Papers

We study the long-time asymptotics of heat kernels on Abelian covers of compact manifolds and on Abelian covers of principal bundles. For Abelian covers, we establish three complementary asymptotic expansions: a distributional expansion describing correlations of the heat semigroup, a local pointwise expansion of the heat kernel on compact subsets, and a global expansion valid on the natural diffusive scale, revealing the Gaussian profile governing heat propagation. We then extend these results to horizontal heat kernels on principal bundles under natural holonomy and curvature assumptions. In this setting, we show that the leading asymptotics are entirely determined by the geometry of the underlying Abelian cover, while the nontrivial fiber modes decay exponentially fast due to a uniform spectral gap. The proof combines Floquet theory on Abelian covers with the Borel–Weil decomposition on principal bundles.

We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.

Let $\mathcal H=-\Delta+|x|^2$ be the Hermite operator on $\mathbb R^2$, and let $\Pi_\lambda$ denote the spectral projection corresponding to $\lambda=2N+2$. We prove the sharp log-free endpoint estimate $||\Pi_\lambda||_{L^2(\mathbb R^2)\to L^{10/3}(\mathbb R^2)}\lesssim\lambda^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $TT^*$ argument across radial scales.

We study the clamped plate eigenvalue problem. We show that clamped plate eigenfunctions can be approximated in the $L^2$-sense by their oscillatory components, which solve the Helmholtz equation, with an error that decays exponentially fast in the frequency on any compact subdomain. Additionally, we establish an upper bound on the boundary localization of clamped plate eigenfunctions analogous to the one observed for membrane eigenfunctions.

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

Kolokolnikov conjectured that, among all simple graphs on \(n\) vertices with exactly \(2(n-2)\) edges, the complete bipartite graph maximizes algebraic connectivity. This paper proves the conjecture. The underlying Lean~4 formalization was generated with MerLean and checked by the Lean kernel.

Let $G$ be any given continuous positive function on $\mathbb{R}_+$. Let $V$ be radial with $|V(x)|\leq G(|x|)$. We prove a Landis-type theorem for any real-valued solution of $\Delta u=Vu$ on $\mathbb{R}^n$. We construct a decay threshold $e^{-g(r)}$, where $g$ is a strictly increasing function which can be computed explicitly in terms of $G$. Under suitable assumptions the exponent in the decay threshold is proportional to the Agmon distance associated with $G$.

We establish a spectral representation theorem for compact operators that are self-adjoint with respect to the duality mapping on smooth Banach spaces. The result is a natural analogue of the classical spectral theorem for compact self-adjoint operators on Hilbert spaces.

In this paper we investigate the spectral properties of certain self-adjoint quadratic pencils, occurring at the separation of the variables in the problem of the oscillation of a semi-infinite cylinder with a pseudo isotropic elastic structure.

We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a $d$-dimensional orthotope, we prove an exact parity law for the maximal $\pi^{-1}$-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first $d$ scalar width moments recover the unordered side vector, and $d-1$ moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies $M_{n,a} = \lambda_n(a)^{1/2}/\pi$. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.

We establish high-energy asymptotic expansions of Titchmarsh–Weyl functions for one-dimensional Schrödinger and Dirac operators in regions whose boundaries approach the spectrum at a prescribed polynomial rate. For bounded potentials with bounded derivatives, the expansions remain uniform up to these boundaries, with an explicit loss in the remainder determined by the rate of approach. We treat both self-adjoint Dirac operators and non-self-adjoint Dirac operators with skew-adjoint potential matrices, keeping track of the distinct half-planes in which the two scalar Weyl coordinates are naturally defined. As an application of the Schrödinger expansion, we verify the high-energy hypothesis in Kotani's construction of KdV flows: for every odd integer $p\geq3$, each real-valued $q\in W^{2p-1,\infty}(\mathbb{R})$ generates a global classical solution of the member of the KdV hierarchy indexed by $(p+1)/2$.

Geodesic flows on closed hyperbolic surfaces are a quintessential example of chaotic dynamics, i.e., systems whose long term behavior is very sensitive to initial conditions. The speed of such chaos is controlled by the spectral gap of the Laplace-Beltrami operator of the underlying hyperbolic surface. In this paper we give an overview of recent breakthroughs of Anantharaman and Monk showing that large genus closed hyperbolic surfaces have optimal spectral gap in a probabilistic sense. On the way we introduce and discuss the foundational works of many authors, from Selberg to Mirzakhani, that play a crucial role in the tour de force proof of Anantharaman and Monk.

The analytic properties of the Dirichlet-to-Neumann operator reflect the geometry of the underlying manifold. As a guiding example of this relationship, consider the Euclidean unit ball. There, the Dirichlet-to-Neumann operator has an explicitly computable spectrum, consisting of the nonnegative integers with multiplicity, and commutes with the boundary Laplacian. In this paper, we study the extent to which these properties distinguish the ball among various classes of manifolds. We show that, in every dimension, Euclidean balls are uniquely determined by their Steklov spectrum among bounded Euclidean domains with smooth and possibly disconnected boundary, answering a well-known open question. We then classify all smooth compact connected oriented surfaces with nonempty boundary whose Dirichlet-to-Neumann operator commutes with the boundary Laplacian. Finally, in dimensions three and above, we prove that, for smooth metrics conformal to the Euclidean ball, the Dirichlet-to-Neumann operator commutes with the Laplacian on the round sphere if and only if the conformal factor is radial.

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in(1,\frac 32)$ the quadratic form $Q^{\rm SP}_s[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto|u|$ provided $u\in\tilde H^s(\Omega)$ changes sign in $\Omega$.

The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\Omega$, then $\Omega$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $\Omega$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.

We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a point-free theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and solve a generalized Sylvester equation that controls the commutator between the differential structure and the highly oscillatory material properties. This abstract perspective unifies stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media–and their respective convergence rates–are entirely captured by the behavior of the spectral measures of the microscopic and macroscopic derivative operators near zero frequency.

We prove $a_n(S) \le e\,(n+1)\, s_n(S)$ for every s-number sequence $(s_n)$, every bounded linear operator $S$ between normed spaces, and every $n \in \mathbb{N}_0$, where $a_n$ are the approximation numbers, which are the largest s-numbers. This is sharp up to the constant and settles conjectures of Mityagin, Henkin, Carl and Pietsch dating back to 1963. We also extend it to widths of convex sets and discuss optimality there. The proof is elementary.

We study a two-dimensional semiclassical Schrödinger operator whose potential admits one reflection symmetry. Under a rational independence assumption on the harmonic frequencies and a nonvanishing condition \(a_{30}\ne0\), we show that the first two layers of the quantum Birkhoff normal form, together with the sign of \(a_{30}\) and the transverse data \(\{a_{1,2k}\}_{k\ge1}\), determine the Taylor series of the potential at the bottom of the well. The proof is constructive: the \(\hbar^2\)-layer gives a triangular recursion for odd-degree terms, while the classical layer recovers even-degree terms from their resonant projections.

Bounded ratios of products of minors of positive definite matrices have a long history, starting with Hadamard's inequality in 1893. It states that for every positive semidefinite matrix $A$ $$ \det A \le A_{11} \cdots A_{nn}. $$ This inequality was subsequently generalized by Fisher and then further by Koteljanskii. The latter states that for every positive semidefinite matrix $A$ and any index sets $\alpha_1, \alpha_2 \subseteq \{1,\ldots, n\}$ one has $$ \det A[\alpha_1 \cup \alpha_2] \det A[\alpha_1 \cap \alpha_2] \le \det A[\alpha_1] \det A[\alpha_2], $$ where $A[\alpha]$ denotes the principal submatrix determined by the indexes in $\alpha$. In a manuscript published only on the arXiv in 2008, Hall and Johnson made three conjectures about ratios of products of principal minors of $4\times4$ positive definite matrices, denoted by $R_i$, $i=1,2,3$, see (2) and (3). They hypothesized that the supremum of $R_1$ was $27/16$, while the supremum of the other two ratios was $1$. Such ratios are called absolutely bounded. The conjecture for $R_1$ was affirmed in [17] and it is the only known bounded determinantal ratio with supremum bigger than one. The goal of this paper is to affirm the conjecture for $R_2$ and $R_3$. It is known that the upper bound for the ratios $R_i$, $i=1,2,3$, does not follow from repeated applications of Koteljanskii's inequality. In addition, Hall and Johnson showed that $R_i$ is bounded above by $4$, for $i=1,2,3$.

We consider the magnetic Neumann Laplacian on rectangles, subject to non-homogeneous fields. We prove that the square is a local minimiser of the lowest eigenvalue among all rectangles of a fixed area for weak magnetic fields . We conjecture that it is a global minimiser both under the area or perimeter constraints and partially prove the conjecture by establishing lower and upper bounds to the principal eigenvalue.

We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.

We study bounded finite-interaction-range operators $H$ on $\ell^2(\Gamma)$, where $\Gamma$ is a uniformly discrete subset of a left-invariant doubling metric measure space $(X,d,\mu)$. Our goal is to approximate spectral information of $H$ from finite sections $H_{L,x}$ supported on balls $B_L(x)\cap\Gamma$. The main technical input is a commutator estimate for Lipschitz ”tent” localisations $W_{L,x}$, which depends only on geometric properties (doubling) and a uniform interaction degree. As a consequence, we obtain explicit two-sided pseudospectral inclusion bounds of the form \[ \gamma_{L,\varepsilon}(H)\subset \sigma_\varepsilon(H)\subset \gamma_{L,\varepsilon+C_0/L}(H), \] and Hausdorff convergence of window pseudospectra to the (global) pseudospectrum as $L\to\infty$. In the self-adjoint/ normal case this yields computable gap tests and spectral sampling schemes with rigorous $O(1/L)$ error control, while in the non-normal case it leads to corresponding approximation results for pseudospectra. The framework isolates the geometric core behind earlier approaches on $\mathbb{R}^n$ and on countable Abelian groups, and covers irregular geometries arising in quasicrystal models, as well as new cases such as discrete (non-Abelian) nilpotent groups (including for example the discrete Heisenberg group).

We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in $\mathrm{L}^2$. For the same velocity field, every ideal solution grows at most linearly in $\mathrm{L}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not $\mathrm{C}^1$.

We develop a perturbative response calculus for spectral problems obtained by changing the speed measure of a fixed symmetric energy form in a Gaussian environment. If Cameron–Martin translation acts by $\mu_{x+f}=e^{\kappa f}\mu_x$, unitary transport identifies the varying $L^2$ spaces and produces a common-domain analytic family. For a positive eigenvalue $\Lambda$, the first-order operator is $-\kappa\Lambda$ times the compression of multiplication by $f$ to the $\Lambda$-eigenspace; its eigenvalues are the derivatives of the analytic branches issuing from $\Lambda$. A countable separation condition and finite-dimensional Gaussian disintegration then give almost-sure simplicity; a response-transversality condition and the inverse function theorem give joint densities for all finite vectors of positive ordered eigenvalues. We verify these hypotheses for every $0<\gamma<2$ in two models: Dirichlet Liouville Brownian motion on an arbitrary bounded connected planar domain, and the Liouville–Cauchy operator on the circle. In the first model the whole spectrum is almost surely simple; in the second the constants form the deterministic zero mode and the positive spectrum is almost surely simple. Transversality follows from a local eigenfunction-square identity in the Brownian case and its nonlocal jump-form analogue in the Cauchy case.

Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)\kappa g$. We introduce a weighted combinatorial Laplacian on $\varepsilon$-discretizations of $M$ and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operator. More precisely, the eigenvalues of the two operators are uniformly comparable with constants depending only on $n,\kappa,\varepsilon$, independently of the injectivity radius. As an application, we prove spectral stability under measured Gromov-Hausdorff convergence. We also recover the Schoen-Wolpert-Yau inequality using the weighted discretization on families of pinching genus-$2$ hyperbolic surfaces.

We obtain closed explicit formulas for the spectral determinants of the smooth hyperbolic Bolza surface and the Klein quartic. In each case, a multiplicative relation expresses the determinant of the surface in terms of determinants of singular quotient orbifolds of genera zero and one. The elliptic factors are evaluated by applying the singular Polyakov anomaly formula to explicit Belyi maps on CM elliptic curves of discriminants -8 and -7, while the genus-zero factors are evaluated by explicit determinant formulas for constant-curvature spheres with conical singularities. The same multiplicative relations hold fibrewise on the corresponding equisymmetric deformation strata and yield determinant and first-variation identities. We also prove that every compact quasiplatonic hyperbolic surface is a critical point of the spectral determinant on its Teichmüller space; in particular, this applies to the Bolza surface and the Klein quartic.

Topological $K$-theory supplies the decisive stable classification principle for gapped free-fermion phases in Euclidean crystals once symmetry and stabilization are fixed. In hyperbolic band theory, by contrast, it is less decisive for the full band problem: generalized momentum sectors vary through moduli spaces, but passing to ordinary $K$-classes collapses their geometric and holomorphic variation. We distinguish kinematical holomorphy, in which complex geometry organizes the sector spaces, from dynamical holomorphy, in which that geometry informs a Hamiltonian or projector, and formulate the resulting loss as an observable-factorization problem. For a compact hyperbolic surface $X$, we prove nonfactorization in three settings: fibrewise $K^0(X)$, occupied-state $K^0(B)$, and operator-algebraic $K$-theory. The affected quantities include spectra and Higgs spectral curves, the Berry holonomy and the quantum metric, the partially filled Hall response, and the Fermi surface and nodal geometries. We also identify the quantized pairings and local charges retained by topology. Up to an explicit area factor, the Kotani–Sunada bottom-band Hessian is the Hodge inner product on $H^1(X;\mathbb R)$. Together with the integral intersection form, it recovers the homology-marked principally polarized Jacobian and hence, by Torelli and uniformization, the underlying complex and hyperbolic surface, though not a full Teichmüller marking. We also separate finite-rank sectors from the thermodynamic bulk. Locally faithful covers reproduce polynomial traces exactly and control continuous spectral observables. In arithmetic congruence towers, analytic observables converge as $O(|G_n|^{-\eta})$ for some $\eta>0$, and $C^s$ observables as $O((\log |G_n|)^{-s})$, while established coherent large-rank limits recover bulk density-of-states moments.

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.