Papers

We show that Payne's conjecture on the comparison between Dirichlet Laplacian eigenvalues and buckling eigenvalues does not hold on flat and hyperbolic cylindrical surfaces and on (round) hemispheres.

We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on the radii and are uniform in the aperture. Real-order angular integration preserves the exact area term, while directional rounding estimates and radial trials control all frequencies. Optimized bounds attain the sharp circular-arc coefficient for the continuous count. We identify the full sector perimeter asymptotic and a low-index obstruction to using its coefficient throughout the spectrum. The strict Neumann inequality also passes to products with bounded Lipschitz domains.

We introduce the Enhanced Neumann Pólya inequality, which is stronger than Pólya's conjecture, and verify it for many classes of domains, including strip-tiling domains, Euclidean balls, thin product domains, and certain higher-dimensional tiling domains. We develop a geometric removal approach for the Enhanced Neumann Pólya inequality. In particular, our results provide new classes of domains satisfying the Neumann part of Pólya's conjecture.

We show that limit-periodic Jacobi matrices in the weak exponential class $EC(\eta)$ have purely absolutely continuous spectral type when $\eta$ is sufficiently large, depending only on the $R$-boundedness constant. The proof first establishes homogeneity of the spectrum from the weak exponential approximation rate. A closure theorem for reflectionless Jacobi matrices then transfers reflectionlessness from the periodic approximants to the limit, and a theorem of Poltoratski and Remling rules out singular spectrum on a homogeneous set.

For a Chern nontrivial Bloch bundle over the three torus, we construct an orthonormal Wannier basis with the optimal pointwise decay $\mathcal{O}(|x|^{-5/2})$. All but one of the generators decay exponentially. This resolves the question left open in arXiv:2505.01999.

We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of $C^k$ domains in $\mathbb{R}^n$, $k > n$, such that the nodal set of any Dirichlet eigenfunction intersects the boundary orthogonally along a $C^{k-1}$ embedded submanifold, whenever the intersection is non-empty. This is achieved by analyzing the boundary asymptotics of Dirichlet eigenfunctions via homogeneous harmonic polynomials and comparing their respective nodal sets. Along the way, we give a complete proof that generic Euclidean domains have simple Dirichlet eigenfunctions with zero as a regular value in the interior and zero as a regular value of the boundary normal derivative. In the special case of planar domains, a different argument provides one-parameter families of perturbed domains with the desired properties.

We consider the linear operator for metamaterial modeling $$L_{c,d }=(\chi_{\RE^{3}\backslash\overline \Omega}+c)\nabla\!\cdot (\chi_{\RE^{3}\backslash\overline \Omega}+d)\nabla\,,$$ where the material parameters $c>0$ and $d$ are constant on each connected component of the bounded domain $\Omega$, $d$ is negative on at least one component of $\Omega$ and transmission boundary conditions hold at the interfaces. It turns out that $L_{c,d }$ has, as in the sign-definite case, an absolutely continuous spectrum equal to $[0,+\infty)$; however, contrary to the latter, it has a non-empty point spectrum; in particular, we find conditions on the parameters so that zero is a threshold eigenvalue. In the case $\Omega$ is connected, we study how such a threshold eigenvalue behaves under $\varepsilon$-small analytic perturbations of the parameters. We show that, depending on the position of the number $\frac12\,\frac{1+d(\ve)}{1-d(\ve)}$ relative to the nearest eigenvalue of the Neumann-Poincaré operator of the interface, the zero eigenvalue splits into two branches (analytic in $\varepsilon^{1/2}$), consisting either of two branches of resonances, or of one branch of negative eigenvalues and one of resonances.

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \[ X=\K(T)\oplus \Hq(T) \] forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if \[ \dim\bigl(\K(T)\cap \Hq(T)\bigr)<\infty \quad\text{and}\quad \codim\bigl(\K(T)+\Hq(T)\bigr)<\infty. \] For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.

We study a one-dimensional single-excitation model of a scalar photon field coupled to a continuum of two-level atoms with quadratic density. The resulting matrix Hamiltonian has an unbounded spatially varying coupling and is not semibounded. We prove that it is essentially self-adjoint and determine its spectrum explicitly. The continuous spectrum is the half-line $(-\infty,\Omega]$, where $\Omega$ is the atomic resonance frequency, while the spectrum above $\Omega$ consists of a strictly increasing sequence of simple eigenvalues. These eigenvalues are characterized by the interlaced zeros of the Airy function and its derivative and satisfy \begin{align*} \omega_n = \left(\frac{3\pi g n}{4}\right)^{1/2} + \frac{\Omega}{4} + O(n^{-1/2}). \end{align*} The analysis is based on a nonlinear spectral reduction that yields a confining massless relativistic oscillator above $\Omega$ and an inverted oscillator below $\Omega$.

We study spectral measures of one-dimensional quasiperiodic Schrödinger operators. For subcritical Schrödinger operator with Diophantine frequency, we prove sharp measurable-set bounds, answering a question of Avila and Jitomirskaya \cite{AJ}; we prove optimal Fourier decay and the sharp decay order of the time-averaged quantum return probability, confirming a conjecture of Aloisio, Carvalho, and de Oliveira \cite{ACO} and a prediction of Zhong and Mosseri \cite{ZhongMosseri}. For the subcritical almost Mathieu operator, we establish summable square-root envelopes for the spectral density, with an optimal density-one formulation of the lower bound, thereby answering a question of Avila \cite{Aqi}. These results follow from a general theorem for reflectionless measures under a summable square-root Craig-type condition. We establish an exact residue expansion of the squared Borel transform, which connects local density blow-up with long-time averaged return probabilities: the endpoint residues determine both the square-root edge profiles and the leading coefficient of the time-averaged return probability.

We study the optimization inverse spectral problem (OISP) for the first $m$ eigenvalues of the Schrödinger operator on a compact metric graph. We prove the existence of a solution for every strictly increasing finite set of prescribed eigenvalues. On a graph without cycles, we obtain a representation of the optimal correction as a linear combination of squared eigenfunctions, provided that the selected eigenvalues are simple and separated from the next eigenvalue. On a graph with cycles, under the same simplicity and separation assumptions, we obtain a strong limiting representation of the optimal potential in terms of eigenfunctions of approximating operators. For a graph with a cycle, we give a counterexample with simple eigenvalues and linearly dependent squared eigenfunctions. We also prove that linear dependence of the squares of nonzero restrictions to an edge of eigenfunctions corresponding to distinct eigenvalues forces the potential on that edge to have an analytic representative

We study the eigenvalues of the negative Dirichlet Laplacian $-\Delta$ in domain $D_{\epsilon,N}$ obtained by removing $N$ Euclidean balls of radius $\epsilon$ from a bounded open set $D\subseteq\mathbb R^d$ ($d\ge2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $\rho$ and for $N\to\infty$ and $\epsilon\downarrow0$ so that the aggregate Newtonian capacity of the removed balls tends to some $\kappa>0$, the eigenvalues of $-\Delta$ in $D_{\epsilon,N}$ homogenize to those of the Schrödinger operator $-\Delta+\kappa\rho$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.

We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional $\Phi^4_2$ and sine-Gordon models and for the three-dimensional $\Phi^4_3$ model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential $L^2$ convergence to equilibrium for the $\Phi^4_2$ dynamics and for sine-Gordon with $\beta^2<2\pi$. For $\Phi^4_2$, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order $m$. This yields a genuine acceleration over the order-$m^2$ relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout $\beta^2<4\pi$, and for $\Phi^4_3$ at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.

We prove Yau's number-theoretic conjecture on positive integral points in real right simplices in every dimension $n\ge3$, and apply the resulting sharp polynomial bound to quantum harmonic oscillators. The lattice-point bound holds for every choice of positive real intercepts for which the count is nonzero, with equality precisely when all intercepts equal the same integer. For oscillators in dimensions $n\ge3$, we obtain explicit upper bounds for eigenvalue counts and spectral remainders for arbitrary positive frequencies at every threshold at or above the lowest eigenvalue. For frequencies with rational ratios in dimensions $n\ge2$, an additional lattice coordinate and interpolation give bounds for energy sums, or first Riesz means, at every threshold. When all frequencies are equal (the isotropic case), the oscillator attains these bounds: the counting bound is exact at its energy levels, and interpolation on its natural energy grid recovers the energy sum at every threshold. In three dimensions, the exact formula displays the periodic contribution associated with the resonance discussed by Fefferman and Seco. The proof builds on the slicing method of Yau and Zhang. Arithmetic and geometric consequences include explicit bounds for smooth numbers and lattice-point discrepancies.

We prove an improved fractal Weyl upper bound for resonances of infinite-area convex cocompact hyperbolic surfaces, expressed in terms of the Hausdorff dimension of the limit set and the Fourier decay exponent of the Patterson–Sullivan measure. This bound also yields a sharper upper bound for the essential spectral gap for these surfaces.

We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant $C^1$ monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where $f'>0$ have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.

We study compact static manifolds with boundary from a spectral point of view, with particular emphasis on the relation between Dirichlet, Neumann, and Robin spectra and the nodal structure of static potentials. We establish comparison inequalities linking Dirichlet and Neumann eigenvalues with the scalar curvature and derive sufficient spectral conditions ensuring that a static potential changes sign. We also obtain bounds on the number of connected components of the zero set, sufficient conditions for its connectedness, and sufficient conditions ensuring that its components intersect the boundary. Finally, we study variational properties of Robin static triples and characterize compact Robin static triples whose metrics are critical for the first mixed Robin–Dirichlet eigenvalue under admissible volume-preserving deformations.

We study the determinant line bundle for a family of elliptic operators on compact manifolds with boundary with the associated Quillen metric and Bismut-Freed connection, and we prove the global anomaly formula in this context. This extends Witten's holonomy theorem to manifolds with boundary. Witten studied the limiting behavior of the eta invariant of a Dirac operator on the total space of a fibration over $\mathbb{S}^1$ when the metric in the base $\mathbb{S}^1$ is blown up in a certain manner, a process known as taking the adiabatic limit. In this paper, we construct the calculus of adiabatic $b$-pseudodifferential operators, which naturally synthesizes the $b$-calculus and the adiabatic calculus of pseudodifferential operators. Using this calculus, and assuming the fiber operators to be invertible, we compute the adiabatic limit of the natural eta invariant in this setting and relate it to the integral of the Bismut-Freed connection and new correction terms coming from the boundary of the fibers.

For $H^1(\mathbb{R}^2)$-orthonormal families $\{u_n\}_{n=1}^N$ with one-body density $\rho=\sum_{n=1}^N |u_n|^2$, we prove the whole-space endpoint Moser–Trudinger inequality $$ \int_{\mathbb{R}^2}\frac{e^{4\pi\rho(x)}-1-4\pi\rho(x)}{\rho(x)}\,dx \le C\left(N-1+\int_{\mathbb{R}^2}\rho(x)\,dx\right), $$ where the integrand on the left-hand side is defined to be zero on $\{x: \rho(x)=0\}$. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient $4\pi$ and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient $4\pi$ and the linear growth in the rank $N$. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term $N-1$ on the right-hand side is indispensable.

In the complex Hilbert space $L_2(\mathbb{R}_+)$ on the half-line $\mathbb{R}_+ = [0, \infty)$, we investigate non-symmetric Schrödinger operators \[ Su = -u'' + qu, \qquad u(0) = 0, \] with complex-valued, highly singular distributional potentials \[ q = s + Q' \in H_{\mathrm{loc}}^{-1}(\mathbb{R}_+), \qquad s \in L_{1,\mathrm{loc}}(\mathbb{R}_+), \quad Q \in L_{2,\mathrm{loc}}(\mathbb{R}_+), \] where the derivative $Q'$ is understood in the sense of Schwartz distributions. A rigorous operator-theoretic setting is established within the framework of Shin–Zettl quasi-derivatives. We prove that every accretive operator $S$ of this class is automatically $m$-accretive in $L_2(\mathbb{R}_+)$ and $C$-self-adjoint with respect to the complex conjugation involution $C : u \mapsto \bar{u}$. In particular, no additional boundary conditions at infinity are needed, meaning that a Weyl limit-point type situation always prevails for accretive realizations. Constructive local conditions on the coefficients $s$ and $Q$ are found under which the generated strongly continuous semigroup $\{e^{-tS}\}_{t \ge 0}$ is contractive or holomorphic in a sector of the complex plane.

We show that the spectrum of the Dirichlet-biLaplacian can be quantitatively bounded from below in terms of the inradius only, for a general open set in dimensions $2$ and $3$. For the same dimensions, we also extend this result to fractional powers $s$ of the Laplacian, with $N/2<s<2$. The resulting estimate comes with a constant having the correct asymptotic behavior as $s$ approaches the two endpoints $N/2$ and $2$. The proofs exploit capacitary techniques in the vein of Maz'ya.

We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter $\gamma$ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals $\gamma^2$ times a positive compression error for a deterministic second-chaos operator, up to $o(\gamma^2)$, with designs allowed to depend on $\gamma$. After enlarging the probe dictionary, the optimal $m$-response coefficient is asymptotic to $\|Lg\|_{L^1(D)}^2/(8m)$, where $L=-(2\pi)^{-1}\Delta_D$ on the unit disk $D$. A quadratic correction of the readouts attains the same fixed-design loss to second order.

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $\lambda$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(\lambda^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\log\lambda)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.

We consider singular Schrödinger operators with an attractive $\delta'$ interaction of strength $\beta$ supported by a star graph and prove that the ground-state eigenvalue is uniquely maximized by stars of the maximum symmetry. Two different proofs are presented, one for an even number of edges using a comparison to the $\delta$ interaction star, the other general based on Birman-Schwinger technique.

Discretization can change a sharp stability threshold even when the continuum threshold is recovered at infinity. We study this phenomenon for the Hardy inequality on the integer lattice in dimensions at least three: its best coefficient agrees with the continuum value through dimension eight and is smaller from dimension nine onwards. We characterize this transition by the total critical defect, the summable error obtained by testing the lattice operator against the critical continuum power. A theta-function representation proves that the defect is negative and strictly decreasing from dimension nine; explicit positive Hardy remainders establish its positivity in dimensions three through eight. An exact energy identity relates this defect to the energy saved by optimizing the profile outside the neighbours of the origin. This exterior remains stable in dimensions eight and nine, locating the first loss of positivity in its coupling to those neighbours. We give self-contained proofs of both dimensional ranges. We then describe how local attraction creates a negative eigenvalue in a general class of lattice operators with critical inverse-square tails. Its magnitude is exponentially small near the threshold, with both the rate and the multiplicative prefactor determined by zero-energy solutions. The result allows connected finite modifications of the lattice and nonradial corrections to the tail. The same analysis controls the spread of the emerging state and gives a two-dimensional return scale for the associated Markov chain in every ambient dimension.

It is a classical fact in spectral geometry that the eigenvalues of the Dirichlet Laplacian depend continuously on the geometry of the underlying domain, under suitable geometric restrictions. We show that this theoretical property is reflected in a concrete numerical situation: the convergence rate of iterative solvers for convection-diffusion problems remains stable under deformations of the domain. The connection is a superlinear estimate for preconditioned GMRES/GCR in which the relative residual errors are bounded by an explicit expression of the eigenvalues of the Laplacian, or of a symmetric positive elliptic operator in divergence form. We prove that this bound depends continuously on the domain, uniformly in the iteration index, under either regularity hypotheses on the boundary or a topological constraint in dimension two. We also establish corresponding statements on the discrete level. Two examples delimit the theory: domains bounded by a fractal curve, where continuity still holds, and perforated domains, where it fails. The theoretical results, including both examples with rough domains, are confirmed by numerical experiments.

In 1997 Guo Wuwen proved that if $(\lambda_1,\lambda_2,\dots,\lambda_n)$ is the spectrum of a nonnegative matrix with Perron eigenvalue $\lambda_1$ and $\lambda_2$ real, then $(\lambda_1+t,\lambda_2\pm t,\lambda_3,\dots,\lambda_n)$ is the spectrum of a nonnegative matrix for every $t\ge 0$. Whether the same holds for symmetric nonnegative matrices has remained an open question. We give an affirmative answer, in a slightly stronger form: if $A$ is a symmetric nonnegative matrix with eigenvalues $\lambda_1\ge\dots\ge\lambda_n$ and $d_1\ge|d_2|+\dots+|d_n|$, then there is a symmetric matrix $B\ge A$ with eigenvalues $\lambda_1+d_1,\dots,\lambda_n+d_n$, and $B$ has the same diagonal as $A$ when $d_1+\dots+d_n=0$. This leads us to call a symmetrically realizable list Guo-extreme if none of Guo's moves can be reversed within the set of symmetrically realizable lists. Every symmetrically realizable list is a Guo-extreme list plus a vector from the cone of Guo's moves. Our second main result is a certificate, modelled on a theorem of Laffey on extreme symmetric matrices: if $A$ is irreducible and every nonnegative symmetric matrix that commutes with $A$ and vanishes wherever $A$ is positive is a multiple of the identity, then the spectrum of $A$ can be moved by a small amount in every trace-preserving direction (in every direction, if $A$ has a positive diagonal entry) and remain symmetrically realizable; in particular it is not Guo-extreme.

We study how the eigenspace associated with a fixed graph Laplacian eigenvalue changes under positive edge updates, allowing both new edges and increases in existing edge weights. We show that a preserved eigenvalue may have inherited eigenvectors from the original eigenspace and emergent eigenvectors outside it. We characterize the inherited eigenspace and show that it is independent of the update weights. We characterize its orthogonal complement within the preserved eigenspace through a symmetric matrix whose size is at most the number of updated edges, yielding a multiplicity formula and an explicit reconstruction of the updated eigenspace. For fixed updated edges, we characterize when emergence is possible and show that, whenever it is, equal update weights suffice. The exceptional weights form the zero set of a nonzero multi-affine polynomial in reciprocal variables and have Lebesgue measure zero. Finally, emergence cannot occur at either of the two smallest Laplacian eigenvalues, counting multiplicities.

We obtain sharp isoperimetric-type upper bounds for Steklov eigenvalues on warped products whose bases are compact Riemannian manifolds with boundary and whose fibres are closed Riemannian manifolds. These bounds involve the volume of the warped product and of its boundary, as well as the eigenvalues of the Laplace operator on the fibre and the Steklov eigenvalues of the base. After separation of variables, these bounds are obtained by comparison with auxiliary Helmholtz-Steklov problems on the base involving weighted Laplace operators naturally induced by the warping function. We also study the effect of various natural geometric constraints, such as the warping function being identically equal to one on the boundary. The proofs make use of various spectral approximation methods, including homogenisation on manifolds when the fibres are one-dimensional.

Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size $n$, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at $n$ quantised momenta from the same limiting band, although separated by a phase mismatch of order $1/n$ which we obtain in closed form. Iterating this correction $K$ times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error $\mathcal{O}(n^{-K-1})$ for any fixed $K$. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on $n$.