In a previous paper, we introduced a class of “semiclassical functions of isotropic type,” starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the “oscillatory functions of Lagrangian type” that have played major role in semiclassical and microlocal analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then, we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual cleanintersection hypothesis). The simplest examples of isotropic states are the “coherent states,” a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920’s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact.We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary 4.5).

In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier paper, for compact abelian groups, i.e. tori. More precisely, Let $\mathsf G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $X$. We will show that for the Schr\"odinger operator $-\hbar^2 \Delta+V$ with $V \in C^\infty(X)^{\mathsf G}$, the potential function $V$ is, in some interesting examples, determined by the $\mathsf G$-equivariant spectrum. The key ingredient in this proof is a generalized Legendrian relation between the Lagrangian manifolds $\mathrm{Graph}(dV)$ and $\mathrm{Graph}(dF)$, where $F$ is a spectral invariant defined on an open subset of the positive Weyl chamber.

In this paper we will show how the Bohr-Sommerfeld levels of a quantum completely integrable system can be computed modulo O ( ħ ∞ ) by an inductive procedure starting at stage zero with the Bohr-Sommerfeld levels of the corresponding classical completely integrable system.

Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold (M,ω). The assignment ring AT(M) is an extension of the equivariant cohomology ring HT(M); it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that AT(M) is a finitely generated S(t⁎)-module, and give a criterion guaranteeing that a given set of assignments generates (alternatively, is a basis for) this module. We define two new types of assignments, delta classes and bridge classes, and show that if the torus T is 2-dimensional, then all assignments of sufficiently high degree are generated by cohomological, delta, and bridge classes. In particular, if M is 6-dimensional, then we can find a basis of such classes.

We introduce a method of geometric quantization for compact b-symplectic manifolds in terms of the index of an Atiyah-PatodiSinger (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg "quantization commutes with reduction" property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.

There are a number of excellent texts available on semi-classical analysis. The focus of the present monograph, however, is an aspect of the subject somewhat less systematically developed in other texts: In semi-classical analysis, many of the basic results involve asymptotic expansions in which the terms can be computed by symbolic techniques, and the focus of this monograph is the symbol created thus. In particular, the techniques involved in this symbolic calculus have their origins in symplectic geometry, and the first seven chapters of the present work are a discussion of this underlying symplectic geometry. Another feature which differentiates this monograph from other texts is an emphasis on the global aspects of the subject: A considerable amount of time is spent here showing that the objects studied are coordinate-invariant and hence make sense on manifolds. Wherever possible, intrinsic coordinate-free descriptions of these objects are given. Topics discussed include wave and heat trace formulas for globally defined semi-classical differential operators on manifolds, and equivariant versions of these results involving Lie group actions.

A $2n$-dimensional Poisson manifold $(M ,\Pi)$ is said to be $b^m$-symplectic if it is symplectic on the complement of a hypersurface $Z$ and has a simple Darboux canonical form at points of $Z$ which we will describe below. In this paper we will discuss a desingularization procedure which, for $m$ even, converts $\Pi$ into a family of symplectic forms $\omega_{\epsilon}$ having the property that $\omega_{\epsilon}$ is equal to the $b^m$-symplectic form dual to $\Pi$ outside an $\epsilon$-neighborhood of $Z$ and, in addition, converges to this form as $\epsilon$ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of $b^m$-manifolds can be more clearly understood by viewing them as limits of analogous properties of the $\omega_{\epsilon}$'s. We will also prove versions of these results for $m$ odd; however, in the odd case the family $\omega_{\epsilon}$ has to be replaced by a family of folded symplectic forms.

The main results of this paper are an asymptotic expansion in powers of ħ for the spectral measure μ_ħ of a semi-classical Toeplitz operator, Q_ħ, and an equivariant version of this result when Q_ħ admits an n-torus as a symmetry group. In addition we discuss some inverse spectral consequences of these results.

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a bm -symplectic manifold.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

This book features a selection of articles by Louis Boutet de Monvel and presents his contributions to the theory of partial differential equations and analysis. The works selected here reveal his cen

In a recent preprint, we showed that for the Dirichlet Laplacian $\Delta$ on the unit disk, the wave trace ${Tr}(e^{it\sqrt{\Delta}})$, which has complicated singularities on $2\pi - \epsilon < t < 2\pi$, is, on the interval $2\pi < t < 2\pi + \epsilon$, the restriction to this interval of a $C^\infty$ function on its closure. In this paper we prove the analogue of this somewhat counter-intuitive result for the Friedlander model. The proof for the Friedlander model is simpler and more transparent than in the case of the unit disk.

We define classes of quantum states associated with isotropic submanifolds of cotangent bundles. The classes are stable under the action of semiclassical pseudo-differential operators and covariant under the action of semiclassical Fourier integral operators. We develop a symbol calculus for them; the symbols are symplectic spinors. We outline various applications.

We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a T^n-invariant Schrodinger operator on R^n determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S^1-invariant metric on S^2 determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action.

Let M be a Riemannian manifold, tau : G x M -> M an isometric action on M of an n-torus G and V : M -> R a bounded G-invariant smooth function. By G-invariance the Schrodinger operator, P = -h(2)Delta(m) + V, restricts to a self-adjoint operator on L-2(M)(alpha/h), a being a weight of G and 1/h a large positive integer. Let [c(alpha), infinity) be the asymptotic support of the spectrum of this operator. We will show that c(alpha) extend to a function, W : g* -> R and that, modulo assumptions on tau and V one can recover V from W, i.e. prove that V is spectrally determined. The main ingredient in the proof of this result is the existence of a 'generalized Legendre transform' mapping the graph of dW onto the graph of dV.

We consider a class of perturbations of the 2D harmonic oscillator, and of some other dynamical systems, which we show are isomorphic to a function of a toric system (a Birkhoff canonical form). We show that for such systems there exists a quantum normal form as well, which is determined by spectral data.

We study Hamiltonian actions on b-symplectic manifolds with a focus on the effective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classifies these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus.

In this note we calculate the variation of the Rayleigh quotient under specific variations of the metric. We consider two cases. First we study conformal variations and secondly variations within a Kähler class. We use the first variational formula to reprove an important result of Uhlenbeck’s: a generic metric can be perturbed by a conformal factor into a metric whose eigenspaces for the Laplace operator have dimension 1 1 . In the case of a Kähler manifold, we write an explicit variational formula for the Rayleigh quotient under Kähler deformations. We relate our calculation with the following question: Is the spectrum of a generic Kähler metric simple?

Let M-2n be a Poisson manifold with Poisson bivector field Pi. We say that M is b-Poisson if the map Pi(n) : M -> Lambda(2n)(TM) intersects the zero section transversally on a codimension one submanifold Z subset of M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M, Pi) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology. (C) 2014 Elsevier Inc. All rights reserved.

The concept of assignments was introduced in [GGK99] as a method for extracting geometric information about group actions on manifolds from combinatorial data encoded in the infinitesimal orbit-type stratification. In this paper we will answer in the affirmative a question posed in [GGK99] by showing that the equivariant cohomology ring of M is to a large extent determined by this data.