Lower bounds to ground state eigenvalues of the Schr�dinger equation via optimized inner projection: Application to quartic and sextic anharmonic oscillators

Abstract

Inner projection combined with bracketing techniques represents a very powerful method of calculating lower bounds to eigenvalues of the Schrodinger equation. The Hamiltonian is expressed as a sum of an “unperturbed” operator (whose eigenvalue equation is soluble) and a “perturbed” operator that should be positive definite. For small values of the coupling constant, the conventional splitting of the Hamiltonian is adequate. For strong coupling, however, it is advantageous to partition the Hamiltonian in an optimized fashion, accomplished by an appropriate scaling of the space coordinates and momenta. This optimized inner projection (OIP) method yields remarkably good results for all values of the coupling constant, even for manifolds of minimal dimensionality.