Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let \tau:D(A)\to\X, X a Banach space, be a surjective linear map such that \|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range} (\tau')\cap\H' =\{0\}, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{\tau=0\}.}$. Any $\phi$ in the operator domain $D(A^\tau_\Theta)$ is characterized by a sort of boundary conditions on its univocally defined regular component \phireg, which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These boundary conditions are written in terms of the map $\tau$, playing the role of a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension parameter $\Theta$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in which $A\phi=T*\phi$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and Applications, vol. 13

Self-adjoint operators with inner singularities and pontryagin spaces

Let A(0) be an unbounded self-adjoint operator in a Hilbert space H-0 and let chi be a generalized element of order -m -1 in the rigging associated with Ag and the inner product (., .)(0) of H-0. In [S1, S2, S3] operators H-t, t epsilon R U {infinity}, are defined which serve as an interpretation for the family of operators A(0) + t(-1)(. , chi)(0) chi. The second summand here contains the inner singularity mentioned in the title. The operators H-t act in Pontryagin spaces of the form Pi(m) = H(0)circle plus C-m circle plus C-m where the direct summand space C-m circle plus C-m is provided with an indefinite inner product. They can be interpreted both as a canonical extension of some symmetric operator S in Pi(m) and also as extensions of a one-dimensional restriction S-0 of A(0) in H-0 and hence they can be characterized by a class of Straus extensions of S-0 as well as via M.G. Krein's formulas for (generalized) resolvents. In this paper we describe both these realizations explicitly and study their spectral properties. A main role is played by a special class of Q-functions. Factorizations of these functions correspond to the separation of the nonpositive type spectrum from the positive spectrum of H-t. As a consequence, in Subsection 7.3 a family of self-adjoint Hilbert space operators is obtained which can serve as a nontrivial quantum model associated with the operators Ag + t(-1)(. , chi)(0) chi.</p

The Spectrum Of Periodic Point Perturbations And The Krein Resolvent Formula

this paper that under certain natural conditions the Krein formula works for the case of point perturbations of elliptic operators on a manifold, too. With the help of this formula we prove that the gaps of a periodic point perturbation of such an operator are labelled by the elements of the K 0 -group of an appropriate