The long title of this paper already explains what the authors did, so I will just tell you how they did it. The paper describes a computationally elegant hypermatrix approach, first used by J. H. Argyris at Stuttgart in the early 1960s. In fact, as I read the paper, I wished that I still had a copy of ArgMat, the Argyris Matrix code, so that I could have entered the authors’ hypermatrix equations and done a computation. The two-dimensional Schrödinger equation can be solved by discretizing both the x and y variables, to produce an algebraic eigenvalue problem. The authors of this paper preferred to discretize only the y variable, to produce a second-order ordinary differential equation that may be solved by the “exponential fitting” method that they have been developing, as described in a series of recent papers, all of which they cite.
Since the wave function approaches zero away from the origin, the authors can approximate its second derivative in a discretized finite interval centered at the origin by difference quotients. Substituting the latter for the second partial derivative of the wave function, with respect to y, in Schrödinger’s equation, and reorganizing the now partially discretized equation into hypermatrix form (which the authors call block matrix form), that is, as a matrix equation in which the components of each matrix are also matrices, produces a remarkably compact representation of the authors’ computational problem. This problem can be reduced to a generalized algebraic eigenvalue problem by assuming exponential solutions. And what a (numerical) whopper of a problem it is, too. The equation is
(P + E h2 Q - E2 h4 R)&PSgr; = 0
in which E is the eigenvalue, h is the mesh size, and P, Q and R are long hypermatrix expressions. &PSgr; is, of course, the wave function discretized as a vector.
Numerical results, and comparisons to a fully discretized solution, are mentioned, but neither is included nor described. This otherwise very elegant paper is marred just slightly by several typographical errors, none of which limit the reader’s ability to follow the algorithmic logic of the presentation.
