Quasi-exactly-solvable problems in quantum mechanics

Exactly solvable problems in quantum mechanics are important; they serve as a basis for modelling various physical situations although these models are rather rough and do not reproduce many essential properties. So-called 'quasi-exactly-solvable' quantal problems of two types are described: when total information is known on the first N eigenstates (N=1,2,3,...) which are related to each other by analytic continuation; and when there are N potentials of the same sort which are different of each other in the magnitude of the potential parameter, with the same i-th eigenvalue of the i-th potential; these potentials are related by analytical continuation. All these problems are non-trivial and with increasing N they approach the well-known exactly solvable problems in the factorization method. Their analytic properties differ strongly from those of exactly solvable problems. In quasi-exactly-solvable problems the calculation of the first N eigenvalues is equivalent to finding the eigenvalues of a certain NxN Jacobi matrix. 11 refs