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New Approach to Bounded Quantum--Mechanical Models

We develop an approach for the treatment of one--dimensional bounded quantum--mechanical models by straightforward modification of a successful method for unbounded ones. We apply the new approach to a simple example and show that it provides solutions to both the bounded and unbounded type of models simultaneously

The roots of the Hankel determinant approach the energies of bound states and resonances as the determinant dimension increases. The RPM does not require explicit specification of the boundary condition, the approach commonly selects the physical one automatically for each problem.
If the potential-energy function has poles at two points, then the roots of the Hankel determinant approach the eigenvalues of the problem with Dirichlet boundary conditions at such coordinate values [8,12,13]. We call these boundary conditions "natural".
In some cases one is interested in that the eigenvalue equation satisfies "artificial" bounday conditions. For that reason, in this paper we propose a modification of the RPM to treat Dirichlet boundary conditions at arbitrary coordinate locations. In Sec. 2 we introduce the RPM for one-dimensional models and suggest how to force the desired boundary conditions. In particular we concentrate on a linear potential that has proved useful for the treatment of some physical problems. In Sec. 3 we show results for the chosen eigenvalue equation, and in Sec. 4 we summarize the main features of the RPM and draw some conclusions.

The method
We introduce the RPM by means of a simple one-dimensional problem of the form that depends on an adjustable parameter that is necessary to satisfy the other boundary condition which we will specify below. For example, in the case of a dimensionless Schrödinger-like equation the energy E is the adjustable parameter. For concreteness we restrict to this case from now on.
In order to apply the RPM we define the modified logarithmic derivative where the function g(x) is chosen so that f (x) is analytic at x = 0 and therefore can be expanded in a Taylor series Notice that the coefficients f j depend on E. The RPM is based on the transformation of the power series (3) into a rational function or Padé approximant that satisfies where M = N + d, d = 0, 1, . . .. Notice that the rational ansatz has just M + N + 1 adjustable parameters a j and b j to fit the first M + N + 2 coefficients of the Taylor series (3). The additional requirement determines the value of E as a root of the Hankel determinant:  [12,13].
In some cases one wants to force boundary conditions that are not related to singular points in the potential-energy function. Suppose that we are interested in the differential equation (1) with the boundary conditions We can force such "artificial" boundary conditions by means of a properly chosen function g(x) in equation (2). In fact, the func- introduces poles at x = 0 and x = 1 into the differential equation for f (x) that we can rewrite as In this way we expect to obtain the eigenvalues consistent with those boundary conditions.
For simplicity we consider A motivation for this choice is that the resulting differential equation and boundary conditions are related to a simple model for the study of electrons in a crystal under the effect of an electric field [14]. The Schrödinger equation provides the states and energy levels of an electron of mass m and charge e in a box of impenetrable walls at X = 0 and X = L (that mimics the finite size of the crystal) under de effect of an electric field of strength F [14]. This extremely simple model has also been useful in the study of the tail of the density of states of a disordered system in the presence of an electric field [15].
Another reason for the choice of such example is that one can write its solutions exactly in terms of the Airy functions Ai(z) and Bi(z): where N is a normalization factor, and the dimensionless eigenvalues ǫ n , n = 0, 1, . . . are given by the quantization condition

Results
The application of the RPM is straightforward: we obtain as many coefficients f j (ǫ) as necessary from the differential equation for f (x), construct the Hankel determinants H d D (ǫ) D = 2, 3, . . . and calculate their roots. We expect these roots to converge towards the eigenvalues of the differential equation with the boundary conditions mentioned above. Table 1 shows sequences of roots of the Hankel determinants that already converge towards the exact eigenvalues given by equation (10) Table 2 shows some of these eigenvalues for λ = 1. Curiously, more roots cluster around a given eigenvalue of the unbounded model than of the bounded one.

Discussion
Simple models of bounded quantum-mechanical systems have proved useful for the study of several physical phenomena [16] (and references therein).
The modification to the RPM proposed here is suitable for bounding a system between impenetrable walls that force Dirichlet boundary conditions at their locations. The numerical results of the preceding section show that the convergence rate of the modified RPM is as remarkable as in the case of the unbounded and naturally bounded systems [1][2][3][4][5][6][7][8][9][10][11][12][13]. A curious feature of present application of the RPM to a bounded model is that the approach also provides the eigenvalues of the unbounded one. This outcome is a consequence of the fact that the RPM automatically selects the correct asymptotic behaviour at infinity of the solution to the differential equation.
The transformation of the Schrödinger equation into a Riccati one has proved suitable for the application of the quasilinearization method (QLM) to quantum mechanics [17][18][19][20][21][22]. Regarding the calculation of the resonances of a quartic anharmonic oscillator the RPM [6] proves to be more accurate than the QLM [22].
The main ideas behind the RPM have recently proved useful for the treatment of two-point nonlinear equations [23] of interest in some fields of physics [24][25][26]. The resulting approach called Hankel-Padé method (HPM) appears to be an alternative accurate tool for the determination of unknown parameters of the theory that are consitent with the desired asymptotic behaviour of the solution of the nonlinear differential equation [23].