Journal of Operator Theory

Volume 38, Issue 2, Fall 1997 pp. 297-322.

The conjugate operator method for locally regular hamiltonians

Authors: Jaouad Sahbani
Author institution:Université Paris 7 Denis Diderot, Equipe de Physique, Mathématique et Géométrie, Case 7012, 2, Place Jussieu, F-75251 Paris Cedex 05, FRANCE

Summary: We develop a version of the conjugate operator method for an arbitrary pair of self-adjoint operators: the hamiltonian H and the conjugate operator A. We obtain optimal results concerning the regularity properties of the boundary values $(H - \lambda \mp i0)^{- 1}$ of the resolvent of H as functions of $\lambda$. Our approach allows one to eliminate the spectral gap hypothesis on H without asking the invariance of the domain or of the form domain of the hamiltonian under the unitary group generated by A (previous versions of the theory assume at least one of theses conditions). In particular one may treat singular hamiltonians with spectrum equal $\mathbb R$, e.g. strongly singular perturbations of Stark hamiltonians or simply characteristic operators.

Keywords: Mourre estimate, conjugate operator, boundary values of the resolvent.

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