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Roberto Paoletti

Publications and source records attributed to Roberto Paoletti.

At least 19 recordsLinked to original sources
arXiv2025

Eigenfunction asymptotics in the complex domain for a compact Lie group

Let $(G,\kappa)$ be a compact connected Lie group endowed with a biinvariant Riemannian metric, and let $\tilde{G}$ be the complexification of $G$. We apply Grauert tube techniques to the near-diagonal scaling asymptotics of certain operator kernels, which are defined in terms of the matrix elements of an irreducuble representation drifting to infinity along a ray in weight space. These kernels are the equivariant components of Poisson and Szeg\H{o} kernels on a fixed sphere bundle in $\tilde{G}$, when the latter is identified with the tangent bundle of $G$ in an appropriate way.

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arXiv2024

Equivariant scaling asymptotics for Poisson and Szeg\H{o} kernels on Grauert tube boundaries

Let $(M,\kappa)$ be a closed and connected real-analytic Riemannian manifold, acted upon by a compact Lie group of isometries $G$. We consider the following two kinds of equivariant asymptotics along a fixed Grauer tube boundary $X^\tau$ of $(M,\kappa)$. 1): Given the induced unitary representation of $G$ on the eigenspaces of the Laplacian of $(M,\kappa)$, these split over the irreducible representations of $G$. On the other hand, the eigenfunctions of the Laplacian of $(M,\kappa)$ admit a simultaneous complexification to some Grauert tube. We study the asymptotic concentration along $X^\tau$ of the complexified eigenfunctions pertaining to a fixed isotypical component. 2): There are furthermore an induced action of $G$ as a group of CR and contact automorphisms on $X^\tau$, and a corresponding unitary representation on the Hardy space $H(X^\tau)$. The action of $G$ on $X^\tau$ commutes with the homogeneous \lq geogesic flow\rq\, and the representation on the Hardy space commutes with the elliptic self-adjoint Toeplitz operator induced by the generator of the goedesic flow. Hence each eigenspace of the latter also splits over the irreducible representations of $G$. We study the asymptotic concentration of the eigenfunctions in a given isotypical component. We also give some applications of these asymptotics.

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arXiv2023

Poisson and Szeg\"{o} kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

We review and elaborate on recent work of Chang and Rabinowitz on scaling asymptotics of Poisson and Szeg\"{o} kernels on Grauert tubes, providing additional results that may be useful in applications. In particular, focusing on the near-diagonal case, we give an explicit description of the leading order terms, and an estimate on the growth of the degree of certain polynomials describing the rescaled asymptotics. Furthermore, we allow rescaled asymptotics in a range $O\left(\lambda^{\epsilon-1/2}\right)$ in all the variables involved, where $\lambda\rightarrow+\infty$ is the asymptotic parameter, rather than rescale according to Heisenberg type.

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arXiv2022

The symplectic structure of a toric conic transform

Suppose that a compact $r$-dimensional torus $T^r$ acts in a holomorphic and Hamiltonian manner on polarized complex $d$-dimensional projective manifold $M$, with nowhere vanishing moment map $\Phi$. Assuming that $\Phi$ is transverse to the ray through a given weight $\boldsymbol{\nu}$, associated to these data there is a complex $(d-r+1)$-dimensional polarized projective orbifold $\hat{M}_{\boldsymbol{\nu}}$ (referred to as the $\boldsymbol{\nu}$-th \textit{conic transform} of $M$). Namely, $\hat{M}_{\boldsymbol{\nu}}$ is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polarization of $M$. With the aim to clarify the geometric significance of this construction, we consider the special case where $M$ is toric, and show that $\hat{M}_{\boldsymbol{\nu}}$ is itself a K\"{a}hler toric obifold, whose moment polytope is obtained from the one of $M$ by a certain "transform", operation (depending on $\Phi$ and $\boldsymbol{\nu}$).

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arXiv2021

Conic reductions for Hamiltonian actions of $U(2)$ and its maximal torus

Suppose given a Hamiltonian and holomorphic action of $G=U(2)$ on a compact Kähler manifold $M$, with nowhere vanishing moment map. Given an integral coadjoint orbit $\mathcal{O}$ for $G$, under transversality assumptions we shall consider two naturally associated 'conic', reductions. One, which will be denoted $\overline{M}^G_{\mathcal{O}}$, is taken with respect to the action of $G$ and the cone over $\mathcal{O}$; another, which will be denoted $\overline{M}^T_{\boldsymbolν}$, is taken with respect to the action of the standard maximal torus $T\leqslant G$ and the ray $\mathbb{R}_+\,\imath\boldsymbolν$ along which the cone over $\mathcal{O}$ intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some results in this directions. Furthermore, we give explicit transversailty criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatoric data associated to the action and the orbit.

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arXiv2021

Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions

Suppose that a compact and connected Lie group $G$ acts on a complex Hodge manifold $M$ in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle $A$ on $M$. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szegö kernels near certain loci defined by the moment map.

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arXiv2020

Polarized orbifolds associated to quantized Hamiltonian torus actions

Suppose given an holomorphic and Hamiltonian action of a compact torus $T$ on a polarized Hodge manifold $M$. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of $T$ on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight $\boldsymbolν$ the $\boldsymbolν$-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through $\boldsymbolν$, we give a gometric interpretation of the isotypical components associated to the weights $k\,\boldsymbolν$, $k\rightarrow +\infty$, in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of $M$ in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map.

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arXiv2018

Equivariant Asymptotics of Szegö kernels under Hamiltonian $SU(2)$-action

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local and global asymptotic properties the equivariant projector associated to a weight $k \, \boldsymbol{ ν}$, when $\boldsymbol{ ν}$ is fixed and $k\rightarrow +\infty$.

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arXiv2018

Equivariant Asymptotics of Szegö kernels under Hamiltonian $U(2)$ actions

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical component are all finite dimensional, they are generally not spaces of sections of some power of $A$. One is then led to study the local and global asymptotic properties the isotypical component associated to a weight $k \, \boldsymbolν$, when $k\rightarrow +\infty$. In this paper, part of a series dedicated to this general theme, we consider the case $G=U(2)$.

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arXiv2016

Local trace formulae for commuting Hamiltonians in Töplitz quantization

Let $(M,J,ω)$ be a quantizable compact Kähler manifold, with quantizing Hermitian line bundle $(A,h)$, and associated Hardy space $H(X)$, where $X$ is the unit circle bundle. Given a collection of $r$ Poisson commuting quantizable Hamiltonian functions $f_j$ on $M$, there is an induced Abelian unitary action on $H(X)$, generated by certain Töplitz operators naturally induced by the $f_j$'s. As a multi-dimensional analogue of the usual Weyl law and trace formula, we consider the problem of describing the asymptotic clustering of the joint eigenvalues of these Töplitz operators along a given ray, and locally on $M$ the asymptotic concentration of the corresponding joint eigenfunctions. This problem naturally leads to a \lq directional local trace formula\rq, involving scaling asymptotics in the neighborhood of certain special loci in $M$. Under natural transversality assumption, we obtain asymptotic expansions related to the local geometry of the Hamiltonian action and flow.

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arXiv2016

Local scaling asymptotics for the Gutzwiller trace formula in Berezin-Töplitz quantization

Under certain hypothesis on the underlying classical Hamiltonian flow, we produce local scaling asymptotics in the semiclassical regime for a Berezin-Töplitz version of the Gutzwiller trace formula on a quantizable compact Kähler manifold, in the spirit of the near-diagonal scaling asymptotics of Szegö and Töplitz kernels. More precisely, we consider an analogue of the \lq Gutzwiller-Töplitz kernel\rq\, previously introduced in this setting by Borthwick, Paul and Uribe, and study how it asymptotically concentrates along the appropriate classical loci defined by the dynamics, with an explicit description of the exponential decay along normal directions. These local scaling asymptotics probe into the concentration behavior of the eigenfunctions of the quantized Hamiltonian flow. When globally integrated, they yield the analogue of the Gutzwiller trace formula.

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arXiv2014

Equivariant local scaling asymptotics for smoothed Töplitz spectral projectors

Let $X$ be the unit circle bundle of a positive line bundle on a Hodge manifold. We study the local scaling asymptotics of the smoothed spectral projectors associated to a first order elliptic Töplitz operator $T$ on $X$, possibly in the presence of Hamiltonian symmetries. The resulting expansion is then used to give a local derivation of an equivariant Weyl law. It is not required that $T$ be invariant under the structure circle action, that is, $T$ needn't be a Berezin-Töplitz operator.

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arXiv2013

Lower order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions

We consider a natural variant of Berezin-Toeplitz quantization of compact Kähler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szegö and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient. When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin-Toeplitz quantization. This continues previous work on near-diagonal scaling asymptotics of equivariant Szegö kernels in the presence of Hamiltonian torus actions.

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arXiv2013

Local scaling asymptotics in phase space and time in Berezin-Toeplitz quantization

This paper deals with the local semiclassical asymptotics of a quantum evolution operator in the Berezin-Toeplitz scheme, when both time and phase space variables are subject to appropriate scalings in the neighborhood of the graph of the underlying classical dynamics. Global consequences are then drawn regarding the scaling asymptotics of the trace of the quantum evolution as a function of time.

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arXiv2011

Scaling asymptotics for quantized Hamiltonian flows

In recent years, the near diagonal asymptotics of the equivariant components of the Szegö kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map.

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arXiv2011

Asymptotics of Szegö kernels under Hamiltonian torus actions

Let $X$ be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on $X$ may be seen as a local manifestation of the decomposition of the (generalized) Hardy space $H(X)$ into isotypes for the $S^1$-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of $H(X)$ over the irreducible representations of the group. We focus here on the case of compact tori.

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arXiv2011

Local trace formulae and scaling asymptotics in Toeplitz quantization, II

In the spectral theory of positive elliptic operators, an important role is played by certain smoothing kernels, related to the Fourier transform of the trace of a wave operator, which may be heuristically interpreted as smoothed spectral projectors asymptotically drifting to the right of the spectrum. In the setting of Toeplitz quantization, we consider analogues of these, where the wave operator is replaced by the Hardy space compression of a linearized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the local asymptotics of these smoothing kernels, and specifically how they concentrate on the fixed loci of the linearized dynamics.

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