Spectral properties of Schrödinger operators on compact manifolds: Rigidity,flows, interpolation and spectral estimates

This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.     

Norm convergence of multiple ergodic averages on amenable groups

We consider abstract non-negative self-adjoint operators on L²(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on Rⁿ, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.     

Asymptotic properties of the heat kernel on conic manifolds

We derive asymptotic properties for the heat kernel of elliptic cone (or Fuchs type) differential operators on compact manifolds with boundary. Applications include asymptotic formulas for the heat trace, counting function, spectral function, and zeta function of cone operators. The author was supported in part by a Ford Foundation Fellowship.     

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.     

Quadratic estimates and functional calculi of perturbed Dirac operators

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge–Dirac operator on compact manifolds depend analytically on L_∞ changes in the metric. We also recover a unified proof of many results in the Calderón program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces.     

Spectral Convergence of Quasi-One-Dimensional Spaces

We consider a family of non-compact manifolds X_ε (“graph-like manifolds”) approaching a metric graph X₀ and establish convergence results of the related natural operators, namely the (Neumann) Laplacian and the generalized Neumann (Kirchhoff) Laplacian on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations. Communicated by Claude Alain Pillet Submitted: December 21, 2005 Accepted: January 30, 2006     

On gravity,torsion and the spectral action principle

We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine–Connes spectral action, deduce the equations of motions and discuss critical points.     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

The eta invariant and parity conditions

We give a formula for the η-invariant of odd-order operators on even-dimensional manifolds and even-order operators on odd-dimensional manifolds. Second-order operators with nontrivial η-invariants are found. This solves a problem posed by Gilkey.     

Intrinsic Covering Dimension for Nuclear C*-Algebras with Real Rank Zero

For operators belonging either to a class of global bisingular pseudodifferential operators on \({{\mathbb{R}^{m}} \times {\mathbb{R}^{n}}}\) or to a class of bisingular pseudodifferential operators on a product \({M \times N}\) of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain operator-valued, homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to larger classes of Toeplitz type operators.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Chernoff approximations of Feller semigroups in Riemannian manifolds

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.     

Non-Gaussian Aspects of Heat Kernel Behaviour

A large number of papers written over the last ten years haveconcerned the spectral theory of Laplace–Beltrami operatorson complete Riemannian manifolds, and of other self-adjointsecond order elliptic operators. Much of the interest has centredon the relationship between various types of Sobolev inequality,parabolic Harnack inequalities and the Liouville property onthe one hand, and Gaussian heat kernel bounds on the other.For manifolds of bounded geometry there is an important connectionbetween this problem and a corresponding one for discrete Laplacianson graphs. Standard references are [9, 37] and more recent literaturecan be traced via [5, 16, 32].     

Myers-Type Theorems and Some Related Oscillation Results

In this paper we study the behavior of solutions of a second-order differential equation. The existence of a zero and its localization allow us to get some compactness results. In particular we obtain a Myers-type theorem even in the presence of an amount of negative curvature. The technique we use also applies to the study of spectral properties of Schr?dinger operators on complete manifolds.     

An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K¹ representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.     

Spectral bounds for Dirac operators on open manifolds

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces.      

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

On Harmonic and C-Harmonic 1-Differentiable Forms on Sasakian Manifolds

In this paper, we firstly extend some classical operators on Sasakian manifolds acting to 1-differentiable forms on Sasakian manifolds. Next in a similar manner with the study of C-harmonic forms, we define and extend such a study for the case of 1-differentiable forms on Sasakian manifolds.     

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

A new definition of canonical conformal differential operators P _k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P _k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P _k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P _k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.