130 research outputs found
Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems
For a Dirac-type operator D with a spectral boundary condition, the
associated heat operator trace has an expansion in powers and log-powers of t.
Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case.
We here investigate the effect of perturbations of D, by use of a
pseudodifferential parameter-dependent calculus for boundary problems. It is
shown that the first k log-terms are stable under perturbations of D vanishing
to order k at the boundary (and the nonlocal power coefficients behind them are
only locally perturbed). For perturbations of D from the APS product case by
tangential operators commuting with the tangential part A, all the
log-coefficients vanish if the dimension is odd.Comment: Published. Abstract added, small typos correcte
Remarks on nonlocal trace expansion coefficients
In a recent work, Paycha and Scott establish formulas for all the Laurent
coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a
formula for the zero'th coefficient at s=0, in terms of two functions
generalizing, respectively, the Kontsevich-Vishik canonical trace density, and
the Wodzicki-Guillemin noncommutative residue density of an associated
operator. The purpose of this note is to provide a proof of that formula
relying entirely on resolvent techniques (for the sake of possible
generalizations to situations where powers are not an easy tool).
- We also give some corrections to transition formulas used in our earlier
works.Comment: Minor corrections. To appear in a proceedings volume in honor of K.
Wojciechowski, "Analysis and Geometry of Boundary Value Problems", World
Scientific, 19 page
Spectral asymptotics for Robin problems with a discontinuous coefficient
The spectral behavior of the difference between the resolvents of two
realizations and of a second-order strongly elliptic
symmetric differential operator , defined by different Robin conditions and , can in the case where all
coefficients are be determined by use of a general result by the
author in 1984 on singular Green operators. We here treat the problem for
nonsmooth . Using a Krein resolvent formula, we show that if and
are in , the s-numbers of satisfy for
all ; this improves a recent result for by Behrndt et al., that
. A sharper estimate is obtained when
and are in for some , with jumps at a
smooth hypersurface, namely that for , with
a constant defined from the principal symbol of and .
As an auxiliary result we show that the usual principal spectral asymptotic
estimate for pseudodifferential operators of negative order on a closed
manifold extends to products of pseudodifferential operators interspersed with
piecewise continuous functions.Comment: 20 pages, notation simplified. To appear in J. Spectral Theor
The sectorial projection defined from logarithms
For a classical elliptic pseudodifferential operator P of order m>0 on a
closed manifold X, such that the eigenvalues of the principal symbol p_m(x,\xi)
have arguments in \,]\theta,\phi [\, and \,]\phi, \theta +2\pi [\, (\theta
<\phi <\theta +2\pi), the sectorial projection \Pi_{\theta, \phi}(P) is defined
essentially as the integral of the resolvent along {e^{i\phi}R_+}\cup
{e^{i\theta}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have
pointed out that there is a flaw in several published proofs that \P_{\theta,
\phi}(P) is a \psi do of order 0; namely that p_m(x,\xi) cannot in general be
modified to allow integration of (p_m(x,\xi)-\lambda)^{-1} along
{e^{i\phi}R_+}\cup {e^{i\theta}R_+} simultaneously for all \xi . We show that
the structure of \Pi_{\theta, \phi}(P) as a \psi do of order 0 can be deduced
from the formula \Pi_{\theta, \phi}(P)= (i/(2\pi))(\log_\theta (P) - \log_\phi
(P)) proved in an earlier work (coauthored with Gaarde). In the analysis of
\log_\theta (P) one need only modify p_m(x,\xi) in a neighborhood of
e^{i\theta}R_+; this is known to be possible from Seeley's 1967 work on complex
powers.Comment: Quotations elaborated, 6 pages, to appear in Mathematica Scandinavic
Integration by parts and Pohozaev identities for space-dependent fractional-order operators
Consider a classical elliptic pseudodifferential operator on
of order ( with even symbol. For example, where
is a second-order strongly elliptic differential operator; the
fractional Laplacian is a particular case. For solutions of
the Dirichlet problem on a bounded smooth subset , we
show an integration-by-parts formula with a boundary integral involving
, where
. This extends recent results of
Ros-Oton, Serra and Valdinoci, to operators that are -dependent,
nonsymmetric, and have lower-order parts. We also generalize their formula of
Pohozaev-type, that can be used to prove unique continuation properties, and
nonexistence of nontrivial solutions of semilinear problems. An illustration is
given with . The basic step in our analysis is a
factorization of , , where we set up a calculus for the
generalized pseudodifferential operators that come out of the
construction.Comment: Final version to appear in J. Differential Equations, 42 pages.
References adde
The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates
For a second-order symmetric strongly elliptic operator A on a smooth bounded
open set \Omega in R^n with boundary \Sigma, the mixed problem is defined by a
Neumann-type condition on a part Sigma_+ of the boundary and a Dirichlet
condition on the other part Sigma_-. We show a Krein resolvent formula, where
the difference between its resolvent and the Dirichlet resolvent is expressed
in terms of operators acting on Sobolev spaces over Sigma_+. This is used to
obtain a new Weyl-type spectral asymptotics formula for the resolvent
difference (where upper estimates were known before), namely s_j j^{2/(n-1)}\to
C_{0,+}^{2/(n-1)}, where C_{0,+} is proportional to the area of Sigma_+, in the
case where A is principally equal to the Laplacian.Comment: 29 pages, proofreading corrections, to appear in J. Math. Anal. App
Extension theory for elliptic partial differential operators with pseudodifferential methods
This is a short survey on the connection between general extension theories
and the study of realizations of elliptic operators A on smooth domains in R^n,
n > 1. The theory of pseudodifferential boundary problems has turned out to be
very useful here, not only as a formulational framework, but also for the
solution of specific questions. We recall some elements of that theory, and
show its application in several cases (including recent results), namely to the
lower boundedness question, and the question of spectral asymptotics for
differences between resolvents.Comment: 26 pages, style changed to LaTeX, new material added at the end, to
appear in the Lecture Notes Series of the London Math. Soc. published by
Cambridge Univ. Pres
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