4 research outputs found
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 of self-adjoint
operators which are extensions of the symmetric operator .
Any in the operator domain 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 , playing the role of
a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension
parameter 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 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
Spectral Theory for Schrödinger Operators with -Interactions Supported on Curves in
The main objective of this paper is to systematically develop a spectral and
scattering theory for selfadjoint Schr\"odinger operators with
-interactions supported on closed curves in . We provide
bounds for the number of negative eigenvalues depending on the geometry of the
curve, prove an isoperimetric inequality for the principal eigenvalue, derive
Schatten--von Neumann properties for the resolvent difference with the free
Laplacian, and establish an explicit representation for the scattering matrix.Comment: to appear in Annales Henri Poincar