Papers

Motivated by scattering theory, this paper proves general results about the distribution of zeros in logarithmic neighborhoods of the real axis for a certain class of functions holomorphic in the closed lower half-plane. We define a new indicator function that measures the growth of the holomorphic function along logarithmic curves, and connect this to the distribution of zeros of the function. This is related to classical results on the distribution of zeros in sectors for entire functions of completely regular growth. These results can be applied to the determinant of the scattering matrix of a Schrodinger operator on odd-dimensional Euclidean space, yielding bounds on resonance counting functions for logarithmic neighborhoods of the real axis. As a further application, we study a certain family of potentials in one dimension having jump-like singularities. Using our complex-analytic results we show that the singularities can lead to many – and even infinitely many – strings of resonances along logarithmic curves. We connect the properties of these strings of resonances, including their location and linear density, both to the parameters describing the singularities of the potential and to the asymptotic behavior of the scattering determinant.

The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth $h>0$. Specifically, given $n$ i.i.d. samples from a probability measure $\mu$ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator $\Delta_{\mu_n}^h$ to those of the population counterpart $\Delta_\mu^h$ under a spectral gap condition, bounding the relative error by $1/\sqrt{nv_\mu(h)}$ for eigenvalues of order smaller than $h^{-2}$, where $v_\mu(h)$ is the smallest mass of a ball of radius $h$. This bound requires very weak regularity conditions on $\mu$: it is satisfied if $\mu$ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale $O(h)$. Even for measures having densities of regularity $s>2$ on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.

Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $\kappa>0$ must satisfy $\sigma_1\geq \kappa$. We disprove this conjecture for every $n\geq 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2t\Phi}g_{\mathbb{R}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and $\sigma_1(\mathbb{B}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.

We prove that the law of one smooth scalar process in a finite Wiener chaos completely determines finite families of symmetric tensors on arbitrary real separable Hilbert spaces, up to simultaneous orthogonal equivalence. Gaussian graph characters recover all contractions, while an intrinsic trace class Gram operator reduces the problem to finite dimensional invariant theory. Applied at a single fixed positive time to the canonical quadratic and quartic heat packet, this principle reconstructs the heat generator and forces every reconstructed unitary to be spatial. Consequently, one universal scalar process determines closed Riemannian manifolds, compact $\operatorname{RCD}$ spaces of finite dimension, and Euclidean bundles with metric connection and self adjoint potential. The heat packet functor is fully faithful, and its tensor automorphisms are precisely the geometric ones.

We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local Hölder continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky–Spencer and Bourgain–Schlag for Schrödinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the “shift” of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the “shift” outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy–Foia\c s theory and the operator ${\bf A}_2$ condition.

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

We prove that every Steklov isospectral family of compact smooth planar domains is compact in the $C^\infty$ topology, answering an open question of Colbois, Girouard, Gordon, and Sher. The proof has two main parts. First, a trace comparison principle for Dirichlet-to-Neumann operators yields monotonicity of negative Steklov zeta values and compactness within a fixed conformal class for genus-zero flat surfaces. Second, we analyze the normalized Steklov determinant on degenerating hyperbolic surfaces with geodesic boundary. Its asymptotics are expressed in terms of shrinking boundary components and small Neumann and Dirichlet eigenvalues, which in genus zero are compared with weighted graph Laplacians. This rules out degeneration under a bound on the total hyperbolic boundary length. We finally establish this bound for planar domains by geometric non-collapse estimates.

Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\leq T^\beta$, $\beta<1/2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)/2}(\log T)^{-r/2}$. The proof makes use of a novel kernel argument and a uniform bound on the spherical function.

We prove a Poincaré inequality, uniform in the particle number, for repulsive (one-component) Coulomb gases in dimensions $d\ge3$, with inverse temperature $\beta_N$ that is allowed to depend on $N$. We assume a weak-coupling condition expressed in terms of the effective interaction strength $\Theta_N$ seen by a single particle. This yields exponential relaxation for the associated overdamped Langevin dynamics. The main ingredient is a one-site Poincaré inequality uniform in the number and positions of the Coulomb poles, including coincident poles. A moving-pole estimate and Dobrushin-Wu tensorization yield the many-particle lower bound, while the center-of-mass coordinate gives the matching upper bound.

The power graph $P(G)$ of a finite group $G$ is the graph with vertex set $G$ and edge set $E(P(G))=\{uv:\ u,v \in G,\ u \neq v,\ u \in \langle v \rangle \ \text{or}\ v \in \langle u \rangle\},$ where $\langle x\rangle$ denotes the cyclic subgroup generated by $x$. In this paper, we characterise all the finite groups with exponent $q$ whose power graphs are friendship graphs, firefly-type graphs, or torch graphs. We prove that the power graph of a finite group $G$ with exponent $q$ is a friendship graph if and only if $q=3$. In particular, in the abelian case, this is equivalent to $G\cong\mathbb{Z}_3^{n}$. We further show that, among all the symmetric and alternating groups, only $S_3$ and $A_4$ have firefly-type power graphs, whereas no finite group has a power graph isomorphic to a torch graph. Finally, we determine the generalised distance spectra $D_{\alpha}$-spectra of these graph classes.

We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so called non critical and critical cases. In the non-critical case the essential spectrum is unchanged - it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator - while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.

We study reducibility of the characteristic polynomials of a class of finite Jacobi pencils. We classify the diagonal equality patterns for which generic connected couplings yield reducibility and use this classification to prove that for $n\ge4$, the connected reducible locus has codimension at least two in the connected parameter space. For $n=2m+1$, we also prove that for every fixed connected coupling vector, the locus of diagonal potentials that admit a constant branch has codimension $m$.

For normal matrices A and B, the classical Hoffman-Wielandt theorem bounds the optimal matching distance between their spectra by the Frobenius norm of A-B. We prove the sharper estimate d(\sigma(A),\sigma(B))^2 \le \sum_{k=1}^{\lfloor (n+1)/2 \rfloor} s_k(A-B)^2, where s_k(M) are the singular values of M, involving only the first \lfloor (n+1)/2 \rfloor singular values of the difference. Consequently, d(\sigma(A),\sigma(B)) \le \sqrt{\lfloor (n+1)/2 \rfloor}\,\|A-B\|, which improves the classical dimension-free bound for all 3 \le n \le 16. The proof combines a min-max duality for optimal matching distances with spectral subspace overlaps and the monotonicity of singular values under rectangular compressions.

Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann–Poincaré operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min–max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}\lambda_j^+(\Omega)\ge \frac{d-2}{2}.$ It follows that $\lambda_1^+(\Omega)\ge \frac{d-2}{2d},$ with equality if and only if $\Omega$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin–Shtrikman inequality $|\Omega|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partial\Omega}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $\Omega$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.

Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Schäffer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subsequent work showed that $S_n$ grows like $\sqrt n$, but the sharp asymptotic constant has remained open for more than five decades. We resolve this problem by proving \[ \lim_{n\to\infty}\frac{S_n}{\sqrt n}=\sqrt e. \] Thus Schäffer's upper bound is asymptotically sharp, including its constant. Our proof is constructive, providing explicit Banach-space norms through duality and explicit matrices through the theory of model operators. At the analytic core of the argument, an extremal formulation of Schäffer's problem in the Wiener algebra reduces the matching asymptotic lower bound for $S_n$ to uniformly controlling the Taylor coefficients of products $QB_n$, where $B_n$ is a finite Blaschke product of degree $n$ and $Q$ is a polynomial factor. We optimize simultaneously the zero distribution of $B_n$ and the choice of $Q$. The resulting zeros follow a logarithmic asymptotic distribution, and a sharp uniform asymptotic analysis of the Taylor coefficients of $QB_n$ yields the constant $\sqrt e$. The corresponding model operators then yield matrices with these spectra that asymptotically attain Schäffer's bound.

This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincaré-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.

In this paper, dissipative singular $q$-Dirac operators are examined in Hilbert space $\mathcal{L}_{q}^{2}(q^{{\mathbb{Z}}};\mathbb{C}^{2})$, where they arise as extensions of a minimal symmetric operator in the limit-point case. We develop a selfadjoint dilation and get its incoming and outgoing spectral representations, enabling the explicit computation of the scattering matrix associated with the dilation. Furthermore, we construct a functional model for the dissipative operator and express its characteristic function in terms of the Titchmarsh-Weyl function of the corresponding selfadjoint operator. Finally, we establish results concerning the completeness of the system of eigenvectors and associated vectors for these dissipative Dirac operators.

We present a new impossibility region for the $5\times5$ symmetric nonnegative inverse eigenvalue problem. The region lies in the low-trace regime and, to the best of our knowledge, has not been identified previously. The proof uses a suitable diagonal shift to transform the problem to a critical high-trace boundary and then reduces a complementary commuting matrix to a weighted five-cycle. The characteristic polynomial and spectral identities of this five-cycle provide the main tools for deriving the resulting contradiction.

Representations of the Kohno-Drinfeld Lie algebra associated with several classical multivariate distributions have played an important role in recent developments in the theory of quantum superintegrable systems. In this work, we analyze the corresponding Gaudin models for the multivariate Hahn, Dirichlet, multinomial, and negative multinomial distributions. More precisely, using tools from representation theory, we construct multivariate orthogonal polynomials with respect to these distributions as common eigenfunctions of Gaudin operators. The polynomials are parametrized by solutions to the Bethe ansatz equations or, equivalently, by the roots of Heine-Stieltjes polynomials.

For $\tfrac12<\alpha<1$ we construct a closed realization of $\mathcal L u=\mathcal D_1^\alpha D_0^\alpha u$ in $L^2(0,1)$ whose domain contains the singular mode $x^{\alpha-1}$. The domain, defined via the distributional Caputo composition, is characterized by $I_0^{1-\alpha}u\in H^1(0,1)$ and $D_0^\alpha u-c\in\operatorname{ran}I_1^\alpha$; we prove a Volterra representation for its elements, graph-norm completeness, and that $\mathcal L$ is the adjoint of its zero-trace restriction. Four bounded endpoint traces give an ordinary boundary triplet, so all self-adjoint extensions correspond to Lagrangian planes in $\mathbb C^4$, including coupled endpoint conditions. For each we give an exact zero-eigenvalue criterion and, when $0\in\rho(\mathcal L_\omega)$, an inverse of the form $I_0^\alpha I_1^\alpha$ plus an explicit finite-rank correction. Sharp singular-value asymptotics $s_n(\mathcal L_\omega^{-1})=(\pi n)^{-2\alpha}\bigl(1+O(n^{-1})\bigr)$ yield $\mathcal L_\omega^{-1}\in\mathfrak S_p\iff p>1/(2\alpha)$, with weak-Schatten endpoint and Weyl asymptotics — now for every invertible boundary plane, not only the positive reference problem. As an application we obtain a mild-solution theorem for a Caputo-time diffusion equation in $C([0,T];L^2(0,1))$.

We extend the notion of Plancherel convergence, introduced in the study of compact quotients of locally compact groups, to finite-area hyperbolic surfaces and establish a quantum ergodicity theorem for Plancherel sequences. We show that the expectation values of integral operators whose kernels are uniformly bounded and have uniformly bounded propagation become asymptotically equidistributed, on average, for eigenvalues in a fixed compact interval. Our result recovers the case of multiplication operators studied by Le Masson and Sahlsten, corresponding to the degenerate case of propagation bound zero and kernels supported on the diagonal. Moreover, our result allows the systole to shrink without a uniform lower bound, with the allowed degeneration constrained by Plancherel convergence. This identifies Plancherel convergence as a natural spectral framework for quantum ergodicity on degenerating hyperbolic surfaces.

We prove comparison inequalities for Dirichlet-to-Neumann maps corresponding to different non-positive Helmholtz parameters. For convex domains our bounds are sharp, and the resulting eigenvalue inequalities partially confirm an earlier conjecture, which we show does not hold in full generality. We further obtain geometry-dependent versions for arbitrary sufficiently regular domains, together with extensions to compact Riemannian manifolds with boundary. We also discuss analogous questions for metric graphs.

We study the spectral behavior of the Fourier blocks of the elastic Neumann-Poincaré operator on a torus. For the full elastic Neumann-Poincaré operator, it is known that eigenvalues accumulate at three points, namely zero and a symmetric pair of nonzero points determined by the Lamé parameters. It remains unclear whether this spectral structure persists within each individual Fourier block. We prove that, for every fixed Fourier mode, each of these three points is approached by infinitely many discrete eigenvalues from both sides. Moreover, we establish precise one-sided eigenvalue counting asymptotics with explicit and strictly positive leading coefficients. Consequently, the essential spectrum of every Fourier block is exactly given by these three points. The proof combines a rotating-frame Fourier decomposition and the Plemelj symmetrization principle to obtain a self-adjoint realization of each block. A cubic polynomial transformation removes the order-zero principal part and reduces the problem to compact pseudodifferential operators of order $-1$. The eigenvalue counting asymptotics are then determined by the principal symbols of these reduced operators.

For the intersection of two disks meeting at angle $2\alpha$, let $C(\alpha)$ be the least constant in the associated spectral-set inequality. We give a self-contained M"obius reduction to the corresponding numerical-range problem on a sector and determine the sharp constant for affine square-zero operators $B=\lambda I+N$, $N^2=0$: $C_{\mathrm{sq0}}(\alpha)=\pi\sin\alpha/(2\alpha)$. A $2\times2$ matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural $\sqrt2$ bound in arbitrary dimension for the full palindromic quadratic family, and give exact rational certificates for several larger parameter families, including a complex post-automorphism disk, the complete imaginary diameter, boundary-reaching phase arcs, and symmetric and asymmetric three-node admissible-kernel problems. We also obtain a strict central bound $|w^2|\leq\kappa_0<\sqrt2$, a two-small-zero extension, an exact two-moment criterion for the remaining boundary layer, and a verified angle-dependent envelope. Every computer-assisted assertion has an exact rational verifier. These results are dimension-free but do not determine the unrestricted fixed-lens constant.

We prove a limit multiplicity conjecture of Hee Oh for principal congruence covers of geometrically finite hyperbolic manifolds, with an explicit power-saving rate. The rate is governed by the return of Patterson–Sullivan shadows through a fixed compact core, giving a geometric interpretation to the exceptional eigenvalue density. For convex-cocompact groups, a packing argument for enlarged shadows gives a stronger rate.

We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<\gamma<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.

We study the sine-kernel time–band limiting operator $S_c$ on $L^2(0,c)$ through the entropy-type trace $\operatorname{Tr}\varphi_u(S_c)$, where $\varphi_u(x)=\log(1+(e^u-1)x)-ux$. A Fourier factorization gives $S_c=A_c^*A_c$ and $\operatorname{Tr}S_c=c$. We derive an exact identity for $\operatorname{Tr}(S_c-S_c^2)$ and the lower bound $\operatorname{Tr}(S_c-S_c^2)\ge\pi^{-2}\log c-\pi^{-3}$, which in turn yields $\operatorname{Tr}\varphi_u(S_c)\ge u\log c/(4\pi^2)$. We also obtain a positive moment expansion $\operatorname{Tr}\varphi_u(S_c)=\sum_{n\ge1}A_n(u)T_n(c)$ and, for each fixed $n$, the Landau–Widom asymptotic $T_n(c)=\log c/(\pi^2n)+o_n(\log c)$. Three natural uniformity hypotheses, denoted (LC), (MT), and (QM), lead to a quadratic lower bound of order $u^2\log c$ on logarithmically growing windows. We prove the implications among these hypotheses and the resulting bounds, with explicit constants, and explain why the fixed-index asymptotic alone does not provide the required uniformity. The hypotheses themselves remain open. Finally, a signed growing-parameter sine-kernel determinant theorem from a companion paper gives the quadratic lower bound independently of (LC), (MT), and (QM). This separates the conclusions available from elementary operator methods from those that use Riemann–Hilbert asymptotics.

Motivated by the geometric properties of curved tubes $T\left(P,a\right)$ defined by closed curves $P$, we compute the Cheeger constant $h\left(T\left(P,a\right)\right)$ in the real space forms of arbitrary dimensions with constant sectional curvature. The structure and properties of the system of Fermi coordinates allows us to parametrize the curved tube and straightforwardly compute the upper bound of $h\left(T\left(P,a\right)\right)$ using the exact formulas for the area and volume of $T\left(P,a\right)$. Next, we derive the lower bound by combining the geometry of tubes with a calibration-type argument. The key idea is to describe the tube using geodesic spheres moving along the underlying curve $P$, which provides a natural outward direction and makes the estimate geometrically transparent. This allows us to compute the lower bound via the divergence theorem. Finally, for the class of unbounded curved tubes in noncompact real space forms, we also compute the Cheeger constant and prove that there is no finite-volume Cheeger set.

We study properties of the calculus $\varphi\mapsto\varphi(A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten–von Neumann class $\boldsymbol{S}_p$. It turns out that this calculus defined on the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ in the case $p\le2$ admits a commutator Lipschitz estimate and is multiplicative modulo $\boldsymbol{S}_p$. On the other hand in the case $p>2$ there is no commutator Lipschitz estimate. There is no commutator Lipschitz estimate in the operator norm as well.