Papers

We prove Pólya's inequality for every Neumann eigenvalue of every Euclidean circular annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. It is entirely analytic when the width is at most one quarter of the inner radius. For arbitrary radii, explicit high-frequency and small-hole bounds reduce the problem to a compact parameter region, which is verified by exact rational and outward interval certificates valid on whole parameter rectangles. Established transfer principles yield the Neumann inequality for annular products and congruent annular sectors.

We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A, B$ on a real symplectic Hilbert space $(\mathbf{H}, J)$. Under a natural commutativity condition, there exists a (generally unbounded) operator $\Phi$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A = \Phi^t M_A \Phi$, $B = \Phi^t M_B \Phi$, and maps $J$ to a canonical symplectic operator. The result generalizes Williamson's normal form to infinite dimensions and yields a complete spectral invariant for the pair $(A,B)$.

Linear relations are multivalued generalizations of linear operators whose spectral properties are closely linked to the stability of differential-algebraic equations. Consequently, reliable spectral enclosures are of considerable interest. In this note, we extend the established concept of the block numerical range from operators to linear relations by means of a range representation. Unlike the classical numerical range, the resulting block numerical range may be nonconvex and can therefore detect gaps in the spectrum. We establish fundamental properties of this new notion and derive corresponding spectral enclosure results. Finally, we illustrate the theory with a numerical example comparing the block numerical ranges obtained from different constructions. As an application, we obtain an explicit left-half-plane spectral bound for a class of port-Hamiltonian DAEs.

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to $L^2(\R_+^{3g-3+m})$ with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.

We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on $\mathbb{Z}$ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.

There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools – spectral analysis and mean first-passage time (MFPT) analysis – and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.

We study the focusing one-dimensional cubic nonlinear Schrödinger / Gross–Pitaevskii equation (NLS-GP), where the potential of the underlying linear Schrödinger operator, $H_V=-\partial_x^2+V(x)$, is compactly supported. In \cite{turner2026resonance} the authors proved that purely imaginary zeros (transmission resonances) or purely imaginary poles in the lower half plane (scattering resonances) of a reflection coefficient of $H_V$ seed branches of resonance-induced nonlinear bound states, and that these states bifurcate at a strictly positive $L^2(\mathbb R)$ excitation threshold. This is in contrast to nonlinear bound states which bifurcate at zero $L^2$-norm from point spectra of $H_V$ (corresponding to poles in the upper half plane). In this paper we establish precise criteria for the nonlinear orbital stability and instability of the resonance-induced states near the bifurcation point. A corollary is that resonance-induced nonlinear states are orbitally stable if i) the underlying linear (scattering or transmission) resonance frequency is sufficiently small, and ii) the corresponding linear resonance eigenmode is strictly positive. Numerical simulations are presented to explore regimes not accessible to our theory, and to explore the large time dynamics for initial conditions near stable and unstable resonance-induced nonlinear bound states.

Jin, Milojević, Tomon and Zhang established a powerful recursive estimate relating the positive eigenvalues of a graph whose least eigenvalue is small in absolute value. We establish a refinement of this result, yielding improved estimates across spectral graph theory and discrepancy theory, and for Chowla's cosine problem. Recent results of Janzer, Tomon and Yip allow us to directly transfer least eigenvalue estimates to the corresponding surplus estimates. Using these tools, we fully resolve a conjecture of Räty, Sudakov and Tomon. We show that, for an $n$-vertex graph $G$ with least eigenvalue $\lambda_n$ and surplus $\operatorname{sp}(G)$, if $G$ is $\epsilon$-far from all disjoint unions of cliques, then $|\lambda_n|\geq \Omega_\epsilon(n^{1/4})$ and $\operatorname{sp}(G)\geq \Omega_\epsilon(n^{5/4})$. Furthermore, we show that when $G$ is $n^{-o(1)}$-far from all disjoint unions of cliques, we have $|\lambda_n|\geq n^{1/4 - o(1)}$ and $\operatorname{sp}(G)\geq n^{5/4 - o(1)}$. Finally, we show that the surplus of a $K_t$-free graph with $m$ edges is at least $m^{0.614 - o(1)}$ as $m$ tends to infinity.

We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\hat a(z) = (1-z)^{\alpha/2}(1-z^{-1})^{\alpha/2}$ for $\alpha \in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenvalues. As a consequence, we prove that all eigenvalues are simple. We also obtain a two-term approximation for every eigenvalue, with explicit bounds on the remainder, as the size of the matrix tends to infinity. While known results are limited to $\alpha > 1$, we bridge this gap by covering the full range of $\alpha \in (0,2)$. Our approach uses a construction of approximate eigenvectors that has not previously been applied in this setting.

We consider the semiclassical magnetic Schrödinger operator in dimensions one and two. The first result is that an even one dimensional semiclassical potential is spectrally determined by the Schrödinger operator. The second result states that a two dimensional radial magnetic field is spectrally determined by a purely magnetic Schrödinger operator. This extends the earlier work of Guillemin-Wang and answers a conjecture of theirs therein.

We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set $S$, and the resulting waves are observed on an open receiver set $R$, which may be at a positive distance from $S$. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: $\|\phi|_R\|_{L^2(R)} \ge ce^{-C\lambda^\theta}\|\phi\|_{L^2(M)}$ for every eigenfunction $\phi$, where $\lambda$ is the corresponding eigenvalue, $\theta\in[0,1/2)$, and the constants $c,C>0$ do not depend on $\phi$. If a bound of this type holds also on $S$, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with $\theta=1/2$. Uniqueness nevertheless fails at this endpoint: in every dimension $n\ge 2$, we construct a pair of nonisometric connected, closed Riemannian $n$-manifolds that produce identical wave observations.

We consider multi-dimensional quasiperiodic Schrödinger operators with monotone sawtooth-type potentials in the perturbative localization regime. We prove that a given gap is closed iff a corresponding (via a natural gap labeling) eigenfunction vanishes at the origin. Using this, we show that for a generic choice of the sampling function $f$, the spectrum of such an operator is a Cantor set, with all possible gaps being open. We also provide a class of functions $f$ for which some gaps are closed. Finally, we observe that certain gaps are more difficult to close than the others.

We prove that the positive first Dirichlet eigenfunction of every smooth domain with convex boundary in $\mathbb{CP}^n$ invariant under the standard $U(1)\times U(n-1)$-action is strictly log-concave for $n=2,3,4$.

We study the distribution of the zeros of $\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely weakly converges as $N\to\infty$ to a deterministic measure, depending on just the variance and covariance of the entries of the $A_j$. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where $d=1$. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name “the petaloid law”. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.

A symplectic spectral function of a $2n \times 2n$ real positive definite matrix $A$ is a function of its symplectic eigenvalues $0< d_1(A) \leq \cdots \leq d_n(A)$, which is given by a composition $f \circ d$ of some symmetric function $f$ on the set of $n$-vectors with positive entries and the symplectic eigenvalue vector map $d(A)=(d_1(A),\ldots, d_n(A))$. In this work, we rigorously study various types of differentiability and Clarke generalized gradient of symplectic spectral functions, and also provide several applications of our findings. We show that the symplectic spectral function $f \circ d$ is Fréchet differentiable at $A$ if and only if $f$ is Fréchet differentiable at $d(A)$, and we compute the derivative expression explicitly. We also show that $f \circ d$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $A$ if and only if $f$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $d(A)$. We determine the Clarke generalized gradient of a symplectic spectral function $f \circ d$ at $A$, which is given in terms of the Clarke generalized gradient of $f$ at $d(A)$. As an application of our work, we show that the purity, von Neumann entropy, and Rényi entropy of a bosonic faithful Gaussian state are Fréchet differentiable functions of the covariance matrix of the Gaussian state. We also provide explicit expressions of their Fréchet derivatives.

This paper constructs explicit closed-form Green's functions for the Helmholtz equation on annular and spherical-shell domains with two independent Robin impedances on the inner and outer boundaries. Graf's and the hyperspherical addition theorems reduce each angular mode to an explicit $2\times2$ linear system, and the resonance spectrum is governed by a characteristic determinant bilinear in the two impedances. This bilinearity has a direct inverse-problem consequence: two resonant frequencies of one non-radial angular mode generate at most two candidate impedance pairs via an explicit quadratic equation, a third resonance selects the physical pair, and an exact reflection-symmetry obstruction identifies where the radial spherical mode cannot recover the impedances. Spectrally, we prove the branches positive, simple and strictly increasing in both impedances; derive first-order asymptotics at the four corners of the impedance plane, with coefficients given by boundary masses and normal derivatives of the limiting eigenfunctions, and an explicit mixed second-order coefficient at the Dirichlet–Dirichlet corner, which for the radial mode evaluates in closed form to $2\pi/(R_2-R_1)^3$; establish a low-frequency resonance-free band, with a second-order threshold expansion explicit in dimension three and a rational approximation accurate over the whole impedance range; and prove the universal high-frequency spacing law with a shell-curvature correction. Jacobian-based sensitivity and conditioning criteria are included. The kernels and spectra are computable to machine precision, all asymptotic regimes are confirmed numerically, and the kernels provide reference solutions for finite-element and boundary-element validation.

The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is $W^{1,q}$, for $q \in [1,\frac{d}{d-1})$, where $d$ is the degree. This result is stable in the sense that the solution map from $C^d$ coefficients to $W^{1,q}$ roots is continuous. We prove that, after a natural Lipschitz reparameterization, the roots are Lipschitz and the map from $C^d$ coefficients to reparameterization and reparameterized roots is continuous with respect to the $W^{1,q}$ topology on the target spaces, for all $q \in [1,\infty)$. The result is based on a convergence and reparameterization theorem for Almgren's $Q$-valued Sobolev functions.

We prove two conjectures of Laugesen and Siudeja: the equilateral triangle uniquely maximizes the harmonic mean of the first two positive Neumann eigenvalues at fixed perimeter and their geometric mean at fixed area. The proof uses the two equilateral eigenfunctions and four projected polynomial corrections. Triangle symmetry reduces the associated variational matrices to two-by-two operations and two scalar invariants. Angular comparison and horizontal trial functions then yield the global inequalities. The same argument gives explicit quadratic stability bounds in terms of normalized side-length asymmetry.

In a bulk ferromagnet, magnetic energy is six orders of magnitude weaker than elastic energy. Slenderness removes this disparity. In a thin ferromagnetic rod, the two energies scale differently with aspect ratio, and an ordinary nickel wire deforms nonlinearly even under modest remote fields. We develop a three-dimensional variational theory of a ferromagnetic Kirchhoff rod under gravity and magnetic loading and study it through bifurcation analysis, computation, and experiment. A uniform field can stabilize a vertical rod against buckling under its own weight, the classical Greenhill instability. We give sharp criteria for clamped and pinned supports in terms of an Airy-type principal eigenvalue, and give a fine description of the character of the bifurcation landscape, which has qualitatively different features than its non-magnetic counterpart. When subject to the field generated by a permanent magnet, we identify a rich bifurcation landscape as the magnet is rotated quasistatically along a circle of radius slightly larger than the rest length of the rod. We prove that instability occurs only through a planar mode and that each snap is a nondegenerate saddle-node. We also locate the two folds that produce hysteresis. Experiments on $50\,\mu$m nickel wires confirm the Greenhill threshold, restabilization at ~$30$ mT for both clamped and pinned supports, as well as the salient features of the hysteresis loop such as switching angles, width, and geometry of folds accurately. Our analysis and experiments offer principled design rules for field-stabilized filaments and remotely switched bistable actuators. Our analysis also offers a well-conditioned numerical method is valid near the switching events, where the direct numerical minimization is strongly ill-conditioned.

For Bernoulli bond percolation on every regular tree of degree at least three, we prove that infinite clusters appear strictly before absolutely continuous spectrum. Throughout an explicit supercritical interval, the adjacency operator of every open cluster almost surely has no absolutely continuous component. Together with Bordenave's theorem, this gives $p_c<p_{ac}<1$.

Let $u:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if \begin{equation*} u = h \circ \pi \circ \Xi, \end{equation*} where $\Xi:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ is an isometric transformation, $\pi:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ is the Hopf map and $h:\mathbb{CP}^n\to\mathbb{CP}^n$ is a holomorphic map with degree one. When $n=1$, it refines a classical result of Urakawa and a recent result of Rivière. Moreover, we prove that $h\circ\pi\circ \Xi$ has nullity $3n^2+5n$.

Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=\lambda_0\leq\lambda_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{\phi_j\}_{j\geq0}$. Let $\{\omega_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of $\phi_0,\ldots,\phi_p$ is asymptotically optimal for $\omega_p(M,g)$ as $p\to\infty$. We show that this conjecture is false in every dimension, even for positively curved real-analytic metrics. Nevertheless, we show that a less rigid notion of nodal sweepouts recovers the volume spectrum. Let \[ E_N(M,g):=\operatorname{span}\{\phi_0,\ldots,\phi_N\}, \qquad a_N\in H^1\bigl(\mathbb P(E_N(g));\mathbb Z_2\bigr) \] be the generator, and for $p\geq1$ define \[ \mathcal N_p^N(g) := \left\{ \Psi \mid X\text{ is a finite complex},\ \Psi\in C\bigl(X,\mathbb P(E_N(g))\bigr),\ \Psi^*(a_N^p)\neq0 \right\}. \] We define the nodal $(p,N)$-width by \[ \nu_p^N(M,g) := \inf_{\Psi\in\mathcal N_p^N(g)} \sup_{x\in\operatorname{dmn}(\Psi)} \mathcal H_g^{n}\bigl(\{\Psi_x=0\}\bigr). \] We prove the following filtration formula \[ \omega_p(M,g) = \lim_{N\to\infty}\nu_p^N(M,g) \] for every $p\geq1$. We state several questions on nodal geometry and min-max theory.

This work constructs explicit closed-form Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions $d\ge 2$. Despite the canonical geometry, such kernels have remained unavailable because the Robin condition couples the field and its normal derivative, preventing the method of images and obstructing standard eigenfunction expansions. By a decomposition–expansion method, the free-space fundamental solution is expanded via Graf's addition theorem in two dimensions and the hyperspherical addition theorem in higher dimensions; the regular correction is then determined algebraically by matching Robin boundary data mode by mode. The resulting series converge absolutely on compact interior subsets and are computable to machine precision. These explicit kernels yield a complete characterisation of the resonance spectra: each spectral branch increases strictly with the impedance parameter, interpolating between Neumann and Dirichlet eigenvalues. We establish a low-frequency spectral gap with an explicit cutoff estimate, and a universal high-frequency mode spacing with a second-order correction that distinguishes the Robin condition from the Dirichlet and Neumann extremes. The closed-form expressions furnish exact benchmark solutions for wave simulations in impedance-matched cavities, eliminating geometric discretisation error and providing reference data for the validation of finite-element and boundary-element algorithms in acoustics, electromagnetism and quantum mechanics.

We supply the details of the subcritical dry Ten Martini statement announced in [6, Remark 1.3]: for every irrational frequency, every subcritical Type I energy with resonant rotation number is an endpoint of an open spectral gap. The proof combines the all-frequency Puig argument developed there with quantitative almost reducibility. A different proof was recently given by Li [7].

We develop a unified asymptotic theory for Minnaert resonances generated by a small bubble $D_\varepsilon$ placed near an interface $\Gamma$ of material discontinuity in an acoustic background. The bubble has characteristic size $\varepsilon>0$, with its density and bulk modulus depending on the same small parameter through the Minnaert high-contrast scaling. The analysis is organized by the scaled separation \[ \Theta_\varepsilon := \frac{\operatorname{dist}(D_\varepsilon,\Gamma)}{\varepsilon}, \] and covers the three natural regimes \[ \Theta_\varepsilon\to+\infty, \qquad \Theta_\varepsilon\to\Theta_*\in(0,\infty), \qquad \Theta_\varepsilon\to0: \] 1) In the far-separation regime, the interface is invisible at leading order and the classical full-space Minnaert law is recovered. 2) In the critical regime, the rescaled geometry retains both the bubble and the interface at finite separation, and the Newtonian capacitance is replaced by a flat-interface capacitance determined by the limiting tangent transmission problem. 3) In the near-contact regime, this family converges to a contact-limiting capacitance, yielding the same leading-order Minnaert resonance for all gap scalings satisfying $\operatorname{dist}(D_\varepsilon,\Gamma) = o(\varepsilon)$. These three regimes provide a unified description of how the resonance frequency changes as the bubble approaches the interface, from the classical isolated-bubble limit to the contact limit. The analysis relies on a uniform reduction of the resonance problem to a scalar equation and on a variational characterization of the interface-dependent capacitances; in particular, Mosco convergence of the associated transmission energies yields convergence of the capacitary potentials and minimum energies up to the singular contact configuration.

We give a complete characterization of periodic graph operators whose dispersion polynomials are reducible for every potential. Consequently, the reducibility dichotomy for parameter-dependent Laurent polynomials implies that, for every other periodic graph operator, the dispersion polynomial, and hence the Bloch variety, is irreducible for generic potentials. Our proof proceeds through two reductions. We first use the monodromy of the roots of the dispersion polynomial with respect to the potential parameters to reduce the problem to the splitting case. We then show that the splitting property passes to the principal submatrices of the Floquet matrix, reducing the problem further to the case that the fundamental domain contains two vertices.

We investigate the interplay between the geometry of a planar domain and the spectrum of the nonlinear $p$-Laplacian, subject to a homogeneous magnetic field and Dirichlet boundary conditions. First, we establish a lower bound for the bottom of the spectrum on the whole plane, extending the classical Landau-level estimate to the nonlinear regime. Second, we prove Pólya-type bounds for the first eigenvalue on bounded convex planar domains in terms of the area, perimeter, minimal width and magnetic field strength, and show that they are asymptotically sharp along sequences of suitable thinning domains. Finally, we post some remarks with open problems.

We show that for a horizontally stretched Bunimovich billiard, for almost every stretching parameter, there exists a sequence of eigenfunctions concentrating in the rectangular part. Thanks to the work of Markarian–Oliffson Kamphorst–Pinto de Carvalho, we know that for an open set of parameters such billiards are ergodic. The proof is a result of interaction with ChatGPT 6: we were curious if it could establish the existence of bouncing ball modes for Bunimovich or Sinai billiards. That was not successful until we suggested varying the wings. That produced an essentially complete argument for the existence of quasimodes. We then suggested using Hassell's parameter-variation argument which led to the result about eigenfunctions. The argument was then significantly simplified and clarified by the authors.

We show that all normal covers of a fixed finite graph $G$ sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.