NUMERICAL COMPUTATION OF BOUNDED STATES FOR SCHRODINGER-OPERATORS

A method developed previously for computing eigenfunctions of one-dime nsional Schrodinger operators is extended to Schrodinger operators in L2(R3N). It is known that in many cases these operators have not a com pact 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 OMEGA w ith impenetrable walls [Dirichlet problem in L2(OMEGA)]. It is shown t hat the eigenfunctions of the unbounded system can be approximated by those of the confined system when the box OMEGA is expanded. On the ot her hand, it is proved that the Schrodinger operator associated to the confined system has a compact resolvent and its corresponding sesquil inear form is bounded and elliptic in the Sobolev space W2,1(0)(OMEGA) . These properties guarantee the convergence in L2(OMEGA) of the stand ard methods to solve the Dirichlet problem: the Ritz method as well as the finite-element and finite-difference methods. Therefore, the eige nfunctions of the unbounded system can be approximated in L2(R3N) by m eans of the numerical solutions of the Dirichlet problem in L2(OMEGA) with sufficiently large OMEGA. This property guarantees the accurate c omputation of the true expectation values. (C) 1994 John Wiley & Sons, Inc.