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The Laplacian ${\Delta }_g$ of a compact Riemannian manifold $(M,g)$ is called if its eigenvalue multiplicity function $m_g\left(k\right)$ is of maximal growth among metrics of the same dimension and volume. Canonical spheres $(S^n,\mathrm{can})$ and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic integrals of curvature...

Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha )$ of the form $U_{\chi }=\Pi A\chi \Pi$, where $\Pi :H^2\left(X\right)\to L^2\left(X\right)$ is a Szegö projector, where $\chi$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $\mathrm{ind}\left(U_{\chi }\right)$ when the principal symbol is unitary, or equivalently to determine...

Let $M$ be a compact manifold let $G$ be a finite group acting freely on $M$, and let ${ℳ}_G$ be the (Fréchet) space of $G$-invariant metric on $M$. A natural conjecture is that, for a generic metric in ${ℳ}_G$, all eigenspaces of the Laplacian are irreducible (as orthogonal representations of $G$). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim$M\ge$ dim$V$ for all irreducibles $V$ of $G$. As an application, we construct isospectral manifolds with simple eigenvalue...