7 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 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