2,683 research outputs found
Schr\"odinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds
We consider Schr\"odinger operators H=- \d^2/\d r^2+V on
with the Dirichlet boundary condition. The potential may be local or
non-local, with polynomial decay at infinity. The point zero in the spectrum of
is classified, and asymptotic expansions of the resolvent around zero are
obtained, with explicit expressions for the leading coefficients. These results
are applied to the perturbation of an eigenvalue embedded at zero, and the
corresponding modified form of the Fermi golden rule.Comment: 17 pages, 2 figure
A complete classification of threshold properties for one-dimensional discrete Schr\"{o}dinger operators
We consider the discrete one-dimensional Schr\"{o}dinger operator ,
where and is a self-adjoint operator on
with a decay property given by extending to a compact
operator from to
for some . We give a complete
description of the solutions to , and ,
. Using this description we give
asymptotic expansions of the resolvent of at the two thresholds and
. One of the main results is a precise correspondence between the solutions
to and the leading coefficients in the asymptotic expansion of the
resolvent around . For the resolvent expansion we implement the expansion
scheme of Jensen-Nenciu \cite{JN0, JN1} in the full generality.Comment: 51 page
Metastable states when the Fermi Golden Rule constant vanishes
Resonances appearing by perturbation of embedded non-degenerate eigenvalues
are studied in the case when the Fermi Golden Rule constant vanishes. Under
appropriate smoothness properties for the resolvent of the unperturbed
Hamiltonian, it is proved that the first order Rayleigh-Schr\"odinger expansion
exists. The corresponding metastable states are constructed using this
truncated expansion. We show that their exponential decay law has both the
decay rate and the error term of order , where is
the perturbation strength.Comment: To appear in Commun. Math. Phy
Memory effects in non-interacting mesoscopic transport
Consider a quantum dot coupled to two semi-infinite one-dimensional leads at
thermal equilibrium. We turn on adiabatically a bias between the leads such
that there exists exactly one discrete eigenvalue both at the beginning and at
the end of the switching procedure. It is shown that the expectation on the
final bound state strongly depends on the history of the switching procedure.
On the contrary, the contribution to the final steady-state corresponding to
the continuous spectrum has no memory, and only depends on the initial and
final values of the bias.Comment: 17 pages, submitte
Schrödinger operators on the half line: Resolvent expansions and the Fermi Golden Rule at threshold
On L<sup>p</sup> boundedness of wave operators for four dimensional Schrödinger operators with threshold singularities
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