Self-adjoint Elliptic Operators Research Articles

A family index theorem for periodic Hamiltonian systems and bifurcation

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah–Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theorem to study bifurcation of branches of periodic solutions for families of nonlinear Hamiltonian systems.

English

We investigate Hormander spectral multiplier theorems as they hold on $X = L^p(\Omega),\: 1 < p < \infty,$ for many self-adjoint elliptic differential operators $A$ including the standard Laplacian on $\R^d.$ A strengthened matricial extension is considered, which coincides with a completely bounded map between operator spaces in the case that $X$ is a Hilbert space. We show that the validity of the matricial Hormander theorem can be characterized in terms of square function estimates for imaginary powers $A^{it}$, for resolvents $R(\lambda,A),$ and for the analytic semigroup $\exp(-zA).$ We deduce Hormander spectral multiplier theorems for semigroups satisfying generalized Gaussian estimates.

Notes on the non self-adjoint elliptic differential operators

Notes on the non self-adjoint elliptic differential operators

Carleman estimates for anisotropic elliptic operators with jumps at an interface

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.

The regularity of the $$\eta $$ η function for the Shubin calculus

We prove the regularity of the $$\eta $$ function for classical pseudodifferential operators with Shubin symbols. We recall the construction of complex powers and of the Wodzicki and Kontsevich–Vishik functionals for classical symbols on $$\mathbb{R }^{n}$$ with these symbols. We then define the $$\zeta $$ and $$\eta $$ functions associated to suitable elliptic operators. We compute the $$K_{0}$$ group of the algebra of zero-order operators and use this knowledge to show that the Wodzicki trace of the idempotents in the algebra vanishes. From this, it follows that the $$\eta $$ function is regular at $$0$$ for all self-adjoint elliptic operator of positive order.

Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $\Omega\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(\Omega)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partial\Omega)$.

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.

On the Convergence to the Continuum of Finite Range Lattice Covariances

In (J. Stat. Phys. 115:415–449, 2004) Brydges, Guadagni and Mitter proved the existence of multiscale expansions of a class of lattice Green’s functions as sums of positive definite finite range functions (called fluctuation covariances). The lattice Green’s functions in the class considered are integral kernels of inverses of second order positive self-adjoint elliptic operators with constant coefficients and fractional powers thereof. The rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing L −n and satisfies uniform bounds. Our main result in this note is that the sequence of these terms converges in appropriate norms at a rate L −n/2 to a smooth, positive definite, finite range continuum function.

ON THE SINGULARITIES OF THE ZETA AND ETA FUNCTIONS OF AN ELLIPTIC OPERATOR

Let P be a self-adjoint elliptic operator of order m &gt; 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form [Formula: see text], where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators

We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.

A collocation method for solving some integral equations in distributions

A collocation method is presented for the numerical solution of a typical integral equation Rh:=∫DR(x,y)h(y)dy=f(x),x∈D¯ of the class R, whose kernels are of positive rational functions of arbitrary selfadjoint elliptic operators defined in the whole space Rr, and D⊂Rr is a bounded domain. Several numerical examples are given to demonstrate the efficiency and stability of the proposed method.

A 𝐾-theoretic proof of the Morse index theorem in semi-Riemannian geometry

We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K K -theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.

Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

Two-Grid Finite Element Discretization Schemes Based on Shifted-Inverse Power Method for Elliptic Eigenvalue Problems

This paper discusses highly efficient discretization schemes for solving self-adjoint elliptic differential operator eigenvalue problems. Several new two-grid discretization schemes, including the conforming and nonconforming finite element schemes, are proposed by combining the finite element method with the shifted-inverse power method for matrix eigenvalue problems. With these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$ and the solution of a linear algebraic system on the fine grid $\pi_{h}$. Theoretical analysis shows that the schemes have a high efficiency. For instance, the resulting solution can maintain an asymptotically optimal accuracy by using the conforming linear element or the nonconforming Crouzeix–Raviart element by taking $H=O(\sqrt[4]{h})$. Numerical experiments are presented to support the theoretical analysis. In addition, this paper establishes multigrid discretization schemes and proves their efficiency.

EIGENFUNCTION EXPANSIONS AND SPACETIME ESTIMATES FOR GENERATORS IN DIVERGENCE-FORM

Let [Formula: see text] be a formally self-adjoint (elliptic) operator in L2 (ℝn), n ≥ 2. The real coefficients aj, k(x) = ak, j(x) are assumed to be bounded and to coincide with -Δ outside of a ball. The paper deals with two topics: (i) An eigenfunction expansion theorem, proving in particular that H is unitarily equivalent to -Δ, and (ii) Global spacetime estimates for the associated inhomogeneous wave equation, proved under suitable ("nontrapping") additional assumptions on the coefficients. The main tool used here is a Limiting Absorption Principle (LAP) in the framework of weighted Sobolev spaces, which holds also at the threshold.

Homogenization of the Spectral Problem for Periodic Elliptic Operators with Sign-Changing Density Function

The paper deals with the asymptotic behaviour of spectra of second order self-adjoint elliptic operators with periodic rapidly oscillating coefficients in the case when the density function (the factor on the spectral parameter) changes sign. We study the Dirichlet problem in a regular bounded domain and show that the spectrum of this problem is discrete and consists of two series, one of them tending towards +∞ and another towards −∞. The asymptotic behaviour of positive and negative eigenvalues and their corresponding eigenfunctions depends crucially on whether the average of the weight function is positive, negative or equal to zero. We construct the asymptotics of eigenpairs in all three cases.

Eta forms and the odd pseudodifferential families index

Let A(t) be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration φ with base Y. The standard example is A + it where A is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and t ∈ R is the ‘suspending’ parameter. Let πA : A(φ) −→ Y be the infinite-dimensional bundle with fibre at y ∈ Y consisting of the Schwartz-smoothing perturbations, q, making Ay(t) + q(t) invertible for all t ∈ R. The total eta form, ηA, as described here, is an even form on A(φ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family: (*) dηA = π ∗ AγA, Ch(ind(A)) = [γA] ∈ H (Y ). The 1-form part of this identity may be interpreted in terms of the τ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family A with (*) giving the ‘curving’ as the 3-form part of the Chern character of the index. We also give ‘universal’ versions of these constructions over a classifying space for odd K-theory. Introduction Eta forms, starting with the eta invariant itself, appear as the boundary terms in the index formula for Dirac operators [2], [5], [4], [10], [11]. One aim of the present paper is to show that, with the freedom gained by working in the more general context of families of pseudodifferential operators, these forms appear as universal transgression, or connection, forms for the cohomology class of the index. That these forms arise in the treatment of boundary problems corresponds to the fact that boundary conditions amount to the explicit inversion of a suspended (or model) problem on the boundary. The odd index of the boundary family is trivial and the eta form is an explicit trivialization of it in cohomology. To keep the discussion within bounds we work here primarily in the ‘odd’ setting of a family of self-adjoint elliptic pseudodifferential operators, taken to be of order 1, on the fibres of a fibration of compact manifolds

Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.

An L2 model for selfadjoint elliptic differential operators with constant coefficients on bounded domains

AbstractThe selfadjoint realization of a second order elliptic differential expression with Dirichlet boundary conditions is shown to be unitarily equivalent to the maximal multiplication operator with the independent variable in an explicit L2 model space. (© 2009 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)