On Multi-Parameter Families of Hermitian Exactly Solvable Matrix Schrodinger Models
Stanislav Spichak · 2002
Here V (x) is an 2×2 matrix whose entries are smooth complex-valued functions of x. Hereafter we denote d/dx as ∂x. The well-known procedure of constructing a ES matrix (scalar) model is based on the concept of a Lie-algebraic Hamiltonian [1, 2] (the Turbiner–Shifman approach). We call a second-order operator in one variable Lie-algebraic if the following requirements are met: • the Hamiltonian is a quadratic form with constant coefficients of first-order operators Q1, Q2, . . . , Qn forming a Lie algebra g; • the Lie algebra g has a finite-dimensional invariant subspace I of the whole representation space. Now if a given Hamiltonian H[x] is Lie-algebraic, then after being restricted to the space I it becomes a matrix operator H whose eigenvalues and eigenvectors are computed in a purely algebraic way. This means that the Hamiltonian H[x] is exactly solvable. In the paper [3] we have extended the Turbiner–Shifman approach to the construction of quasi-exactly solvable (QES) models on line for the case of matrix Hamiltonians. In this paper we suggested the method for construction of exactly solvable matrix models, which based on the idea explained in [3]. Let us remind, the method consists in supplementing a set of operators Q1, Q2, . . . , Qn, forming a representation of some algebra, so that the obtained set of operators left an appropriate subspace I invariant. However, there is a difference between the approaches suggested in this paper and in [3]. Namely, the obtained set of operators does not form a Lie algebra, in contrast to a set found in [3]. So, let us realize this method considering the set of the operators