88 research outputs found
Analyticity and uniform stability of the inverse singular Sturm--Liouville spectral problem
We prove that the potential of a Sturm--Liouville operator depends
analytically and Lipschitz continuously on the spectral data (two spectra or
one spectrum and the corresponding norming constants). We treat the class of
operators with real-valued distributional potentials in the Sobolev class
W^{s-1}_2(0,1), s\in[0,1].Comment: 25 page
On closed embeddings of free topological algebras
Let be a complete quasivariety of completely regular universal
topological algebras of continuous signature (which means that
is closed under taking subalgebras, Cartesian products, and
includes all completely regular topological -algebras algebraically
isomorphic to members of ). For a topological space by
we denote the free universal -algebra over in the class
. Using some extension properties of the Hartman-Mycielski
construction we prove that for a closed subspace of a metrizable (more
generally, stratifiable) space the induced homomorphism
between the respective free universal algebras is a closed topological
embedding. This generalizes one result of V.Uspenskii concerning embeddings of
free topological groups.Comment: 3 page
Eigenvalue asymptotics for Sturm--Liouville operators with singular potentials
We derive eigenvalue asymptotics for Sturm--Liouville operators with singular
complex-valued potentials from the space W^{\al-1}_{2}(0,1), \al\in[0,1],
and Dirichlet or Neumann--Dirichlet boundary conditions. We also give
application of the obtained results to the inverse spectral problem of
recovering the potential from these two spectra.Comment: Final version as appeared in JF
Norm resolvent convergence of singularly scaled Schr\"odinger operators and \delta'-potentials
For a real-valued function V from the Faddeev-Marchenko class, we prove the
norm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon of
one-dimensional Schr\"odinger operators on the line of the form S_\epsilon:=
-D^2 + \epsilon^{-2} V(x/\epsilon). Under certain conditions the family of
potentials converges in the sense of distributions to the first derivative of
the Dirac delta-function, and then the limit of S_\epsilon might be considered
as a "physically motivated" interpretation of the one-dimensional Schr\"odinger
operator with potential \delta'.Comment: 30 pages, 2 figure; submitted to Proceedings of the Royal Society of
Edinburg
Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra
We solve the inverse spectral problem of recovering the singular potentials
of Sturm-Liouville operators by two spectra. The
reconstruction algorithm is presented and necessary and sufficient conditions
on two sequences to be spectral data for Sturm-Liouville operators under
consideration are given.Comment: 14 pgs, AmS-LaTex2
Inverse spectral problems for Sturm-Liouville operators with singular potentials, IV. Potentials in the Sobolev space scale
We solve the inverse spectral problems for the class of Sturm--Liouville
operators with singular real-valued potentials from the Sobolev space
W^{s-1}_2(0,1), s\in[0,1]. The potential is recovered from two spectra or from
the spectrum and norming constants. Necessary and sufficient conditions on the
spectral data to correspond to the potential in W^{s-1}_2(0,1) are established.Comment: 16 page
- β¦