PT-symmetric, quasi-exactly solvable matrix Hamiltonians

Abstract

Matrix quasi exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses a or several finite dimensional invariant vector spaces. New examples of 2 × 2matrix quasi exactly solvable operators are constructed with the emphasis set on PT-symmetric Hamiltonians.

Matrix Quasi-Exactly Solvable Jacobi Elliptic Hamiltonian

“…A general test to check whether a -matrix differential operator H (in a variable 2 2  x ) preserves a vector space whose components are polynomials is proposed [9]. After a gauge transformation and a change of variable on the operator H lead to a new operator H  which can be decomposed as follows…”

“…More recently, interesting tools for classification of 2 × 2-matrix QES operators in one spatial dimensional [7][8][9] and in creation and annihilation operators [10] have been constructed.…”

“…[9], PT-symmetric, QES 2 × 2-matrix Hamiltonians are analyzed with the emphasis set on the reality properties of the eigenvalues. The authors considered both trigonometric and hyperbolic 2 × 2-matrix Hamiltonians.…”

“…This paper is organized as follows: In the Section 2, based on the Ref. [9], we briefly recall the QES analytic method used to investigate the quasi-exact solvability of 2 × 2-matrix operators. In Section 3, along the same lines as in the Ref.…”

“…In Section 3, along the same lines as in the Ref. [9], we apply the QES analytic method in order to construct a new 2 × 2-matrix QES Hamiltonian depending on Jacobi elliptic functions. We will consider two values of the constant δ: the case δ = 1 and the case δ = 2.…”