Numerical computation of bounded states for schröudinger operators

A method developed previously for computing eigenfunctions of one‐dimensional Schrödinger operators is extended to Schrödinger operators in L2(R3N). It is known that in many cases these operators have not a compact resolvent; therefore, the convergence in L2(R3N) of the more used methods for computing the eigenfunctions is not guaranteed. The idea of the present method consists of replacing the eigenvalue problem in L2(R3N) by one corresponding to the system confined into a box Ω with impenetrable walls [Dirichlet problem in L2(Ω)]. It is shown that the eigenfunctions of the unbounded system can be approximated by those of the confined system when the box Ω is expanded. On the other hand, it is proved that the Schrödinger operator associated to the confined system has a compact resolvent and its corresponding sesquilinear form is bounded and elliptic in the Sobolev space W2,10(Ω). These properties guarantee the convergence in L2(Ω) of the standard methods to solve the Dirichlet problem: the Ritz method as well as the finite‐element and finite‐difference methods. Therefore, the eigenfuncions of the unbounded system can be approximated in L2(R3N) by means of the numerical solutions of the Dirichlet problem in L2(Ω) with sufficiently large Ω. This property guarantees the accurate computation of the true expectation values.