10,985 research outputs found
Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold
We consider the index problem for a wide class of nonlocal elliptic operators
on a smooth closed manifold, namely differential operators with shifts induced
by the action of an isometric diffeomorphism. The key to the solution is the
method of uniformization: We assign to the nonlocal problem a
pseudodifferential operator with the same index, acting in sections of an
infinite-dimensional vector bundle on a compact manifold. We then determine the
index in terms of topological invariants of the symbol, using the Atiyah-Singer
index theorem.Comment: 16 pages, no figure
Pseudospectra in non-Hermitian quantum mechanics
We propose giving the mathematical concept of the pseudospectrum a central
role in quantum mechanics with non-Hermitian operators. We relate
pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint
operators, and basis properties of eigenfunctions. The abstract results are
illustrated by unexpected wild properties of operators familiar from
PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion
excluding basis property (Proposition 6) added, unbounded time-evolution
discussed, new reference
Differential systems of pure Gaussian type
We give the transformation rule for the Stokes data of the Laplace transform
of a differential system of pure Gaussian type.Comment: 31 pages. V2: final version to appear in Izv. Mat
Elliptic regularity and solvability for partial differential equations with Colombeau coefficients
The paper addresses questions of existence and regularity of solutions to
linear partial differential equations whose coefficients are generalized
functions or generalized constants in the sense of Colombeau. We introduce
various new notions of ellipticity and hypoellipticity, study their
interrelation, and give a number of new examples and counterexamples. Using the
concept of \G^\infty-regularity of generalized functions, we derive a general
global regularity result in the case of operators with constant generalized
coefficients, a more specialized result for second order operators, and a
microlocal regularity result for certain first order operators with variable
generalized coefficients. We also prove a global solvability result for
operators with constant generalized coefficients and compactly supported
Colombeau generalized functions as right hand sides
Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback
We use the backstepping method to study the stabilization of a 1-D linear
transport equation on the interval (0, L), by controlling the scalar amplitude
of a piecewise regular function of the space variable in the source term. We
prove that if the system is controllable in a periodic Sobolev space of order
greater than 1, then the system can be stabilized exponentially in that space
and, for any given decay rate, we give an explicit feedback law that achieves
that decay rate
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