Quasi-exactly solvable spin 1/2 Schrödinger operators

The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave functions with polynomial components to be equivalent to a Schrodinger operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of new examples of multi-parameter QES spin 1/2 Hamiltonians in one dimension