8 research outputs found
Translation Representations and Scattering By Two Intervals
Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that
a model for obstacle scattering can be built, up to unitary equivalence, with
the use of translation representations for L2-functions in the complement of
two finite and disjoint intervals.
The model encompasses a family of systems (U (t), H). For each, we obtain a
detailed spectral representation, and we compute the scattering operator, and
scattering matrix. We illustrate our results in the Lax-Phillips model where (U
(t), H) represents an acoustic wave equation in an exterior domain; and in
quantum tunneling for dynamics of quantum states
Equiconvergence of Spectral Decompositions for Sturm–Liouville Operators: Triples of Lebesgue Spaces
Equiconvergence of spectral decompositions of Hill operators
We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = -d (2)/dx (2) + v(x), x a L (1)([0, pi], with H (per) (-1) -potential and the free operator L (0) = -d (2)/dx (2), subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that
parallel to S-N - S-N(0) : L-a -> L-b parallel to -> 0 if 1 < a <= b < infinity, 1/a - 1/b < 1/2,
where S (N) and S (N) (0) are the N-th partial sums of the spectral decompositions of L and L (0). Moreover, if v a H (-alpha) with 1/2 < alpha < 1 and , then we obtain the uniform equiconvergence aEuro-S (N) -S (N) (0) : L (a) -> L (a)aEuro- -> 0 as N -> a