Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Johannes Brasche
TU Clausthal
On the absolutely continuous spectra of selfadjoint extensions
Let S be a symmetric operator in a Hilbert space H . We give conditions which are sufficient in order that
for every selfadjoint operator Aaux in H there exists a selfadjoint extension A of S such that the absolutely
continuous parts of A and Aaux are unitarily equivalent and show how to construct such an extension A.
Jonathan Eckhardt
Universität Wien
Inverse spectral theory for Schrödinger operators with strongly singular potentials:
a uniqueness theorem
Consider self-adjoint Schrödinger operators
H = − d2
+ q(x),
dx2 in the Hilbert space L2 (a,b), where q ∈ L1 loc (a,b) is real-valued. Given some nontrivial real entire solution
φ of
−φ ′′ (z,x) + q(x)φ(z,x) = zφ(z,x), x ∈ (a,b), z ∈ C,
which is square integrable near a and satisfies the boundary condition there (if any), it is possible
to construct an associated spectral measure. Utilizing de Branges’ theory of Hilbert spaces of entire
functions, we give conditions under which this spectral measure uniquely determines the operator H. As
an example we apply our result to perturbed Bessel operators and show that in this case the associated
spectral measure uniquely determines the operator.
185