1,016 research outputs found
When is a non-self-adjoint Hill operator a spectral operator of scalar type?
We derive necessary and sufficient conditions for a one-dimensional periodic
Schr\"odinger (i.e., Hill) operator H=-d^2/dx^2+V in L^2(R) to be a spectral
operator of scalar type. The conditions demonstrate the remarkable fact that
the property of a Hill operator being a spectral operator is independent of
smoothness (or even analyticity) properties of the potential V.Comment: 5 page
On Local Borg-Marchenko Uniqueness Results
We provide a new short proof of the following fact, first proved by one of us
in 1998: If two Weyl-Titchmarsh m-functions, , of two Schr\"odinger
operators H_j = -\f{d^2}{dx^2} + q_j, j=1,2 in , , are exponentially close, that is, |m_1(z)- m_2(z)|
\underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a}), 0<a<R, then
a.e.~on . The result applies to any boundary conditions at x=0 and x=R
and should be considered a local version of the celebrated Borg-Marchenko
uniqueness result (which is quickly recovered as a corollary to our proof).
Moreover, we extend the local uniqueness result to matrix-valued Schr\"odinger
operators.Comment: LaTeX, 18 page
On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and
the associated Green's matrices for whole-line and half-line self-adjoint
Hamiltonian finite difference systems with separated boundary conditions.Comment: 30 pages, to appear in J. Comput. Appl. Mat
Symmetrized Perturbation Determinants and Applications to Boundary Data Maps and Krein-Type Resolvent Formulas
The aim of this paper is twofold: On one hand we discuss an abstract approach
to symmetrized Fredholm perturbation determinants and an associated trace
formula for a pair of operators of positive-type, extending a classical trace
formula. On the other hand, we continue a recent systematic study of boundary
data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more
generally, Robin-to-Robin maps, associated with one-dimensional Schr\"odinger
operators on a compact interval [0,R] with separated boundary conditions at 0
and R. One of the principal new results in this paper reduces an appropriately
symmetrized (Fredholm) perturbation determinant to the 2\times 2 determinant of
the underlying boundary data map. In addition, as a concrete application of the
abstract approach in the first part of this paper, we establish the trace
formula for resolvent differences of self-adjoint Schr\"odinger operators
corresponding to different (separated) boundary conditions in terms of boundary
data maps.Comment: 38 page
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