On Form-Sum Approximations of Singularly Perturbed Positive Self-adjoint Operators

✍ Scribed by Sergio Albeverio; Volodymyr Koshmanenko

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 227 KB
• Volume: 169
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3506

✦ Synopsis

We discuss singular perturbations of a self-adjoint positive operator A in Hilbert space H formally given by A T =A+T, where T is a singular positive operator (singularity means that Ker T is dense in H). We prove the following result: if T is strongly singular with respect to A in the sense that Ker T is dense in the Hilbert space H 1 (A)=D(A 1Â2 ) equipped by the graph-norm, then any suitable approximation by positive operators, T n Ä T, gives a trivial result, i.e., A T n Ä A in the strong resolvent sense, where A T n is defined as a form-sum of A and T n . A corresponding statement is true for operators T, T n of finite rank which are not necessarily positive. This can be considered as an abstract version of the well known result for the perturbation by a point interaction of the Laplace operator in L 2 (R 3 ). In the more general case, where the singular operator T has a nontrivial regular component T r in H 1 (A), we prove that A T n Ä A T r in the strong resolvent sense. We give applications to the case of perturbations of the Laplace operator by a positive Radon measure with a nontrivial singular component.