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Victor Guillemin

Publications and source records attributed to Victor Guillemin.

At least 37 records · Page 2Linked to original sources
arXiv2011

Balanced fiber bundles and GKM theory

Let $T$ be a torus and $B$ a compact $T-$manifold. Goresky, Kottwitz, and MacPherson show in \cite{GKM} that if $B$ is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring $H_T^*(B)$ as a subring of $H_T^*(B^T)$. In this paper we prove an analogue of this result for $T-$equivariant fiber bundles: we show that if $M$ is a $T-$manifold and $π\colon M \to B$ a fiber bundle for which $π$ intertwines the two $T-$actions, there is a simple combinatorial description of $H_T^*(M)$ as a subring of $H_T^*(π^{-1}(B^T))$. Using this result we obtain fiber bundle analogues of results of \cite{GHZ} on GKM theory for homogeneous spaces.

math.AT↗
arXiv2011

A Fulling-Kuchment theorem for the 1D harmonic oscillator

We prove that there exists a pair of "non-isospectral" 1D semiclassical Schrödinger operators whose spectra agree modulo h^\infty. In particular, all their semiclassical trace invariants are the same. Our proof is based on an idea of Fulling-Kuchment and Hadamard's variational formula applied to suitable perturbations of the harmonic oscillator. Keywords: Inverse spectral problems, semiclassical Schrödinger operators, trace invariants, Hadamard's variational formula, harmonic oscillator, Penrose mushroom, Sturm-Liouville theory.

math.SP↗
arXiv2011

Band invariants for perturbations of the harmonic oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form $S = S_0 +\h^2V$, where $S_0 = \frac 12 \Bigl(-\h^2Δ_{\bbR^n} + |x|^2\Bigr)$ is the harmonic oscillator and $V:\bbR^n\to\bbR$ is a tempered smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as $\h$ tends to zero, and compute the first two associated "band invariants". We derive several inverse spectral results for $V$, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation).

math.SP↗
arXiv2011

Equivariant inverse spectral theory and toric orbifolds

Let O be a symplectic toric 2n-dimensional orbifold with a fixed T^n-action and with a toric Kahler metric g. We previously explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator acting on smooth functions on (O,g) determines the moment polytope of O, and hence by Delzant's theorem determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results and show that the moment polytope of a generic toric orbifold is determined by its equivariant spectrum, up to two possibilities and up to translation. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature.

math.DG↗
arXiv2011

Codimension one symplectic foliations and regular Poisson structures

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we consider in a later paper.

math.SG↗
arXiv2011

Cohomology of GKM Fiber Bundles

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.

math.CO↗
arXiv2010

Singularities of the wave trace near cluster points of the length spectrum

Let $-λ_j$ be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution $h(t) = \sum_j e^{i\sqrtλ_j t}$ is the trace of the solution operator of the wave equation on the disk. It is well known that $h$ has isolated singularities at the lengths of the reflecting geodesics. In particular, $h$ is singular at $t_k$, the perimeter of the regular inscribed polygon with $k$ sides. Evidently, $t_k < 2π$, the perimeter of the circle, and $t_k$ tends to $2π$. In this paper, we show that $h(t)$ is infinitely differentiable as $t$ tends to $2π$ from the right.

math.AP↗
arXiv2009

Semiclassical Spectral Invariants for Schrödinger Operators

In this article we study the semiclassical spectral measures associated with Schrödinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give an alternative proof of Colin de Verdiere's result on recovering one dimensional potential wells from semiclassical spectral data. We also study the relation of semiclassical spectra measures to Birkhoff normal forms and describe a generalization of the asymptotic expansion above to the Schrödinger operator in the presence of a magnetic field.

math.SP↗
arXiv2009

Stability Functions

In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to problems in toric geometry. In particular we show how one can use these techniques to recover results of Burns-Guillemin-Uribe and Shiffman-Tate-Zelditch on asymptotic properties of sections of holomorphic line bundles over toric varieties.

math.SG↗
arXiv2009

Some Remarks About Semiclassical Trace Invariants and Quantum Normal Forms

In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrodinger operators. In particular we give a "non-symbolic" operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semi-classical BCF.

math-ph↗
arXiv2008

Geodesics on weighted projective spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can "hear" the weights of a weighted projective space.

math.SP↗
arXiv2008

A semi-classical inverse problem I: Taylor expansions

In dimension 1, we show that the Taylor expansion of a potential near a generic non degenerate critical point can be recovered from the knowledge of the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value.

math-ph↗
arXiv2007

The Mellin transform and spectral properties of toric varieties

In this article we apply results of \cite{W} on the twisted Mellin transform to problems in toric geometry. In particular we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, $z^k$, $k \in \ZZ^d$, in Bargmann space and prove an "upstairs" version of the spectral density theorem of \cite{BGU}. We also obtain for the $z^k$'s, "upstairs" versions of the results of \cite{STZ} on distribution laws for eigenstates on toric varieties.

math.SG↗
arXiv2006

Riemann sums over polytopes

We show that the Euler-MacLaurin formula for Riemann sums has an n-dimensional analogue in which intervals on the line get replaced by convex polytopes.

math.CO↗
arXiv2006

Determinants of zeroth order operators

We show that the Szego regularized determinant of a zeroth order operator differs from the zeta regularized determinant of the same operator by a local term. In addition, we compute multiplicative anomalies for the zeta regularized determinant.

math.SP↗
arXiv2004

Signature quantization, representations of compact Lie groups, and a q-analogue of the Kostant partition function

We discuss some applications of signature quantization to the representation theory of compact Lie groups. In particular, we prove signature analogues of the Kostant formula for weight multiplicities and the Steinberg formula for tensor product multiplicities. Using symmetric functions, we also find, for type A, analogues of the Weyl branching rule and the Gelfand-Tsetlin theorem. These analogues involve a q-analogue of the Kostant partition function. We show that in type A, this q-analogue is polynomial in the relative interior of the cells of a complex of cones. This chamber complex can be taken to be the same as the chamber complex of the usual Kostant partition function. We present the case of A_2 as a detailed example.

math.CO↗