The spectrum of a quantum potential

We apply a method recently devised by one of the authors to obtain an approximate analytical formula for the spectrum of a quantum anharmonic potential. Due to its general features the method can be applied with minimal effort to general quantum potentials thus allowing very promising applications. PACS numbers: 03.65.Ge,02.30.Mv,11.15.Bt,11.15.Tk ‡ [email protected] The spectrum of a quantum potential 2 Recently one of the authors has developed a method which allows to obtain analytical approximations to a certain class of integrals with arbitrary precision [1]. It has been proven that the expansion proposed in [1] converges uniformly and that it yields excellent results already to first order. Such method has been applied to a large class of problems in Classical Mechanics and General Relativity, allowing to obtain simple analytical formulas which work even in the non-perturbative regime. In this letter we deal with a further application of the method, showing that it is possible to obtain accurate analytical expressions for the spectrum of a quantum potential by using it in conjunction of the WKB expansion. Although our calculations are obtained for the anharmonic potential, the procedure that we suggest applies with minor changes to any potential. We will use [2] as our main reference: in that paper Bender and collaborators show how to solve the one-dimensional eigenvalue problem for analytic potential to all orders in the WKB approximation. They obtain a recursion relation for the WKB corrections and they are able to sum the WKB series in two special cases, recovering the exact energy eigenvalues. However, in the general case of a potential V (x) the calculation of the WKB correction to a given order can be difficult and, depending on the potential itself, it is possible that no analytical expression can be found for the WKB integrals. By applying the method of [1] to the anharmonic oscillator to order h̄ we show that one can obtain a precise analytical approximation to the entire spectrum of the potential. The precision of the approximation depends both on the order to which the WKB expansion is considered and on the order to which our method is applied. Let us briefly review the method by considering the integral J1 =