Asymptotic Perturbation Theory for the Eigenvalues of Schroedinger Operators in a Strong Coupling Limit.

The eigenvalues of Schrodinger operators of the form H((lamda)) = -(DELTA) + U + (lamda)V (defined on a dense domain of the Hilbert space L('2)((//R)('3))) are studied in the limit (lamda) (--->) (INFIN). Here U and V are spherically symmetric potentials, V (GREATERTHEQ) 0, and the complement of the support of V is a ball of radius a centered at the origin. The discussion concentrates on the Dirchlet problem on the half-line which is obtained by separation of variables; the corresponding Schrodinger operator is. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). when the angular momentum is zero. When V(x) has leading behavior like V(x) = (x -a)('P) X(,(a,(INFIN))) (p a nonnegative integer), it is shown that H((lamda)) can be advantageously viewed as a perturbed operator associated with the unperturbed operator. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). The eigenvalues of H(,0)((lamda)) are found to be analytic functions of the variable (lamda)('-1/(p+2)). Asymptotic expansions in this variable for the eigenvalues of H((lamda)) are then developed in the case p = 0 by a perturbation procedure. Lastly, various generalizations are discussed and the relation of the above problem to a treatment of resonances is noted.