We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely the singular anharmonic potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations. A hidden $sl(2)$ algebraic structure is also discovered in these models.
Over the years, much efforts have been devoted to the determination of closed form solutions to (Schrödinger) differential equations of the form HΨ = EΨ, where H is the Hamiltonian and E the eigenvalue. In certain cases, the concerned equation can be reduced via suitable substitutions and transformations to a well known differential equation, from which solutions to the original problem can be easily obtained. For instance, closed form solutions to many exactly solvable models in quantum mechanics have been studied in connection with some well known classical differential equations. However, due to limited applications of exactly solvable systems in quantum mechanics, recent attentions have been on systems with partially, algebraically solvable spectra. Such systems are said to be quasi-exactly solvable. Thus a quantum mechanical system is called quasi-exactly solvable if only a finite number of eigenvalues and corresponding eigenvectors can be obtained exactly through algebraic means [1,2,3].
An essential feature of a quasi-exactly solvable system is that the coefficients of the power series solutions to the underlying differential equations satisfy three-or more step recursion relations, in contrast to the two-step recursions for the exactly solvable cases. The complexity of the three-or more step recursion relations makes it very hard (if not impossible) to get exact power series solutions of such systems. However, one can terminate the series at certain power of the variable by imposing certain constraints on the system parameters. By so doing, exact (polynomial) solutions to the system can be obtained, but only for certain energies and for special values of the parameters of the problem.
Solutions to quasi-exactly solvable systems have mostly been discussed in terms of the recursion relations of the power series coefficients. However, this approach does not generally allow one to give explicit, closed form algebraic expressions for the allowed potential parameters of the systems. In this paper, we give a systematic and unified algebraic treatment to a class of four quantum mechanical systems, namely, 1. Singular anharmonic potential [4,5] :
Generalized quantum isotonic oscillator [6] :
Soft core Coulomb potential [7] :
Model 1 was investigated in [4] using the Laurent series ansatz and continued fractions, and its ground state was found in [5]. Solutions to models 2, 3 and 4 were studied in [16]- [20] on a case by case basis by means of the recursion relations. We will show that all four models are reducible to the same basic differential equation, which is quasi-exactly solvable. We solve these models exactly by using the functional Bethe ansatz method in [21]. Our method allows us to obtain the explicit, closed form expressions of the energies and the allowed potential parameters for all cases once and for all in terms of the roots of the algebraic (Bethe ansatz) equations. Particularly interesting are our exact results for model 1, in that to our knowledge this model was not previously recognized to be quasiexactly solvable. We also find a underlying sl(2) algebraic structure in all these models, which is responsible for the quasi-exact solvability.
The work is organized as follows. In section 2, we define the four models and transform the corresponding differential equations into the same basic form. In section 3, we present exact polynomial solutions to the basic equation underlying the four cases by using the Bethe ansatz method. We then give the general, closed form expressions of both the energies and the allowed parameters for each of the four cases. Hidden sl(2) algebraic structures underlying the four models are presented in section 4. In section 5 we provide some concluding remarks.
We consider a quantum mechanical model with the singular anharmonic potential [4,5],
where ω, e, d > 0 are constant parameters. The corresponding radial Schrödinger equation is given by
where ℓ = -1, 0, 1, • • • , and E is the energy eigenvalue.
Making the substitution,
we obtain
Then the change of variable t = r 2 transforms the above differential equation into the form,
(2.5)
The generalized quantum isotonic oscillator has recently attracted a number interests. This system is interesting because it is endowed with properties closely related to those of the harmonic oscillator. The radial Schrödinger equation reads
where ω, g > 0 and a are constant parameters of the systems; ℓ = -1, 0, 1, • • • , and E is the energy eigenvalue. Making the variable change z = ωr 2 , (2.6) becomes
This equation can be transformed, by the substitution,
into the form
where
We find it convenient to work with the new variable t = z + ωa 2 . Then Eq. (2.9) becomes
(2.11)
We now consider the soft-core Coulomb potential [7] V
where G = Z > 0 and β > 0 are constant parameters. Such potential is of interest in atomic and molecular physics. The G = 0 case simulates the field of a smeared charge and is useful in
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