919 research outputs found
A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators
Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator
are studied under the assumption that the weight function has one turning
point. An abstract approach to the problem is given via a functional model for
indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues
are obtained. Also, operators with finite singular critical points are
considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4,
and 3.12 extended, details added in subsections 2.3 and 4.2, section 6
rearranged, typos corrected, references adde
The similarity problem for -nonnegative Sturm-Liouville operators
Sufficient conditions for the similarity of the operator with an indefinite weight r(x)=(\sgn x)|r(x)| are
obtained. These conditions are formulated in terms of Titchmarsh-Weyl
-coefficients. Sufficient conditions for the regularity of the critical
points 0 and of -nonnegative Sturm-Liouville operators are also
obtained. This result is exploited to prove the regularity of 0 for various
classes of Sturm-Liouville operators. This implies the similarity of the
considered operators to self-adjoint ones. In particular, in the case
r(x)=\sgn x and , we prove that is similar to a
self-adjoint operator if and only if is -nonnegative. The latter
condition on is sharp, i.e., we construct such that is -nonnegative with the singular critical
point 0. Hence is not similar to a self-adjoint operator. For periodic and
infinite-zone potentials, we show that -positivity is sufficient for the
similarity of to a self-adjoint operator. In the case , we prove
the regularity of the critical point 0 for a wide class of weights . This
yields new results for "forward-backward" diffusion equations.Comment: 36 pages, LaTeX2e, version 2; addresses of the authors added, the
reference [38] update
Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints
Part I of this paper deals with two-dimensional canonical systems
, , whose Hamiltonian is non-negative and
locally integrable, and where Weyl's limit point case takes place at both
endpoints and . We investigate a class of such systems defined by growth
restrictions on H towards a. For example, Hamiltonians on of the
form where
are included in this class. We develop a direct and inverse spectral theory
parallel to the theory of Weyl and de Branges for systems in the limit circle
case at . Our approach proceeds via - and is bound to - Pontryagin space
theory. It relies on spectral theory and operator models in such spaces, and on
the theory of de Branges Pontryagin spaces.
The main results concerning the direct problem are: (1) showing existence of
regularized boundary values at ; (2) construction of a singular Weyl
coefficient and a scalar spectral measure; (3) construction of a Fourier
transform and computation of its action and the action of its inverse as
integral transforms. The main results for the inverse problem are: (4)
characterization of the class of measures occurring above (positive Borel
measures with power growth at ); (5) a global uniqueness theorem (if
Weyl functions or spectral measures coincide, Hamiltonians essentially
coincide); (6) a local uniqueness theorem.
In Part II of the paper the results of Part I are applied to Sturm--Liouville
equations with singular coefficients. We investigate classes of equations
without potential (in particular, equations in impedance form) and
Schr\"odinger equations, where coefficients are assumed to be singular but
subject to growth restrictions. We obtain corresponding direct and inverse
spectral theorems
- …