17. Dezember 2024
Ian Wood (Kent University)
Complete non-selfadjointness for Schroedinger operators on the semi-axis
An operator is completely non-selfadjoint if it has no non-trivial reducing subspace on which it acts as a selfadjoint operator.Complete non-selfadjointness is an important property of an operator on a Hilbert space which, in particular, plays a crucial role in the construction of a selfadjoint dilation of a maximally dissipative operator. It is often a surprisingly difficult property to prove.
In this talk we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential. We show that all proper maximally dissipative extensions (that preserve the differential expression) are completely non-selfadjoint. However, it is possible for non-proper maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterisation of these extensions and the corresponding subspaces and present a specific example. This is joint work with Christoph Fischbacher and Sergey Naboko.