Papers

An inverse spectral problem for a $(1+1)$-dimensional wave equation with wave speed $c(x)$ is considered. The spectral data consist of the eigenvalues of the Neumann–Dirichlet problem for the operator $-c^2(x)\partial_x^2$ together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map $\Lambda$ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed $\Lambda$ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of $L^2$ and $H^1$ penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.

In an earlier study of the Laplacian eigenvalues with strongly attractive Robin boundary conditions in polygons it was found that if each corner is either concave or non-resonant, then the asymptotic behavior of the individual eigenvalues is determined by an effective one-dimensional operator on the boundary with Dirichlet boundary conditions at the corners. The non-resonance condition was formulated in terms of an eigenvalue bound on truncated sectors, and it was only shown to be satisfied by the right and obtuse opening angles. In the present work, we determine the full range of non-resonant angles: it turns out that the non-resonance condition fails only for the opening angles $\frac{\pi}{2n+1}$ with $n=1,2,3,\dots$, and these critical values are equivalently characterized in terms of the jumps of the eigenvalue counting function and the existence of bounded generalized eigenfunctions. This resolves the numerically supported conjecture stated by Dauge and Lafranche.

We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.

We derive approximation rates for Sobolev functions defined on the complex unit sphere and complex projective space, representing quantum states, and on the special and projective unitary groups, representing quantum unitary operations. By using harmonic analysis on compact groups and compact homogeneous spaces, we characterize the corresponding Peter–Weyl decompositions and Sobolev spaces via irreducible representations and the Laplace–Beltrami operator. Combining spectral truncation with finite-depth quantum-circuit approximation, invoking the Solovay–Kitaev theorem, we establish quantitative rates that separate the spectral truncation error from the circuit approximation error.

We derive second order perturbation expansions of volume-normalized sums of the first $n$ Dirichlet eigenvalues for near-circular and near-spherical domains. The disk (resp. ball) is only a critical shape when $n$ is the last index of an eigenvalue cluster, and in dimension two and three our expansions yield a finitely checkable criterion deciding whether the disk or ball is a weak local minimizer or a saddle point. In two dimensions, this allows us to show that there is an infinite number of cases in which the disk is a saddle point for finite eigenvalue sums.

In 1952, Pólya proved the inequality $\mu_2 < \lambda_1$ for the eigenvalues of Dirichlet and Neumann problems in domain $\Omega \subset \mathbb{R}^d$, $d \ge 2$. We show that $\mu_3 < \lambda_1$ if $d\ge 3$. In the two-dimensional case, in 2025, Rohleder proved that $\mu_3 < \lambda_1$ for simply connected domains. We prove that this inequality holds also true for double connected domains symmetric with respect to the origin.

We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.

We study half-line CMV matrices whose Verblunsky coefficients are decaying perturbations of periodic sequences. We quantify the stability of fine properties of the absolutely continuous spectral measure under such perturbations. More precisely, we establish Hölder continuity of the perturbed spectral measure and derive sharp time-averaged Fourier estimates governing the associated quantum dynamics. These results remain valid up to the edges of the absolutely continuous spectrum. Since the spectral measure of a half-line CMV matrix is precisely the orthogonality measure for the corresponding orthogonal polynomials on the unit circle (OPUC), our results also yield new statements for OPUC whose Verblunsky coefficients are decaying perturbations of periodic sequences.

Let $\Lambda_n$ be the set of all eigenvalues of standardized Laplacian matrices of order $n$. In 2005, a polygon $S_n\subseteq\Lambda_n$ was constructed, and it was conjectured that $\Lambda_n=S_n$. We observe that $S_n$ is a zonotope, which gives a short proof of $S_n\subseteq\Lambda_n$, and we describe $\Lambda_n$ as a union, over all vectors $v$, of intersections of zonotopes $Z_i(v)$ built from the ratios $v_j/v_i$. For a vector whose components are the $n$ distinct $n$th roots of unity, every $Z_i(v)$ equals $S_n$. We then show that the conjecture is false for $n=7$: the directed $6$-cycle with one vertex duplicated has an eigenvalue lying $2.1\cdot10^{-4}$ beyond one edge of $S_7$. For the analogous digraph of order $n$, the corresponding eigenvalue lies at signed distance $\delta_n\sim\frac{\pi^2}{2n^4}$ beyond the supporting line of the analogous edge of $S_n$, so the construction yields counterexamples for all sufficiently large $n$; numerically, it does so for every $7\le n\le300$. We also prove that $S_m\subseteq S_n$ whenever $m\mid n$, with $S_m\setminus\{0,1\}$ lying in the interior of $S_n$ for $n\ge3m$.

Consider the Mathieu operator $L=-d^2/dx^2+2a\cos(2x)$ for fixed $a\in\mathbb{C}\setminus\{0\}$. For large enough $n$, the periodic ($n$ even) or anti-periodic ($n$ odd) problem on $[0,\pi]$ has exactly two algebraically simple eigenvalues $\lambda_n^-,\lambda_n^+$ in the disc $|\lambda-n^2|<1$. We prove that, as $n\to\infty$, $\lambda_n^+-\lambda_n^-=\pm\frac{8(a/4)^n}{[(n-1)!]^2}\exp\left[-\frac{a^2(n+2)}{4(n^2-1)^2}-\frac{15a^4}{128n^7}+O(n^{-8})\right]$, where the sign depends on the labelling. For real $a$ with $\lambda_n^-<\lambda_n^+$, this difference is the length of the $n$th spectral gap. Expanding the exponential determines the relative corrections of orders $n^{-4}$ to $n^{-7}$, beyond the correction $-a^2/(4n^3)$ obtained by Anahtarci and Djakov (2012) with a relative $O(n^{-4})$ remainder. By isospectrality, the same formula holds for the potential $ce^{-2ix}+de^{2ix}$ with $a^2=cd\ne0$.

We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.

In this paper we study Koopman operators associated with finite periodic realizations of elementary cellular automata. The spectra of these operators encompass various properties of the underlying dynamical systems, independently of any initial condition. A characterization of these spectra is proposed, leading to a spectral classification of all elementary cellular automata. This work is thought as a preliminary step for studying the spectra of Koopman operators for infinite realizations of elementary cellular automata.

Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet–Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac–Hodge operator $D_m$, we establish a spectral reduction expressing $\sigma(D_m)$ through the graph Laplacian on vertices $-\Delta_V$ and the Laplacian on faces $-\Delta_F$. As a consequence, $D_m$ carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator $D_m$ on Archimedean lattices and contrast the outcome with that on the Lieb lattice.

We prove a sharp Szegö-Weinberger-type inequality for balanced doubly connected domains in $\mathbb S^2$, $\mathbb R^2$ and $\mathbb H^2$. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the area of the smallest component of the complement. Equality holds only for the rotationally symmetric annulus. We also show that the balancing assumption is essential: if we drop it, the inequality fails in general.

We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator $h^2\Delta+V(x)$ that is perturbed by a smooth potential $h^\alpha V_\omega$ with $\alpha\in (0,1)$, and $V_\omega$ is a random potential that decorrelates on distances $h^\beta$, with $0<\beta<2\alpha$. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.

We study graph transformations and their effects on the eigenvectors of three fundamental graph matrices: the adjacency matrix $A$, the signless Laplacian $Q$, and the Laplacian $L$. We extend the Laplacian eigenvector edge and matching principles introduced by Merris to $Q$ and $A$. We also extend some of the known $L$ graph transformations to $Q$ and $A$ and prove that graphs admitting a common eigenvector $\mathbf{x}$ for $A, L$ and $Q$ are such that vertices corresponding to nonzero components of $\mathbf{x}$ have same degree. Following our previous work for $L$, we investigate bivalent graphs for $A$ and $Q$, that is graphs admitting an eigenvector with all components in $\{-1, 1\}$. In particular, for trees, we characterize $Q$-bivalence and establish that $A$-bivalence forces all vertex degrees to be odd.

We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov–Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets $A,B\subset\mathbb{R}^d$ of positive measure with finite upper codimension-one Minkowski boundary content, the number $\Lambda_\varepsilon(cA,B)$ of eigenvalues in $(\varepsilon,1-\varepsilon)$ satisfies $\Lambda_\varepsilon(cA,B)\le C(d,A,B,\beta)c^{d-1}D(1+\log_+(c/D))$, where $D=\log(1/(\varepsilon(1-\varepsilon)))$, for every fixed $\beta>0$, $c\ge2$, $0<\varepsilon<1/2$, and $D\le\beta c$, with $\log_+t=\max(0,\log t)$. The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant $d-1$ for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For $C^1$ convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is $2$. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.

We study the regularity of eigenfunctions of the Dirichlet Laplacian on domains with corners. The specific question we consider is whether the eigenfunctions can be smooth, e.g. trigonometric, on the closure of their domain. We determine how this regularity depends on the local geometry of the corner. Local asymptotic analysis provides detailed information about eigenfunctions near corners and allows us to quantify the Hölder regularity of the normal derivative. We demonstrate the limitations of some of the methods used to study these problems in finding a complete set of eigenvalues on polygons.

Let $P_h$ be a semiclassical pseudodifferential operator on a closed Riemannian manifold $(M,g)$, with principal symbol $p$. We consider perturbations $P_h+h^\alpha Q$, with $0<\alpha\le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^\beta$, with $2\beta <\alpha$. Assuming that the classical dynamics of the unperturbed system on a regular energy surface $p^{-1}(E)$ is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order $|\log h|$. This formula relates the distribution of energy levels of the quantum system near the energy $E$ to the periodic orbits of its classical counterpart. The perturbation $h^\alpha Q$ affects only the phases appearing in the trace formula.

We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local $H^s$ condition with $s>2$. The multiplier operators $F(\mathcal L)$ are of weak type $(1,1)$ and bounded on $L^p$ for every $1<p<\infty$. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in $(0,\infty)$, we also determine the leading term and first correction for the localized kernels. Whether the weak type $(1,1)$ estimate holds under the local $H^2$ condition remains open.

In this paper, we prove the strict Faber-Krahn inequality for non-negative-mass Dirac operators on simply connected domains with infinite-mass boundary condition. Denoting the first positive eigenvalue of the corresponding Dirac operator with mass $m\geq 0$ as $\lambda_1^{+}(\Omega;m)$, our main result shows that \begin{equation*} \lambda_1^{+}(\Omega;m)\geq \lambda_1^{+}(B_{|\Omega|};m), \end{equation*} where $B_{|\Omega|}$ is the disk with the same area as the domain $\Omega$, and the equality is strictly valid if and only if $\Omega=B_{|\Omega|}$. The key observation leading to the proof is that the upper component of the Dirac spinor associated with $\lambda_1^{+}(\Omega;m)$ is non-vanishing over $\Omega$, which is proved by exploiting its Dirac current and the associated stream function. Based on this global non-vanishing property, we construct a sphere-valued spinor map of degree one, which allows a sharp comparison with the radial disk profile using the planar isoperimetry.

We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{\lambda_j\}_{j=0}^\infty$. For a domain $\Omega\subset M$, consider $$ N_\Omega(\lambda) = \sum_{\lambda_j\leq\lambda} \left|\int_\Omega e_j\,dV_g\right|^2. $$ Suppose that $\partial\Omega$ is $s$-Ahlfors regular, with $s\in{[d-1,d)}$, and that $\Omega$ satisfies a two-sided corkscrew condition. We prove that $$ N_\Omega(\lambda) = \operatorname{vol}(\Omega)+O\bigl(\lambda^{s-d}\bigr), $$ and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order $\lambda^{s-d}$ holds precisely when the boundary admits a natural $s$-dimensional crossing volume $\mathcal V^s(\partial\Omega)$, in which case $$ N_\Omega(\lambda) = \operatorname{vol}(\Omega) - C_{d,s}\, \mathcal V^s(\partial\Omega)\, \lambda^{s-d} + o\bigl(\lambda^{s-d}\bigr). $$ For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.

The Kannan–Lovász–Simonovits (KLS) conjecture asserts that isotropic log-concave probability measures have Poincaré constants bounded by a universal constant, independently of dimension. We give a deterministic variational proof with an explicit bound on the Poincaré constant $C_P(\mu)\le25$, where $\mu$ is any isotropic log-concave probability measure. Starting from elliptic moment estimates and a quadratic variance inequality, we establish geometric bounds on normalized Appell coefficient norms through joint maximization over the measure and test function. Variation of the measure gives a maximum-principle inequality, while stationarity in the test function controls the highest-order cumulant terms. Two concave barriers constructed from quadratic and cubic polynomials close the induction. A weighted Helmholtz–Hodge decomposition then controls the curl correction of weighted divergence, yielding a curvature estimate for compatible symmetric tensor fields that is uniform in rank. Quadratic duality converts this estimate into an operator comparison for centered integration, linking the coefficient bounds to control of the inverse gradient. A spectral-radius estimate and a scalar growth inequality for adjoint iterates then yield the Poincaré bound. The final estimate is independent of the auxiliary positive curvature, allowing approximation to complete the proof for general isotropic log-concave measures.

We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.

We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.

We prove an asymptotically sharp lower bound for the eigenvalues $\lambda_k$ of the Dirichlet Laplacian on bounded open sets in the plane with area $A>0$. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain $(2/3)\cdot 4\pi k/A\leq \lambda_k$ for every $k$, which is two thirds of the bound conjectured by Pólya.

For $d\ge3$, let $\lambda_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on $\mathbb Z^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient \[ C_d=\frac{(d-2)^2}{4} \] is sharp in dimensions three and four, while it is not sharp for $d\ge9$. We prove $\lambda_d=C_d$ for the remaining cases $5\le d\le8$. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.

Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least $g$ and the standard second-moment bound for graph spectra. We prove that the supremum of $(\lambda_1+\lambda_n)/n$ over these lists lies between $(1-o(1))(\log g)^2/g^3$ and $(10/7+o(1))(\log g)^2/g^3$ as $g\to\infty$ through the odd integers. The upper bound applies in particular to graphs of odd girth at least $g$. For regular graphs of odd girth $g\geq5$, we obtain a stronger bound $(\lambda_1+\lambda_n)/n \leq 12/((g-1)(g-2)(g-3))$, which is optimal up to a constant factor.

We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval $[\mu,L]$. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian perturbations can differ. We prove that the sine-Jacobi recurrence of Part I is asymptotically optimal for this response norm within a factor of $1.000004$, uniformly over $0<\mu<L$. Every terminal-exact recurrence satisfies $A_N\ge(2\sqrt{c_{cert}}-o(1))N^{3/2}\epsilon_N^\star/(L-\mu)$ with $c_{cert}>1.14268353624$, whereas the sine-Jacobi recurrence attains the constant $2\sqrt{c_{\sin}}$ with $c_{\sin}<1.14269218$. The proof reduces the gain to the diagonal energy of a Jacobi Green matrix. A centering lift represents this matrix by an exponential kernel; packing its diagonal into a density on the line preserves the energy and controls the change in the Birman-Schwinger spectrum. A calibrated identity gives a lower bound using the ground parameter and one Fredholm determinant. Two further determinants and a trace observation improve the bound through a pointwise inequality on an explicit compact state region. One rational certificate verifies this inequality by interval arithmetic. A second certificate constructs a density satisfying the selected observations and both normalizations with energy below the sine value. Finally, no $C^1$ function of finitely many observations of the specified kinds gives a locally sharp lower bound at the sine density.