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Using mixed data in the inverse scattering problem
Consider the fixed- inverse scattering problem. We show that the zeros
of the regular solution of the Schr\"odinger equation, , which are
monotonic functions of the energy, determine a unique potential when the domain
of the energy is such that the range from zero to infinity. This
suggests that the use of the mixed data of phase-shifts
, for which the zeros of the regular solution are monotonic in both domains,
and range from zero to infinity, offers the possibility of determining the
potential in a unique way.Comment: 9 pages, 2 figures. Talk given at the Conference of Inverse Quantum
Scattering Theory, Hungary, August 200
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
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