We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.
We study resolvent estimates for non-selfadjoint semiclassical pseudodifferential operators with double characteristics. Assuming that the quadratic approximation along the double characteristics is elliptic, we obtain polynomial upper bounds on the resolvent in a suitable region inside the pseudospectrum.
For certain Dirac operators $ð_ϕ$ associated to a fibered boundary metric $g_ϕ$, we provide a pseudodifferential characterization of the limiting behavior of $(ð_ϕ+kγ)^{-1}$ as $k\searrow 0$, where $γ$ is a self-adjoint operator anti-commuting with $ð_ϕ$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $ð_ϕ^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give …
The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with $C^{1,1}$ Hölder smoothness, by use of a recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols.
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.
A phase-space anisotropic operator in H=L^2(R^n) is a self-adjoint operator whose resolvent family belongs to a natural C*-completion of the space of Hörmander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schrödinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its "phase-space asymptotic localizations", obtained as limits over diverging ultrafilters of R^{n}\times R^n-translations of the operator. The…
We consider a non-self-adjoint $h$-pseudodifferential operator $P$ in the semi-classical limit ($h\to 0$). If $p$ is the leading symbol, then under suitable assumptions about the behaviour of $p$ at infinity, we know that the resolvent $(z-P)^{-1}$ is uniformly bounded for $z$ in any compact set not intersecting the closure of the range of $p$. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance ${\cal O}(1)((h\ln \frac{1}{h})^{k/(k+1)})$ from certain boundary points, where $k\in \{2,4,...\}$. This is a slight improvement of a result b…
We consider self-adjoint realizations of a second-order elliptic differential expression on ${\mathbb R}^n$ with singular interactions of $δ$ and $δ^\prime$-type supported on a compact closed smooth hypersurface in ${\mathbb R}^n$. In our main results we prove spectral asymptotics formulae with refined remainder estimates for the singular values of the resolvent difference between the standard self-adjoint realizations and the operators with a $δ$ and $δ^\prime$-interaction, respectively. Our technique makes use of general pseudodifferential methods, classical results on spectral asymptotics o…
For a strongly elliptic second-order operator $A$ on a bounded domain $Ω\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(Ω)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of Hölder $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show…
We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary. For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C_0(A,B) of (-λ)^{-N} in the asymptotic expansion of the resolvent trace Tr(A(B-λ)^{-N}) (with N large) in powers and log-powers of λas λtends to infinity in a sui…
The spectral behavior of the difference between the resolvents of two realizations $\tilde A_1$ and $\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $νu=b_1γ_0u$ and $νu=b_2γ_0u$, can in the case where all coefficients are $C^\infty$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ of $(\tilde A_1 -λ)^{-1}-(\tilde A_2 -λ)^{-1}$ s…
This paper is a continuation of the investigation of resolvents of elliptic operators on conic manifolds from math.AP/0410178 and math.AP/0410176 to the case of manifolds with boundary and realizations of operators under boundary conditions. We prove the existence of sectors of minimal growth for realizations of boundary value problems for cone operators under natural ellipticity conditions on the symbols. Special attention is devoted to the clarification of the analytic structure of the resolvent.
We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter varies in cones; one of the two branches of boundary for the cones being given by the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.
Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C_0(A,P) in the trace expansion of A(P-λ)^{-N} (with an auxiliary elliptic operator P), as determined in a joint work with Seeley 1995. The definition of TR A is extended from the cases of noninteger order, or integer order and even-even parity on odd-dimensional manifolds, to the case of even-odd parity on even-dimensional manifolds. For the ge…
In asymptotic expansions of resolvent traces $\Tr(A(P-λ)^{-1})$ for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-λ)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Oki…
For $2a$-order strongly elliptic operators $P$ generalizing $(-Δ)^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $Ω\subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\infty $, when $Ω$ is $C^{τ+1}$ with a finite $τ>2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar resu…
Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω\subset R^n, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) s_j(G) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**)…
Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $Δ$ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of $Δ$. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis fo…
We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm-Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the zeta-determinant of an infinite direct sum of Sturm-Liouville operators. The Laplace-Beltrami operator on a surface of revo…
We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable window in the complex spectral plane, provided that the strength of the non-selfadjoint perturbation $\gg h$ (or sometimes $\gg h^…
We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schrödinger operators with a radially symmetric potential falling off like $-γr^{-2},\;γ>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $γ>(l+d/2-1)^2$. In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic form…
For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between zero and lambda when lambda tends to infinity, under an additional dynamical condition. This is a well known result of J.J.Duistermaat and V.W.Guillemin from 1975. In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in …
In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{χ,Σ_+}$ in $L_2(Ω)$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $Ω\subset R^n$; here the boundary $\partialΩ=Σ$ is partioned …
