New ways to solve the Schrödinger equation (Q1772397)

The paper is devoted to a new approach, presented by the same authors in their previous investigations [see for example, Ann. Phys. 308, 263--284 (2003; Zbl 1037.81042)], to solve the low lying states of the Schrödinger equation. The present paper consists of five sections and an appendix. In the introduction the authors' new method to solve the Schrödinger equation is shortly described. In this method it is important to construct trial functions for the equation. In section 2 the details of how to construct a good trial function for the \(N\)-dimensional problem is discussed. In section 3 the hierarchy theorem, i.e. convergence of the iterative solutions, for the one-dimensional problem is proved, in which the potential \(V(x)\) is even and the potential-difference function \(w(x)\) is assumed to satisfy \(w'(x)<0\) for \(x>0\). Section 4 is devoted to the proof of the hierarchy theorem for the asymmetric case \(V(x)\neq V(-x)\). It should be noted that in this section the potential is chosen as a specific asymmetric quadratic double well potential. Section 5 is devoted to the generalization of the results for the higher dimensional setting. The attempt fails at present because the proof of one of the lemma does not appear to generalize in higher dimension. In Appendix a solvable example in one-dimension is given.