MULTIPLE TIME SCALES

By KENNETH WHITE

CONTENTS NOTATIONS SETS, SUBSETS, POINTS FUNCTIONS AND MAPPINGS SPACES TOPOLOGICAL SPACES METRIC SPACES LINEAR TOPOLOGICAL SPACES LINEAR NORMED SPACES BANACH AND HILBERT SPACES FUNCTION SPACES SOBOLEV SPACES LINEAR TRANSFORMATIONS CONTINUOUS LINEAR FUNCTIONALS LINEAR OPERATORS BOUNDED SESQUILINEAR FORMS UNBOUNDED OPERATORS, CLOSABILITY RESOLVENT, SPECTRUM, DE?CIENCY INDICES ADJOINTS AND SELF-ADJOINT OPERATORS THE CONSTRUCTION OF THE ADJOINT OPERATOR SELF-ADJOINTNESS BASIC CONCEPTS OF EXTENSION THEORY EXAMPLES CLOSABLE QUADRATIC FORMS IN A HILBERT SPACE OPERATOR REPRESENTATION OF CLOSED QUADRATIC FORM THE SPECTRAL THEOREM THE OPERATOR PARAMETRIZATION DESCRIPTION OF EXTENSIONS IN TERMS OF QUADRATIC FORMS +(A) ABSTRACT BOUNDARY CONDITIONS THE WEYL FUNCTION CONSTRUCTION OF A RIGGED HILBERT SPACE CONNECTIONS WITH SELF-ADJOINT OPERATORS A-SCALES OF HILBERT SPACES PROPERTIES OF THE A-SCALE QUADRATIC FORMS SYMMETRIC QUADRATIC FORMS, CLOSABILITY SINGULAR QUADRATIC FORMS ON HILBERT SPACE A CANONICAL DECOMPOSITION FOR QUADRATIC FORMS PROPERTIES OF SINGULAR QUADRATIC FORMS OPERATOR REPRESENTATION FOR SINGULAR FORMS SINGULAR FORMS AND OPERATORS IN THE A-SCALE DENSELY EMBEDDING OF SUBSPACE CONSTRUCTION OF DENSE SUBSPACES PRELIMINARIES AND NOTATIONS THE SHIFT METHOD THE INTERSECTION METHOD OTHER VERSIONS OF DENSENESS CONDITIONS DENSE SUBSPACES IN SCALES OF THE SOBOLEV SPACES A NON-DENSENESS DEFECT THE DENSENESS PROBLEM IN TERMS OF AN A-SCALE ORIENTATION SINGULAR PERTURBATIONS THE FORM-SUM METHOD THE GENERALIZED OPERATOR SUM THE UNIQUENESS PROBLEM RIGGED SPACES AND SINGULAR PERTURBATIONS THE SINGULARITY PHENOMENON E?ECTS OF THE SINGULARITY PHENOMENON ˜A-SCALES GENERATED BY SINGULAR QUADRATIC FORMS SINGULAR RANK-ONE PERTURBATIONS OF HIGHER ORDERS ON S-SIMILARITY OF HILBERT SCALES THE OPERATOR ASSOCIATED WITH A DENSE SUBSPACE THE SETTING OF THE PROBLEM ONCE MORE ON RIGGED SPACES AGAIN ABOUT DENSENESS OF EMBEDDED SUBSPACES THE OPERATOR ?A CONSTRUCTION OF THE OPERATOR ?D IDEA OF THE METHOD DETAILS OF THE METHOD NEW RIGGINGS BY MEANS OF SINGULAR QUADRATIC FORMS PARAMETRIZATION OF SUPER-SINGULAR PERTURBATIONS THE POINT SPECTRUM OF SINGULARLY PERTURBED OPERATORS ON THE POINT SPECTRUM ARISING UNDER SINGULAR ?NITE-RANK PERTURBATIONS THE INVERSE EIGENVALUE PROBLEM A GENERAL CONSTRUCTION THE EIGENVALUE PROBLEM FOR RANK-ONE SINGULAR PERTURBATIONS THE EIGENVALUE PROBLEM FOR SINGULARLY PERTURBED RANK-ONE OPERATORS THE INDUCTIVE METHOD IN THE INVERSE EIGENVALUE PROBLEM REFERENCES