13 research outputs found
On the class SI of J-contractive functions intertwining solutions of linear differential equations
In the PhD thesis of the second author under the supervision of the third
author was defined the class SI of J-contractive functions, depending on a
parameter and arising as transfer functions of overdetermined conservative 2D
systems invariant in one direction. In this paper we extend and solve in the
class SI, a number of problems originally set for the class SC of functions
contractive in the open right-half plane, and unitary on the imaginary line
with respect to some preassigned signature matrix J. The problems we consider
include the Schur algorithm, the partial realization problem and the
Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence
between elements in a given subclass of SI and elements in SC. Another
important tool in the arguments is a new result pertaining to the classical
tangential Schur algorithm.Comment: 46 page
Multi-operator colligations and multivariate characteristic functions
In the spectral theory of non-self-adjoint operators there is a well-known
operation of product of operator colligations. Many similar operations appear
in the theory of infinite-dimensional groups as multiplications of double
cosets. We construct characteristic functions for such double cosets and get
semigroups of matrix-valued functions in matrix balls.Comment: 15p
Super-Radiant Dynamics, Doorways, and Resonances in Nuclei and Other Open Mesoscopic Systems
The phenomenon of super-radiance (Dicke effect, coherent spontaneous
radiation by a gas of atoms coupled through the common radiation field) is well
known in quantum optics. The review discusses similar physics that emerges in
open and marginally stable quantum many-body systems. In the presence of open
decay channels, the intrinsic states are coupled through the continuum. At
sufficiently strong continuum coupling, the spectrum of resonances undergoes
the restructuring with segregation of very broad super-radiant states and
trapping of remaining long-lived compound states. The appropriate formalism
describing this phenomenon is based on the Feshbach projection method and
effective non-Hermitian Hamiltonian. A broader generalization is related to the
idea of doorway states connecting quantum states of different structure. The
method is explained in detail and the examples of applications are given to
nuclear, atomic and particle physics. The interrelation of the collective
dynamics through continuum and possible intrinsic many-body chaos is studied,
including universal mesoscopic conductance fluctuations. The theory serves as a
natural framework for general description of a quantum signal transmission
through an open mesoscopic system.Comment: 85 pages, 10 figure
Higher cohomology of parabolic actions on certain homogeneous spaces
We show that for a parabolic R^d-action on a compact quotient of PSL(2,R)^d, the cohomologies in degrees 1 through d-1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers we have established these results for certain Anosov systems. The present work extends the methods of those papers to systems that are not Anosov. The main new idea is in Section 4, where we define special elements of representation spaces that allow us to modify the arguments from the previous papers. In Section 7 we discuss how one may generalize this strategy to R^d-systems coming from a product of Lie groups, like in the systems we have here