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Resonance Theory for Schroedinger Operators
Resonances which result from perturbation of embedded eigenvalues are studied
by time dependent methods. A general theory is developed, with new and weaker
conditions, allowing for perturbations of threshold eigenvalues and relaxed
Fermi Golden rule. The exponential decay rate of resonances is addressed; its
uniqueness in the time dependent picture is shown is certain cases. The
relation to the existence of meromorphic continuation of the properly weighted
Green's function to time dependent resonance is further elucidated, by giving
an equivalent time dependent asymptotic expansion of the solutions of the
Schr\"odinger equation. \keywords{Resonances; Time-dependent Schr\"odinger
equation
Time Dependent Resonance Theory
An important class of resonance problems involves the study of perturbations
of systems having embedded eigenvalues in their continuous spectrum. Problems
with this mathematical structure arise in the study of many physical systems,
e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger
states of the helium atom, as well as in spectral geometry and number theory.
We present a dynamic (time-dependent) theory of such quantum resonances. The
key hypotheses are (i) a resonance condition which holds generically
(non-vanishing of the {\it Fermi golden rule}) and (ii) local decay estimates
for the unperturbed dynamics with initial data consisting of continuum modes
associated with an interval containing the embedded eigenvalue of the
unperturbed Hamiltonian. No assumption of dilation analyticity of the potential
is made. Our method explicitly demonstrates the flow of energy from the
resonant discrete mode to continuum modes due to their coupling. The approach
is also applicable to nonautonomous linear problems and to nonlinear problems.
We derive the time behavior of the resonant states for intermediate and long
times. Examples and applications are presented. Among them is a proof of the
instability of an embedded eigenvalue at a threshold energy under suitable
hypotheses.Comment: to appear in Geometrical and Functional Analysi
Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian
and are perturbations of linear dispersive equations. The unperturbed dynamical
system has a bound state, a spatially localized and time periodic solution. We
show that, for generic nonlinear Hamiltonian perturbations, all small amplitude
solutions decay to zero as time tends to infinity at an anomalously slow rate.
In particular, spatially localized and time-periodic solutions of the linear
problem are destroyed by generic nonlinear Hamiltonian perturbations via slow
radiation of energy to infinity. These solutions can therefore be thought of as
metastable states.
The main mechanism is a nonlinear resonant interaction of bound states
(eigenfunctions) and radiation (continuous spectral modes), leading to energy
transfer from the discrete to continuum modes.
This is in contrast to the KAM theory in which appropriate nonresonance
conditions imply the persistence of invariant tori. A hypothesis ensuring that
such a resonance takes place is a nonlinear analogue of the Fermi golden rule,
arising in the theory of resonances in quantum mechanics. The techniques used
involve: (i) a time-dependent method developed by the authors for the treatment
of the quantum resonance problem and perturbations of embedded eigenvalues,
(ii) a generalization of the Hamiltonian normal form appropriate for infinite
dimensional dispersive systems and (iii) ideas from scattering theory. The
arguments are quite general and we expect them to apply to a large class of
systems which can be viewed as the interaction of finite dimensional and
infinite dimensional dispersive dynamical systems, or as a system of particles
coupled to a field.Comment: To appear in Inventiones Mathematica
Nonautonomous Hamiltonians
We present a theory of resonances for a class of non-autonomous Hamiltonians
to treat the structural instability of spatially localized and time-periodic
solutions associated with an unperturbed autonomous Hamiltonian.
The mechanism of instability is radiative decay, due to resonant coupling of
the discrete modes to the continuum modes by the time-dependent perturbation.
This results in a slow transfer of energy from the discrete modes to the
continuum. The rate of decay of solutions is slow and hence the decaying bound
states can be viewed as metastable. The ideas are closely related to the
authors' work on (i) a time dependent approach to the instability of
eigenvalues embedded in the continuous spectra, and (ii) resonances, radiation
damping and instability in Hamiltonian nonlinear wave equations. The theory is
applied to a general class of Schr\"odinger equations. The phenomenon of
ionization may be viewed as a resonance problem of the type we consider and we
apply our theory to find the rate of ionization, spectral line shift and local
decay estimates for such Hamiltonians.Comment: To appear in Journal of Statistical Physic
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