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Yuri A. Kordyukov

Publications and source records attributed to Yuri A. Kordyukov.

At least 19 recordsLinked to original sources
arXiv2026

Determinantal point processes associated with the Bochner-Schrödinger operator

We consider the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry under the assumption of non-degeneracy of the curvature form of $L$. For large $p$, the spectrum of $H_p$ asymptotically coincides with the union $Σ$ of all local Landau levels of the operator at the points of $X$. We study the determinantal point process on $X$ associated with the spectral projection of $H_p$ corresponding to an interval $I=(α,β)$ such that $α,β\not \in Σ$ and compute the asymptotics of its linear statistics as $p$ goes to infinity. When $X$ is compact, this implies the law of large numbers and central limit theorem for the corresponding empirical measures.

math.DG↗
arXiv2026

Monte Carlo methods on compact symplectic manifolds

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

math.DG↗
arXiv2025

Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator

The Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ on high tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry is studied under the assumption of non-degeneracy of the curvature form of $L$. For large $p$, the spectrum of $H_p$ asymptotically coincides with the union of all local Landau levels of the operator at the points of $X$. Moreover, if the union of the local Landau levels over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete. The main result of the paper is the trace asymptotics formula associated with these eigenvalues. As a consequence, we get a Weyl type asymptotic formula for the eigenvalue counting function.

math.SP↗
arXiv2025

A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator

We consider the Schrödinger operator $H(μ) = \nabla_{\bf A}^*\nabla_{\bf A} + μV$ on a Riemannian manifold $M$ of bounded geometry, where $μ>0$ is a coupling parameter, the magnetic field ${\bf B}=d{\bf A}$ and the electric potential $V$ are uniformly $C^\infty$-bounded, $V\geq 0$. We assume that, for some $E_0>0$, each connected component of the sublevel set $\{V<E_0\}$ of the potential $V$ is relatively compact. Under some assumptions on geometric and spectral properties of the connected components, we show that, for sufficiently large $μ$, the spectrum of $H(μ)$ in the interval $[0,E_0μ]$ has a gap, the spectral projection of $H(μ)$, corresponding to the interval $(-\infty,λ]$ with $λ$ in the gap, belongs to the Roe $C^*$-algebra $C^*(M)$ of the manifold $M$, and, if $M$ is not compact, its class in the $K$ theory of $C^*(M)$ is trivial.

math.DG↗
arXiv2025

Long time behavior of leafwise heat flow for Riemannian foliations

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.

dg-ga↗
arXiv2025

Adiabatic limits and spectral sequences for Riemannian foliations

For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of the corresponding eigenforms also leads to a Hodge theoretic description of this spectral sequence. This is an extension of results of Mazzeo-Melrose and R. Forman.

math.DG↗
arXiv2025

Semiclassical trace formula for the Bochner-Schrödinger operator

We study the semiclassical Bochner-Schrödinger operator $H_{p}=\frac{1}{p^2}Δ^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry. For any function $φ\in C^\infty_c(\mathbb R)$, we consider the bounded linear operator $φ(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of $p^{-1}$ in the semiclassical limit $p\to \infty$. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of $φ(H_p)$.

math.DG↗
arXiv2024

Zeta invariants of Morse forms

Let $η$ be a closed real 1-form on a closed Riemannian $n$-manifold $(M,g)$. Let $d_z$, $δ_z$ and $Δ_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian, parametrized by $z=μ+iν\in\mathbb C$ ($μ,ν\in\mathbb{R}$, $i=\sqrt{-1}$). Let $ζ(s,z)$ be the zeta function of $s\in\mathbb{C}$, defined as the meromorphic extension of the function $ζ(s,z)=\operatorname{Str}({η\wedge}\,δ_zΔ_z^{-s})$ for $\Re s\gg0$. We prove that $ζ(s,z)$ is smooth at $s=1$ and establish a formula for $ζ(1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $ζ(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as $μ\to+\infty$, uniformly on $ν$. We describe $\mathbf{z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field $X$ and the Mathai-Quillen current on $TM$ defined by $g$. Any real 1-cohomology class has a representative $η$ satisfying the hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf{z}$ by perturbing $g$, $η$ and $X$, and achieve the same limit as $μ\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $η$. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem of C.~Deninger.

math.DG↗
arXiv2024

Topology of the space of conormal distributions

Given a closed manifold $M$ and a closed regular submanifold $L$, consider the corresponding locally convex space $I=I(M,L)$ of conormal distributions, with its natural topology, and the strong dual $I'=I'(M,L)=I(M,L;Ω)'$ of the space of conormal densities. It is shown that $I$ is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace $K\subset I$ of conormal distributions supported in $L$ and for its strong dual $K'$. We construct a locally convex Hausdoff space $J$ and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to J\to 0$ as well as the transpose sequence $0\to J'\to I'\to K'\to 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty(M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $ϕ=\{ϕ^t\}$ on a compact foliated manifold $(M,F)$. It describes a Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ defined by the induced action $ϕ^*=\{ϕ^{t\,*}\}$ on the reduced cohomologies $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $ϕ$.

math.FA↗
arXiv2024

Exponential localization for eigensections of the Bochner-Schrödinger operator

We study asymptotic spectral properties of the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$ of bounded geometry under assumption that the curvature form of $L$ is non-degenerate. At an arbitrary point $x_0$ of $X$ the operator $H_p$ can be approximated by a model operator $\mathcal H^{(x_0)}$, which is a Schrödinger operator with constant magnetic field. For large $p$, the spectrum of $H_p$ asymptotically coincides, up to order $p^{-1/4}$, with the union of the spectra of the model operators $\mathcal H^{(x_0)}$ over $X$. We show that, if the union of the spectra of $\mathcal H^{(x_0)}$ over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete and the corresponding eigensections decay exponentially away the compact subset.

math.SP↗
arXiv2024

A trace formula for foliated flows

Let $F$ be a transversely oriented foliation of codimension 1 on a closed manifold $M$, and let $ϕ=\{ϕ^t\}$ be a foliated flow on $(M,F)$. Assume the closed orbits of $ϕ$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, $M^1=M\setminus M^0$ and $F^1=F|_{M^1}$. We consider two topological vector spaces, $I(F)$ and $I'(F)$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{F}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$ induced by $ϕ$. Let $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$. We define some kind of Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ of the actions $ϕ^*$ on both $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one being the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\text{\rm dis}}(ϕ)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one.

math.GT↗
arXiv2022

Semiclassical asymptotic expansions for functions of the Bochner-Schrödinger operator

The Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry is studied. For any function $φ\in \mathcal S(\mathbb R)$, we consider the bounded linear operator $φ(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem and describe an asymptotic expansion of its smooth Schwartz kernel in a fixed neighborhood of the diagonal in the semiclassical limit $p\to \infty$. In particular, we prove that the trace of the operator $φ(H_p)$ admits a complete asymptotic expansion in powers of $p^{-1/2}$ as $p\to \infty$.

math.DG↗
arXiv2022

Trace formula for the magnetic Laplacian at zero energy level

The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of the semiclassical Gutzwiller trace formula and reduces to it in the case when the magnetic field form is exact. It differs somewhat from the Guillemin-Uribe trace formula studied in the author's previous work with I.A. Taimanov. Moreover, in contrast to that work, the focus is on the trace formula at the zero energy level, which is a critical energy level. The paper gives an overview of the main notions and results related to the trace formula at the zero energy level, describes various approaches to its proof, and gives concrete examples of its computation. In addition, a brief review of the Gutzwiller trace formula for regular and critical energy levels is given.

math.DG↗
arXiv2022

Trace formula for the magnetic Laplacian on a compact hyperbolic surface

We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level.

math.DG↗
arXiv2021

Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry

We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit.

math.DG↗
arXiv2020

Semiclassical spectral analysis of the Bochner-Schrödinger operator on symplectic manifolds of bounded geometry

We study the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal.

math.DG↗
arXiv2020

Berezin-Toeplitz quantization asssociated with higher Landau levels of the Bochner Laplacian

In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${\mathcal O}(p^{3/4})$ around the points $pΛ$, where $Λ$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.

math.DG↗