Searcharxiv⌕ Search

arXiv subjects

Maxim Braverman

Publications and source records attributed to Maxim Braverman.

At least 19 recordsLinked to original sources
arXiv2026

The classical Weyl law for Schr\"odinger operators on complete Riemannian manifolds

We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard geometric assumptions such as bounded geometry, nor on analytic assumptions such as the doubling condition on the potential. Instead, we identify a geometric-analytic invariant that encodes the precise balance between the geometry of the manifold, the growth of $V$, and the oscillation scale of $V$. This intrinsic quantity, denoted $c_{\delta}(\lambda)$ admits effective quantitative estimates. We prove that the Weyl asymptotic holds provided $\lim_{\lambda\to\infty} c_\delta(\lambda)=0 .$ The sharpness of this criterion is demonstrated through explicit examples showing that the Weyl law can fail when the criterion is violated.

math.DG↗
arXiv2025

A gluing formula for the $Z_2$-valued index of odd symmetric operators

We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index.

math.DG↗
arXiv2024

Background Cohomology and Symplectic Reduction

We consider a Hamiltonian action of a compact Lie group $G$ on a complete \ka manifold $M$ with a proper moment map. In a previous paper, we defined a regularized version of the Dolbeault cohomology of a $G$-equivariant holomorphic vector bundle, called the background cohomology. In this paper, we show that the background cohomology of a prequantum line bundle over $M$ `commutes with reduction', i.e. the invariant part of the background cohomology is isomorphic to the usual Dolbeault cohomology of the symplectic reduction.

math.SG↗
arXiv2024

The Z_2-valued spectral flow of a symmetric family of Toeplitz operators

We consider families $A(t)$ of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary $\mathbb{Z}_2$-valued spectral flow of such families. We prove an analog of the Atiyah-Singer-Robbin-Salamon theorem, showing that this secondary spectral flow of $A(t)$ is equal to the secondary $\mathbb{Z}_2$-valued index of the suspension operator $\frac{d}{dt}+A(t)$. Applying this result, we show that the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold equals the secondary index of a certain Callias-type operator. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel's index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf-Porta module for 2D topological insulators of type AII.

math.DG↗
arXiv2024

On the Z_2-valued index of elliptic odd symmetric operators on non-compact manifolds

We investigate elliptic operators with a symmetry that forces their index to vanish. We study the secondary index, defined modulo 2. We examine Callias-type operators with this symmetry on non-compact manifolds and establish mod 2 versions of the Gromov-Lawson relative index theorem, the Callias index theorem, and the Boutet de Monvel's index theorem for Toeplitz operators.

math.DG↗
arXiv2022

Deformation Spaces, Rescaled Bundles and the Kirillov Character Formula

In this paper, we construct a smooth vector bundle over the deformation to the normal cone $\text{DNC}(V,M)$ through a rescaling of a vector bundle $E\to V$, which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham-Dirac operator using the deformation to the normal cone obtained recently by O. Mohsen.

math.DG↗
arXiv2021

Geometric quantization of b-symplectic manifolds

We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (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.

math.SG↗
arXiv2019

The Atiyah-Patodi-Singer index on manifolds with non-compact boundary

We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold $M$. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds $N_0$ and $N_1$. If the dimension of $M$ is odd we show that the latter index depends only on the restrictions $A_0$ and $A_1$ of $D$ to $N_0$ and $N_1$ and thus is an invariant of the boundary. We use this invariant to define the relative eta-invariant $η(A_1,A_0)$. We show that even though in our situation the eta-invariants of $A_1$ and $A_0$ are not defined, the relative eta-invariant behaves as if it was the difference $η(A_1)-η(A_0)$.

math.DG↗
arXiv2019

An index of strongly Callias operators on Lorentzian manifolds with non-compact boundary

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant introduced by Braverman and Shi. This extends recent results of Bär and Strohmaier, who studied the index of a hyperbolic Dirac operator on a spatially compact globally hyperbolic manifold.

math.DG↗
arXiv2018

APS index theorem for even-dimensional manifolds with non-compact boundary

We study the index of the APS boundary value problem for a strongly Callias-type operator $D$ on a complete even dimensional Riemannian manifold $M$ (the odd dimensional case was considered in our previous paper arXiv:1706.06737). We use this index to define the relative $η$-invariant $η(A_1,A_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the $η$-invariants of $A_1$ and $A_0$ are not defined, the relative $η$-invariant behaves as if it were the difference $η(A_1)-η(A_0)$. We also define the spectral flow of a family of such operators and use it compute the variation of the relative $η$-invariant.

math.DG↗
arXiv2018

The spectral Flow of a family of Toeplitz operators

We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain in $C^n$. This result is similar to the Boutet de Monvel's computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. Finally, we show that the bulk-boundary correspondence in a tight-binding model of topological insulators is a special case of our result. In the appendix, Koen van den Dungen reviewed the main result in the context of (unbounded) KK-theory.

math.DG↗
arXiv2018

The index of a local boundary value problem for strongly Callias-type operators

We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on M. Our result extends an index theorem of D. Freed to non-compact manifolds, thus providing a new insight on the Horava-Witten anomaly.

math.DG↗
arXiv2017

Equivariant APS index for Dirac operators of non-product type near the boundary

We consider a generalized APS boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type operators. In particular, we get a formula for the equivariant signature of a local system over a manifold with boundary. In case of a trivial local system, our formula can be viewed as a new way to compute the infinitesimal equivariant eta-invariant of S. Goette. We explicitly compute all the terms in this formula, which involve the equivariant Hirzebruch L-form and its transgression, for four-dimensional SKR manifolds, a class including many Kaehler conformally Einstein manifolds, in the case where the boundary is given as the zero level set of a certain Killing potential. In the case of SKR metrics which are local Kaehler products, these terms are zero, and we obtain a vanishing result for the infinitesimal equivariant eta invariant.

math.DG↗
arXiv2016

Refined analytic torsion as analytic function on the representation variety and applications

We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a unitary point. Finally we provide a new proof of Bruening-Ma gluing formula for the Ray-Singer torsion associated to a non-Hermitian connection. Our proof is quite different from the one given by Bruening and Ma and uses a temporal gauge transformation.

math.SP↗
arXiv2016

Callias-type operators in von Neumann algebras

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's $L^2$-index theorem, which states that the index of a Callias-type operator on a non-compact manifold $M$ is equal to the $Γ$-index of its lift to a Galois cover of $M$. We also prove the cobordism invariance of the index of Callias-type operators. In particular, we give a new proof of the cobordism invariance of the von Neumann index of operators on compact manifolds.

math.DG↗
arXiv2015

Cobordism Invariance of the Index of Callias-Type Operators

We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two non-compact, but topologically simpler manifolds. As another application we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.

math.DG↗