are established. This PT-symmetric 2 x 2 -matrix Hamiltonian is called quasi-exactly solvable (QES).
Highlights
In quantum physics, one of the main mathematical problems consists in constructing the set of eigenvalues of a linear operator defined on a suitable domain of a Hilbert space
Three necessary and sufficient quasi-exactly solvable (QES) conditions for the 2 × 2 -matrix trigonometric Hamiltonian to have a finite dimensional invariant vector space will be established for two cases: ψ =t and ψ =t
Our purpose is to establish the three necessary and sufficient QES conditions for the gauge Hamiltonian given by the Equation (26) to have a finite dimensional invariant vector space [1] [2] [3] [4]: 1) The first QES condition is obtained as follows
Summary
One of the main mathematical problems consists in constructing the set of eigenvalues of a linear operator defined on a suitable domain of a Hilbert space. In most cases, this type of problem cannot be explicitly solved, that is to say that the spectrum of the Hamiltonian cannot be found algebraically. It means that a finite part of eigenvalues associated to this type of operators can be found algebraically [1]-[10] Another concept that we consider throughout this paper is the PT-symmetric operator.
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