25,606 research outputs found
Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
The abstract theory of self-adjoint extensions of symmetric operators is used
to construct self-adjoint realizations of a second-order elliptic operator on
with linear boundary conditions on (a relatively open part of)
a compact hypersurface. Our approach allows to obtain Krein-like resolvent
formulas where the reference operator coincides with the "free" operator with
domain ; this provides an useful tool for the scattering
problem from a hypersurface. Concrete examples of this construction are
developed in connection with the standard boundary conditions, Dirichlet,
Neumann, Robin, and -type, assigned either on a
dimensional compact boundary or on a relatively open
part . Schatten-von Neumann estimates for the difference
of the powers of resolvents of the free and the perturbed operators are also
proven; these give existence and completeness of the wave operators of the
associated scattering systems.Comment: Final revised version, to appear in Journal of Differential Equation
Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities
Let X be a surface with an isolated singularity at the origin, given by the
equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In
particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C)
finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson
homology of Y, as a graded vector space with respect to the weight grading. In
the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson
homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth
Hochschild homology of the n-th symmetric power of the algebra of G-invariant
differential operators on C. That is, the Brylinski spectral sequence
degenerates in this case. In the elliptic case, this yields the zeroth
Hochschild homology of symmetric powers of the elliptic algebras with three
generators modulo their center, for the parameter equal to all but countably
many points of the elliptic curve.Comment: 17 page
Spectral functions of non essentially selfadjoint operators
One of the many problems to which J.S. Dowker devoted his attention is the
effect of a conical singularity in the base manifold on the behavior of the
quantum fields. In particular, he studied the small- asymptotic expansion of
the heat-kernel trace on a cone and its effects on physical quantities, as the
Casimir energy. In this article we review some peculiar results found in the
last decade, regarding the appearance of non-standard powers of , and even
negative integer powers of , in this asymptotic expansion for the
selfadjoint extensions of some symmetric operators with singular coefficients.
Similarly, we show that the -function associated to these selfadjoint
extensions presents an unusual analytic structure.Comment: 57 pages, 1 figure. References added. Version to appear in the
special volume of Journal of Physics A in honor of Stuart Dowker's 75th
birthday. PACS numbers: 02.30.Tb, 02.30.Sa, 03.65.D
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