Papers

We analyse the essential spectrum of ${\mathcal M} = {\operatorname{curl}} {\operatorname{curl}}$ acting on divergence-free vector fields in unbounded domains of $\mathbb{R}^3$. We show that $\sigma_e({\mathcal M}) = [0,+\infty)$ in quasi-conical domains and $\sigma_e({\mathcal M}) \neq \emptyset$ in quasi-cylindrical domains. For horn-shaped domains with circular cross-section and eventually mean-convex boundary, we establish that $\sigma_e({\mathcal M}) = \emptyset$, independently of their volume. For horns with annular cross-section, the transverse normal harmonic field determines an effective one-dimensional Schrödinger operator $H_V$ with $\sigma_e({\mathcal M}) \supseteq \sigma_e(H_V)$. Finally, for a concrete family of perforated exponential horns with a double-exponential hole, we show that depending on the rate of shrinking of the hole at infinity, either $\sigma_e({\mathcal M}) = \emptyset$, or $\sigma_e({\mathcal M}) = [\gamma^2, +\infty)$, or $\sigma_e({\mathcal M}) = [0,+\infty)$.

We develop an encounter-based formulation of restricted diffusion in a bounded domain with multiple targets, resolving separately the boundary local time accumulated on each target. Unlike the single-target problem, the target projection operators do not generally commute with the governing Dirichlet-to-Neumann (DtN) operator, so that its eigenbasis does not diagonalize the local-time dependence. We overcome this difficulty by decomposing the DtN operator into target-restricted blocks. Multi-dimensional Laplace inversion then yields a convergent switching expansion, whose successive terms describe alternating diffusive transfers between the targets. We also derive an equivalent operator-valued renewal equation. For two targets, diagonalizing the target-restricted blocks provides an explicit spectral representation in terms of two DtN block spectra and an inter-target coupling matrix. When the coupling preserves spectral modes, the switching series can be resummed exactly in terms of modified Bessel functions. In general, as the off-diagonal DtN block is smoothing for separated targets, one can resort to low-rank finite-dimensional approximations. We examine an effective two-mode reduction for small, well-separated targets and identify a regime in which repeated inter-target transfers are progressively suppressed. Probabilistic interpretations and implications for diffusion-controlled reactions are discussed.

Building on the point interaction method developed in our previous work, this paper studies inverse eigenvalue problems for regular Sturm-Liouville problems with Dirac weights. More precisely, we explicitly reconstruct the potential of regular Sturm-Liouville problems with the single-point and two-point Dirac weights from the solutions of a class of Cauchy problems which is completely determined by the first eigenvalues of a family of perturbed problems originating from moving point interaction models in quantum mechanics. Finally, the multi-point Dirac weighted case is also discussed.

This work analyzes the spectral zeta function associated with regular and singular Sturm–Liouville operators endowed with $N$ generalized point potentials. The point interactions are characterized as a particular class of self-adjoint extensions of Sturm–Liouville operators defined on a chain of $N+1$ adjacent intervals. Each interaction is described by $SL(2,\R)$ matching conditions relating generalized boundary values across neighboring intervals. We construct the spectral zeta function associated with Sturm–Liouville operators endowed with $N$ generalized point potentials in terms of a contour integral involving an appropriate characteristic function. The process of analytic continuation of the spectral zeta function to a neighborhood of the origin is illustrated by means of specific examples where the $\zeta$-regularized functional determinant is also explicitly computed.

We investigate time-dependent semiclassical measures associated with the magnetic Schrödinger flow on the flat two-dimensional torus in long-time regimes. Under a geometric nonvanishing assumption on an effective magnetic force, we identify two thresholds and prove a dynamical form of quantum unique ergodicity for such systems. If the time scale $\tau_h \gg h^{-1/2}$, then the configuration marginal is the normalized Lebesgue measure on $\mathbb T^2$. In addition, if $\tau_h \gg h^{-1}$, then the momentum marginal is, conditionally on each regular level set, the normalized invariant measure on that curve.

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $\pi$ of $\mathrm{GL}(n,\mathbb{Z})\backslash\mathrm{GL}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $\pi$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.

For every sufficiently large N, we prove that the regular N-gon uniquely minimizes the first Dirichlet eigenvalue among convex polygons of prescribed area with at most N sides, up to rigid motions. Quantitative Faber–Krahn stability localizes minimizers near the disk but leaves their discrete geometry undetermined. We encode that geometry by a centered measure of exterior angles and prove an inverse-cubic Fourier inequality whose equality case is the uniform root configuration. Its defect gives a lower bound for the quadratic spectral excess over the regular polygon. Cyclic cancellation and a common material-coordinate comparison make the nonlinear difference small relative to the same defect. Exact constraint coordinates extend this comparison to arbitrarily small positive defects. Minimality then forces zero defect, which characterizes the regular polygon.

Let $d>1$ be fixed and let $A_n$ be an $n\times n$ matrix with independent $\Ber(d/n)$ entries. For every $0<r<\sqrt d$, we prove that, with high probability, a positive proportion of the eigenvalues of $A_n$ have modulus larger than $r$. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to $\sqrt d$. Our proof works with the Brown measure $\mu_d$ of the adjacency operator of the directed Poisson–Galton–Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of $A_n$ to $\mu_d$ in probability. We prove that the outer radius of its support is $\sqrt d$, using a resolvent recursion on Poisson–Galton–Watson trees.

We study the minimum adjacency spectral radius in the class $\mathcal{SC}_{m+2}(m)$ of strongly connected digraphs with $m$ vertices and $m+2$ edges. Using generating functions for directed paths, we associate with the relevant digraphs topological polynomials whose smallest positive roots determine the corresponding spectral radii. Based on an ear decomposition, we obtain a complete structural classification of $\mathcal{SC}_{m+2}(m)$ by showing that every digraph in this class can be obtained from a butterfly digraph by attaching a single ear. This reduces the extremal problem to the optimization and comparison of finitely many polynomial families subject to their realizability conditions. We prove that the minimum spectral radius is determined by the polynomial $P_{\min}(z)=1-2z^{m-1}-z^m$. If $R_m\in(0,1)$ denotes its unique root, then $\min_{G\in\mathcal{SC}_{m+2}(m)}\rho(G)=R_m^{-1}$. For $m\geq4$, the minimum is attained, up to isomorphism, uniquely by the cross-chorded cycle $\mathcal{C}_m^\times$. For $m=3$, there are exactly two non-isomorphic minimizers, both with spectral radius $(1+\sqrt5)/2$. Finally, we establish the bounds $2^{1/(m-1)}<\rho\left(\mathcal{C}_m^\times\right)<3^{1/(m-1)}$.

We investigate the optimal endpoint $L^2$ restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold $M$. For every fixed smooth codimension-two submanifold $\Sigma$, we prove a little-o improvement $o(\lambda^{1/2}\sqrt{\log\lambda})$ on the classical estimate $O(\lambda^{1/2}\sqrt{\log\lambda})$ of Burq–Gérard–Tzvetkov and Hu. Our proof uses the Bargmann transform and Tataru's phase-space representation to reduce the problem to Stein–Street's estimate for singular Radon transforms. The three-dimensional case can be handled directly by Ricci–Stein's estimate. Moreover, we construct explicit examples to show that the little-o improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery–Smith and Carbery–Hofmann. In particular, we establish the log-free estimate $O(\lambda^{1/2})$ when $(M,g)$ and $\Sigma$ are real analytic, whereas this estimate fails in general in the smooth setting.

We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least $L^2$-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensity set; in the generic standard-Kirchhoff setting, this set is governed by the regular Gauss image of the secular manifold. Primitive cycles and exterior-to-exterior paths consequently determine the positivity threshold, but not, in general, the positive numerical value. We introduce a boundary-aware singular-completion cone and prove that it contains all regular secular edge-energy vectors for trees, unicyclic graphs, and closed graphs of cycle rank two. We then construct a cycle-rank-two graph with two Dirichlet leaves for which this completion principle fails. An exact rational separator, combined with a validated Krawczyk enclosure, yields a nonsingular scalar secular state lying outside every boundary-compatible singular sector. A positive radial derivative identity and recurrence in the compact orbit closure convert this local separation into an exact high-frequency eigensequence for a single fixed metric. For a suitable measurable observation set, the true high-frequency localization constant $C_\infty(\omega;\ell)$ and its singular-completion counterpart $C_{\mathrm{sing}}(\omega;\ell)$ satisfy $$C_\infty(\omega;\ell)<1/2<C_{\mathrm{sing}}(\omega;\ell)$$. Thus, singular completion captures the quantitative localization geometry in several low-complexity classes but does not, in general, determine the high-frequency variational problem on a fixed quantum graph.

We determine exactly how harmonic sums, taken over integers with a given number of occurrences of a fixed block of digits, depend on that number of occurrences. Language factorizations, combined with stochastic radix expansions, relate these harmonic sums to the iteration of an operator whose eigenvectors are the Lebesgue measure and distributional derivatives of singular measures. The dual picture is given by a Riesz basis of eigenpolynomials in a suitable Hardy space. The resulting modal expansion admits a natural interpretation as a block-directed Euler-Maclaurin formula interpolating between Euler-Maclaurin and Taylor expansions.

Let $d\ge1$ and $V\in C(\mathbb{T}^d;\mathbb{C})$. We prove that the Bloch variety of $-\Delta+V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spectral band functions with integer translations of quasimomentum. Using analytic geometry and perturbation estimates along imaginary rays, we show that, after suitable integer translations, all irreducible components contain the same holomorphic spectral branch.

We prove an asymptotic expansion of a weighted heat trace of the Dirichlet Laplacian on a bounded Lipschitz domain. The weight is given by a power of the distance to the boundary times a bounded function that is continuous near the boundary. Depending on the exponent of the weight, we obtain one-term or two-term asymptotics. We present two applications, namely a version of one-term asymptotics of weighted Riesz means for arbitrary orders, as well as convergence results for a family of measures that describes localization of Laplace eigenfunctions.

On a connected closed smooth Riemannian manifold, the algebraic structure of the Laplace eigenfunctions, as described by eigenfunction triple products, uniquely determines the geometry. We refine this correspondence by introducing the notion of an $N$-product eigenbasis, which consists of eigenfunctions whose pairwise products may be written as linear combinations of at most $N$ basis elements. We prove that a manifold admits a $2$-product eigenbasis if and only if it is a flat torus. We also prove an analogous result for Laplace eigenvectors of bounded-degree graphs.

We study eigenvalues of discrete Schrödinger operators $H=\Delta+V$ on $\mathbb{Z}^2$, where $\Delta$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge $\lambda=\pm4$, and the bulk, there is a further distinction between the regular energies $0<|\lambda|<4$ and the interior critical energy $\lambda=0$ stemming from the reducibility of the corresponding Fermi surface in the latter case. For every $0<|\lambda|<4$, we construct potentials $V$ satisfying $|V(n)|\leq C|n|^{-1}$ for which $\lambda$ is an eigenvalue of $\Delta+V$, and prove absence of eigenvalues when $|V(n)|\leq C|n|^{-1-\varepsilon}$ for some $\varepsilon>0$. At $\lambda=0$, the critical power changes and we construct potentials $V$ satisfying $|V(n)|\leq C|n| ^{-2}$ for which $0$ is an eigenvalue of $\Delta+V$, as well as prove absence when $|V(n)|\leq C|n|^{-2-\varepsilon}$ for any $\varepsilon>0$. Finally, at each spectral edge, we show that, for every $K\geq3$, an eigenvalue can be created by potentials supported on exactly $K$ sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.

We solve the robust ten martini problem: Cantor spectrum with no arithmetic restriction on the irrational frequency for Type I analytic one-frequency quasiperiodic Schrodinger operators. We introduce Type I through a new acceleration-type quantity with stability under analytic perturbations. This yields a global open class of analytic potentials whose associated operators have Cantor spectrum for every irrational frequency.

Predict-then-optimize methods learn a mapping from features to unknown edge costs. They then solve the shortest-path problem on the predicted costs. We study a feature-free alternative for the case where the decision-maker observes many noisy cost vectors on a fixed network. The cost vectors are denoised by an eigenvalue decomposition (PCA) of their covariance matrix. Dijkstra's algorithm then runs on the result. We test this on the $5\times5$ grid of \citet{elmachtoub2022}, with $1000$ independent replications and a fresh ground truth in each. We compare against SPO+, against naive Dijkstra on noisy costs, and against two alternatives to a hand-picked rank $k$ (the Gavish-Donoho threshold and a Ledoit-Wolf filter). We also test a hybrid of SPO+ and PCA. Four findings emerge. First, the rank matters. Keeping $k=2$ components performs worse than no denoising. Second, when costs are close to linear in the features, the Gavish-Donoho threshold recovers the true rank without supervision. PCA-denoised Dijkstra outperforms SPO+ at $\deg=1$, but underperforms SPO+ at $\deg\geq 2$. The hybrid works best at $\deg=1$ and is slightly better than SPO+ at $\deg=2$. Third, when costs are strongly nonlinear ($\deg=4$), the signal is no longer low-rank, and only the Ledoit-Wolf filter clearly beats the naive baseline. Fourth, the benefit of denoising grows with network size on grids, but not on a layered non-grid network at high nonlinearity.

The singular value decomposition has been established for matrices, Hilbert–Schmidt operators, trace-class operator, compact operators, and bounded operators, but surprisingly not for unbounded operators. Unfortunately, most interesting operators in applied math are unbounded, as any operators involving some form of derivatives — gradient, exterior derivatives, Laplacians, Fourier and other transforms of derivatives, Hamiltonians, etc. — are likely unbounded. In this article, we fill in this last missing piece by establishing the existence of singular value decompositions for unbounded operators in three natural forms: multiplication-operator, direct-integral, and operator-valued-measure. We show it inherits classical properties of finite-dimensional singular value decomposition including approximation results, relationships with fundamental subspaces, and the Moore–Penrose inverse. This discovery opens the door to the singular value decompositions of a myriad of well-known unbounded operators in mathematics, physics, statistics, and finnance — gradients on Euclidean spaces and manifolds, Petrov–Galerkin method, finite-difference operators, Hilbert–Schmidt operators, Hilbert complexes, supersymmetric quantum mechanics, Sturm–Liouville theory, nonparametric density estimation, and the Black–Scholes equation. The resulting decompositions reveal a number of novel insights, among many others: bosonic and fermionic states in supersymmetric quantum mechanics arise as left and right singular vectors of generalized ladder operators; the Riesz transform appears as the left singular operator of the Euclidean gradient; and the Hodge decomposition follows directly from the singular value decompositions of the exterior derivatives.

We prove a sharp cut-off resolvent estimate for the Laplacian on non-compact Riemannian manifolds whose ends satisfy the Cardoso–Vodev assumption and whose geometry near the trapped set agrees with that of an even asymptotically hyperbolic manifold with the corresponding resolvent estimate. As applications, we show the global Strichartz and spectral projection estimates on negatively curved asymptotically hyperbolic surfaces, without assuming the evenness at infinity.

We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang–bang indefinite weight $\kappa\mathbf 1_D-\mathbf 1_{\Omega\setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=\delta$ should be located. Put $\varepsilon=\delta^{1/N}$ and $\tau_\delta=\alpha_\delta\varepsilon$. We prove that there is a finite threshold $\tau_*=\tau_*(N,\kappa)$, independent of the ambient domain, such that $\delta^{2/N}\Lambda_\delta(\alpha_\delta)\to\Lambda_{\mathbb H}(\tau)$ whenever $\tau_\delta\to\tau<\infty$. If $\tau<\tau_*$, optimal small regions concentrate at the boundary. If $\tau>\tau_*$, their concentration centres move to distances much larger than $\varepsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $\tau=\tau_*$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime $\tau_\delta=\tau_*+\sigma\varepsilon+o(\varepsilon)$, we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary–interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for $N=2$ and $\kappa=1$ they suggest a transition near $\tau_*\approx3.2$.

In this paper, we introduce and study conformal permutation matrices. In particular, for a permutation $\pi_s\in S_n$, we define the corresponding conformal permutation matrix $H_{\pi_s}$ and show that its $d-$th power, where $d=n/\gcd(s,n)$, is a diagonal block matrix whose blocks are products of certain matrix blocks $h_i$. We further show that these diagonal blocks share a common subset of eigenvalues and prove that $H_{\pi_s}$ is block-diagonalizable. We also introduce the notion of conformal circulant matrix and investigate its eigenvalues. Finally, we establish a necessary and sufficient condition for a rectangular matrix to be conformal circulant.

We establish quantitative comparisons between the Laplace–Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.

In this paper, we show that for every p>2, there is an orthonormal basis of spherical harmonics on Sn with n>=2 which achieves the maximal Lp norm growth. More precisely, one basis achieves this growth for all p<=2(n+1)/(n-1), and another for all p>=2(n+1)/(n-1). This extends the results in Han (arXiv:1404.5016) to all dimensions and all p>2, and improves the positive density subsequence there to a whole basis.

Let $\Omega\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operators and, consequently, an $O(\varepsilon^2)$ eigenvalue comparison for every fixed index in every dimension $n\ge2$. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.

We formulate and prove existence of pure quasifree Hadamard states for the Klein–Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution $p\circ G_D$ rather than on the two-point function. This is expressed by a $b$-wave-front set condition on the compressed future cone. The analysis on the manifold with corners $\bar M\times\bar M$ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small $b$-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling–Narcowich–Wald deformation argument this yields the construction of a pure quasi-free Dirichlet–Hadamard state.

We prove optimal trace-norm relaxation for the one-dimensional Lindbladian associated with the Witten differential $a=h\partial_x+V'$ which annihilates the classical Gibbs density: if $\lambda_1(h)$ is the first positive eigenvalue of $H=a^*a$, then the relaxation rate is $\gamma_h=\lambda_1(h)/(2h)$. It applies to initial operators whose Schwartz kernels, after a Gibbs conjugation, satisfy $ L^2 $ estimates for the restriction to the diagonal and for the normal derivative. An appendix by Chat GPT 6 presents a stronger result specialised to positive initial data.

Becker et al. produced a striking trace formula for the sum of fourth powers of generalized magic angles, $ \theta $, for the chiral model of twisted bilayer graphene (TBG) with the exact Bistritzer–MacDonald potential: $\sum \theta^{4} = 8 \pi / \sqrt 3 $. The purpose of this note is to generalize this formula to a larger class of potentials satisfying the symmetries of the model.

A weighted dynamical version $\mathbf{Z}_f$ of the classical Ihara zeta function is presented on connected finite regular graphs, expressed in terms of weighted periodic orbit data. We establish its meromorphic continuation, with poles given by the resonances of the associated non-backtracking transfer operator acting on a suitable Banach space, and compute its residues. At simple spectral parameters, the residue of $\mathbf{Z}_f$ is identified with the invariant Ruelle distribution on the graph phase space. Combining this result with a relation obtained by Arends-Palmirotta, we further derive Patterson-Sullivan and Wigner residue formulae. This provides a finite-graph analogue of residue formulae for weighted dynamical zeta functions for geodesic flow on rank one locally symmetric spaces, in the spirit of Schütte-Barkhofen-Weich.

We study a deformation of the central heat trace on $\mathrm U(N)$ obtained by deforming Schur polynomials to Jack polynomials, and prove that it admits an asymptotic expansion to arbitrary order. Its coefficients are governed by elliptic $b$-Hurwitz numbers, which are genus-one counterparts of the $b$-Hurwitz theory of Chapuy and Dołęga \cite{ChapuyDolega22}. We construct the associated elliptic $b$-Hurwitz theory by means of generalized coverings on a torus and identify it with the genus-one closure of the genus-zero simple $b$-Hurwitz theory. At $b=0$ the construction recovers ordinary elliptic Hurwitz theory, while at $b=1$ it gives an automorphism-weighted geometric interpretation of the connected and disconnected twisted elliptic Hurwitz numbers of Hahn–Markwig \cite{HahnMarkwig26}. Our results extend the topological expansion obtained in the classical case in \cite{LemMai25,LM2} and take the form of a coupling between chiral and antichiral elliptic $b$-Hurwitz generating functions.