0 30 20 40 v 1 1 5 Fe b 20 03 A unified treatment of exactly solvable and quasi - exactly solvable quantum potentials

By exploiting the hidden algebraic structure of the Schrödinger Hamiltonian, namely the sl(2), we propose a unified approach of generating both exactly solvable and quasi-exactly solvable potentials. We obtain, in this way, two new classes of quasi-exactly solvable systems one of which is of periodic type while the other hyperbolic. Tracking down solvable quantum potentials has always aroused interest. Apart from being useful in the understanding of many physical phenomena, the importance of searching for them also stems from the fact that they very often provide a good starting point for undertaking perturbative calculations of more complex systems. Solvable potentials can be broadly classified into two categories : the ones which are exactly solvable[1, 2, 3, 4](including the conditional ones[5, 6]) and others which are quasi-exactly solvable[7, 8, 9, 10]. A spectral problem is said to be exactly solvable(ES) if one can determine the whole spectrum analytically by a finite number of algebraic steps. Factorization hypothesis[11, 12], group-theoretical techniques with a spectrum-generating algebra[13, 14, 15] and use of integral transformations[16, 17] are some of the time-honoured procedures of constructing ES potentials[18]. On the other hand, there exist an infinite number of normal spectral problems which are not amenable to an exact treatment. These are the non-solvable (NS) ones. The quasi-exactly solvable(QES) class is the missing link[19, 20] between the ES and the NS potentials. Actually for a QES system we can only determine a part of the whole spectrum : this essentially means that in an infinite-dimensional space of states there exists a finite-dimensional subspace for which the Schrödinger equation admits partial algebraization. However, in the literature, a common framework that brings together the ES and QES class is still lacking. The purpose of this letter is to fill this gap by exploiting the hidden dynamical symmetry of the Schrödinger equation. We show, in a straightforward way, that by subjecting the Schrödinger equation to some coordinate transformation and adopting for the underlying symmetry group the simplest choice namely the sl(2), it is possible to set up a master equation from which the ES and QES potentials readily follow. Following this line,