Papers

In 1955, L. E. Payne conjectured that for a planar clamped plate, each buckling eigenvalue is at least as large as the Dirichlet Laplacian eigenvalue of the same domain with index shifted by one. We prove this conjecture for every odd buckling index.

Let $\Omega\subset\mathbb Z^d$ be finite and let $S\subset\mathbb T^d$ be measurable. We study the discrete Fourier concentration operator \[ T_{\Omega,S}=P_\Omega B_SP_\Omega, \] which describes simultaneous localization to a finite set of spatial indices and a prescribed spectral region. For independent exponents $\gamma,\eta\in(0,1]$, we introduce translation seminorms for $\Omega$ and $S$ and use them to bound the trace defect \[ \operatorname{tr}(T_{\Omega,S}-T_{\Omega,S}^2). \] The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when $\gamma=\eta$, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for $\gamma\ne\eta$ the sum is controlled by one end of the scale range. For discretizations $$ \Omega_R=(RF)\cap\mathbb Z^d $$ of a bounded measurable set $F\subset\mathbb R^d$ satisfying an upper Minkowski-neighborhood estimate with exponent $\gamma$, we obtain trace-defect bounds of order $R^{d-\gamma}\log R$ in the critical case and $R^{d-\min\{\gamma,\eta\}}$ off the critical line. These estimates yield quantitative eigenvalue-counting and plunge-region bounds and recover the discrete Landau asymptotic. For a box model at $\gamma=\eta=1$, the logarithmic factor is sharp.

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate $\lambda_1^{(2)}\ge(n-1)\sigma/2$ under $\sigma$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive Hölder class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-Hölder continuity of the integrated density of states for the associated quasiperiodic Schrödinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

We compare return probabilities for random walks stopped at an independent geometric time on Lubotzky–Phillips–Sarnak graphs and on the infinite regular tree, with the McKay measure giving the tree value and the Ihara zeta function describing the additional returns caused by cycles. We estimate this correction when the mean walk length is comparable to the length of the shortest cycle, a regime in which walks long enough to traverse a cycle receive non-negligible weight. Unconditionally, we prove an upper bound with a double logarithmic factor and bounded fixed moments over prime levels after multiplication by the number of vertices. Under the Generalized Riemann Hypothesis for quadratic Dirichlet $L$-functions, the correction is of the same order as the reciprocal of the number of vertices.

Let $\gamma\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let $\lambda_\gamma$ be the first eigenvalue of the periodic curvature Schrödinger operator $-d^2/d s^2+\kappa(s)^2$. We obtain \[ \lambda_\gamma\geq \frac{\sqrt{\pi}}{2} \left(\frac{\Gamma(7/6)}{\Gamma(5/3)}\right)^3. \] This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.

Given a centrally symmetric convex body, consider the zero set of the Fourier transform of its characteristic function. We study how large the distance from this set to the origin can be when the volume of the body is fixed. A 2009 conjecture of Benguria, Levitin, and Parnovski asserts that the maximum is attained by a Euclidean ball. We disprove it in all dimensions greater than one. In the plane, every regular centrally symmetric polygon with at least twelve sides outperforms the disk, albeit by a tiny margin, and the regular dodecagon is best among them. In dimensions three and higher, no maximiser exists.

We prove that the best constant in the one-dimensional Lieb-Thirring inequality with exponent one is $4/(3\sqrt{3}\pi)$, confirming the Lieb-Thirring conjecture in this case. Moreover, we extend the inequality to operator-valued potentials and obtain the bound $L_{1,d}\leq(2/\sqrt3)L_{1,d}^{\mathrm{cl}}$ in every dimension.

We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis–Marchioro–Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.

We extend Cuenin's compact-manifold spectral bounds for Schrödinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators \(P+V\), where \(P\) is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and \(V\) is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for \(P\) imply \(L^p\)-\(L^{p'}\) resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and Hölder stability from terminal-time observations for sources in a suitable spectral Gevrey class.

We study internal exact controllability of the free Schrödinger equation on finite compact metric graphs with Kirchhoff conditions at interior vertices and a fixed Dirichlet-Neumann assignment at exterior vertices. For a prescribed primitive cycle or exterior-to-exterior path, we identify an arithmetic criterion ensuring that the corresponding scar is accessible by exact high-frequency eigenfunctions of a fixed target metric. The criterion is formulated by intersecting the regular scar stratum of the secular set with the orbit-closure torus of the target length vector, and it can hold for resonant metrics beyond the rationally independent class. Under an intrinsic Diophantine condition on the orbit-closure flow, the construction is quantitative: along an exact eigensequence the mass outside the prescribed support decays polynomially, yielding a polynomial degeneration law for finite-frequency observation when that support is uncontrolled. Rationally independent metrics have full phase-orbit closure and therefore make every primitive obstruction accessible. Combining this spectral result with the known graph-theoretic characterization and sufficiency of the Graph Geometric Control Condition (GGCC), we prove that, for every rationally independent and hence for Lebesgue-almost every metric, GGCC is necessary and sufficient for exact controllability at every positive time in the free setting.

Let $A$ be a bounded operator on a separable right quaternionic Hilbert space and let $\CA$ be the set of compact operators commuting with $A$. We establish the formula \[ \sigb(A)=\bigcap_{K\in\CA} \sS(A+K), \] a quaternionic analogue of the classical characterization of the complex Browder spectrum. The proof rests on a factorization lemma (Lemma~\ref{lem:fact}), a consequence of the Riesz–Schauder theorem and an ascent/descent argument, and on the characterization $\sigb(A)=\sS(A)\setminus\sd(A)$ in terms of S-eigenvalues of finite type, due to Arzini and Jaatit \cite{AJ26}, for which we give here an independent proof (Lemma~\ref{lem:A}). We also record two consequences of independent interest: the intersection defining $\sigb(A)$ may be restricted, without loss, to finite-rank operators commuting with $A$ (Corollary~\ref{cor:finite}); and, by a separate argument resting on the ascent/descent stability results of \cite{KD24}, the stronger pointwise statement $\sigb(A+K)=\sigb(A)$ holds for a fixed $K\in\CA$ and arbitrary $A\in\B(\V)$ (Theorem~\ref{thm:pointwise}), known in the complex case.

We study sample covariance matrices $K = \frac{1}{N} \sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^* \in \mathbb{R}^{n \times n}$ in the proportional regime $n \asymp N$. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance $\mathbb{E} \mathbf{x}_i \mathbf{x}_i^* = \Sigma$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \mathbf{x}_i^* A \mathbf{x}_i - \mathrm{Tr} \Sigma A | \prec \| A \|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\Sigma$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta:= \mathrm{Im}\, z \geq N^{-1 + \tau}$, \[ \big| \langle \mathbf{u} , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)\Sigma \big)^{-1} \big) \mathbf{v} \> \big| \prec \sqrt{\frac{\mathrm{Im}\, \widetilde m_0 (z)}{N\eta}} + \frac{1}{N\eta} \] for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where $\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.

The work is devoted to the study of Dirac operators with MIT bag boundary conditions in Euclidean domains with compact Lipschitz boundaries in arbitrary dimensions. It is shown that such operators are self-adjoint on suitable definition domains and can be recovered as the norm-resolvent limits of Dirac operators in the whole space with a large mass term outside the domain, under the assumption that an associated Robin-Laplacian eigenvalue has a prescribed asymptotic behavior with respect to a parameter in the boundary condition. This assumption is shown to hold for a class of non-smooth domains, which includes convex domains and, more generally, domains that can be "locally convexified" by suitable diffeomorphisms. To the best of our knowledge, this represents the first infinite mass interpretation for the MIT bag model in the sense of resolvent convergence for non-smooth domains. Most results are extended to the generalized MIT bag boundary conditions with the help of the recently established congruence transform.

We consider the lowest eigenvalue $\lambda(b)$ of the magnetic Neumann Laplacian in the unit disk, for a constant magnetic field of strength $b>0$. We prove that $\lambda$ is strictly increasing on $(0,+\infty)$. This means that strong diamagnetism holds at every field strength, and not only at large ones. We also show that the normalized energies at the successive crossings of angular-momentum branches form a strictly increasing sequence; combined with the strong-field asymptotics, this gives the global bound $\lambda(b)<\Theta_0 b$, where $\Theta_0$ is the de Gennes constant. These results settle the three conjectures formulated by Helffer and Léna for the disk. As a consequence, the local, or spectral, critical field $H_{C_3}^{\mathrm{loc}}$ of Ginzburg–Landau theory is, in the disk, uniquely determined for every value of the Ginzburg–Landau parameter, and not only for large ones. We also give a second proof of the bound $\lambda(b)<\Theta_0 b$, independent of the first and of the results of Helffer and Léna, by a direct variational method: trial states built from the de Gennes ground state for large fields, constant trial states for small fields, and, on the remaining bounded field interval, finite-dimensional spaces of polynomial trial states certified by finitely many exact computations in rational arithmetic. That proof uses no asymptotic input. It yields in addition an explicit upper bound for $\lambda(b)$, valid above an explicit field strength, whose two leading terms are those of the strong-field asymptotics.

We study the problem of perturbing a given matrix X belonging to a subspace S of linearly structured real matrices of order n to its nearest counterpart Y=X+D, where D belongs to S and Y possesses an eigenvalue with prescribed lower bounds on its algebraic multiplicity and geometric multiplicity. The proposed framework takes care of both cases where the target eigenvalue is known and where it is treated as an unknown decision variable to be determined jointly with the structured perturbation. We establish necessary and sufficient conditions for the existence of such matrices by expressing the feasibility constraints through Jordan-chain relations in structural coordinates. This allows us to formulate the problem as a nonlinear constrained nested optimization that minimizes the Frobenius norm of the perturbation. To solve this, we implemented a two-level strategy, utilizing MATLAB's fmincon for continuous inner optimization via a multi-start Sequential Quadratic Programming approach, while the outer level solves a discrete optimization problem. Comprehensive numerical experiments on various classes of structured matrices, including comparisons with existing methods, demonstrate the effectiveness and accuracy of the proposed approach.

Let \(N_M(\lambda)\) be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold \(M=\Gamma\backslash\mathbb H_d\), where $\Gamma$ is a lattice subgroup of the Heisenberg group $\mathbb H_d$. In 2016, Strichartz \cite[J. Geom. Anal.]{Str16} proved the Weyl law with remainder \(R_M(\lambda)=N_M(\lambda)-A_d\operatorname{vol}(M)\lambda^{d+1} = O_M(\lambda^d\log\lambda)\), and conjectured the optimal remainder to be \(O_M(\lambda^d)\). In this work, we establish a new upper bound and the first two-sided lower bounds $$ R_M(\lambda)=O_M\!\left(\lambda^d(\log\lambda)^{2/3}\right), \qquad R_M(\lambda)=\Omega_{M,\pm}\!\left(\lambda^d\log\log\lambda\right). $$ As a result, this implies that the sharp polynomial order is $d$, and disproves Strichartz's conjecture.

We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius $R$, the region where the curvature radius is smaller than $R^{-1}$ occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.

Digraphs with constant cut balance admit nearly linear directed cut sparsifiers. This condition requires the total arc weights in the two directions of every cut to be within a constant factor of each other. We ask whether this condition also permits nearly linear spectral sparsification with respect to the energy of the nonlinear directed Laplacian. For a weighted digraph $G=(V,E,w)$, let \[ Q_G^+(x)=\sum_{(u,v)\in E}w_{uv}(x_u-x_v)_+^2, \qquad (t)_+:=\max\{t,0\}. \] This energy agrees with the outgoing-cut function on binary vectors. A spectral sparsifier is a nonnegatively reweighted subgraph that preserves $Q_G^+(x)$ within a factor of $1\pm\varepsilon$ simultaneously for all $x\in\mathbb R^V$. We show that cut balance alone does not yield nearly linear spectral sparsifiers: for constant error, the worst-case support size for simple unweighted Eulerian digraphs is $\widetilde\Theta(n^{3/2})$, although Eulerian digraphs are perfectly cut-balanced and admit nearly linear directed cut sparsifiers. In contrast, we prove that every $n$-vertex tournament has a spectral sparsifier with $\widetilde O(n/\varepsilon^3)$ arcs, without any assumption on its cut balance. This includes the transitive tournament, whose cut balance is unbounded. Thus perfect balance does not guarantee nearly linear spectral sparsification, while unbounded imbalance does not preclude it. Finally, we use convex duality to show that preserving $Q_G^+$ also preserves, for every feasible demand vector, the optimum quadratic cost of a nonnegative flow. Hence the guarantee contains information beyond directed cut values.

Layer potentials provide a classical approach to boundary value problems for Laplace's equation on Lipschitz domains. Kenig's 1994 spectral-radius conjecture for the double-layer operator would ensure operator-norm convergence of the associated Neumann series on mean-zero $L^2$ densities when the boundary is connected. We disprove this conjecture by constructing a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. More precisely, for every $t>1/2$ sufficiently close to $1/2$, we obtain such a domain with $\pm i t$ in its Fredholm essential spectrum. The construction starts from smooth graphs whose shapes repeat under translation. In the limit of separated scales, refinement makes solutions of adjoint resolvent equations grow with fixed forcing. The graph slopes remain uniformly bounded. A computer-assisted certificate proves this growth through an inequality for Hermitian $2\times2$ matrices. Its strict margin at $- i/2$ persists at nearby spectral parameters. Normalisation and a Floquet transform then give compactly supported densities with small residuals on the full graphs. We insert rescaled segments of successive graphs into one bounded boundary, where these densities form a weakly null sequence of approximate eigenvectors. The same spectral conclusion holds on a single periodic Lipschitz graph. The certificate combines continuous estimates, exact rational arithmetic and rigorous interval enclosures.

We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-\Delta$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. We then specialize to the the Aharonov–Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted $L^p(\R^2,|x|^\beta\,dx)$ spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all $L^{p}$ spaces $1<p<\infty$, while boundedness fails at $p=1,\infty$. In the Krein case, both wave operators and adjoints are bounded precisely when $2/(2-\eta_\alpha)<p<2/\eta_\alpha$, where $\eta_\alpha=\max\{\alpha,1-\alpha\}$ (here $\alpha\in(0,1)$). Thus the boundary condition changes the admissible exponents.

We study the operator Michelson contrast Delta(A) = (kappa(A)-1)/(kappa(A)+1), with kappa(A)=||A|| ||A^{-1}||, for positive invertible operators, and introduce its logarithmic parametrization via arctanh(Delta(A)) = (1/2) ln kappa(A). This defines the Operator Logarithmic Contrast (OLC), extended to all invertible operators via polar decomposition. We prove OLC is subadditive under multiplication: OLC(AB) <= OLC(A)+OLC(B). For positive invertible A, we identify OLC(A) exactly as the projective Thompson distance from the identity ray to the ray of A. We establish a sharp dimensional threshold for equality in subadditivity: for dimensions 2 and 3, equality forces commutativity, while n=4 is the smallest dimension admitting noncommuting equality examples. Moreover, equality still forces commutativity in any total dimension if the operators factor as pure Kronecker products X = tensor X_i and Y = tensor Y_i with each factor dimension <= 3. We derive contraction inequalities, sum bounds, trace estimates for density operators, Lipschitz continuity (including singular limits), and limiting behavior for Cesaro means and infinite products. For density operators, we obtain an exact closed-form bijection between OLC and von Neumann entropy for qubits; prove this bijection does not exist for d >= 3; and establish upper/lower entropy envelopes at fixed condition number, showing the admissible range [S_min(kappa), S_max(kappa)] satisfies ln 2 <= S_max(kappa) < ln 3 for every kappa > 1. The qubit case recovers, as an operator lift, the classical Michelson-Jensen-Shannon equivalence of Bruni, Rossi, and Vitulano, while a counterexample shows this equivalence fails for d > 2 as a direct consequence of the non-bijectivity.

We give a new proof of the existence of absolutely continuous spectrum for the weakly disordered Anderson model on the Bethe lattice. The argument follows a general cyclicity criterion for Anderson-type Hamiltonians and reduces the problem to showing that the spectral measures of two independent copies of the rooted tree are not mutually singular. This is detected by their Hellinger affinity, and the required weak-disorder estimate follows from a short harmonic and compactness argument. We also formulate an analogous quantitative cyclicity criterion for the Anderson model on $\mathbb{Z}^d$, expressed through the Schur complement of the Poisson transform of a $2\times2$ matrix spectral measure.

In the 2004 collection Unsolved Problems in Mathematical Systems and Control Theory, Erik Verriest posed the problem of characterizing Riccati stability “without invoking additional matrices.” We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency–direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.

Let $T_n$ be the infinite $n$-regular tree, $n\ge3$. We prove that every finite connected vertex set $\Omega\subset T_n$ satisfies the sharp inequality \[ \sigma_1(\Omega)\le \frac{n}{(n-1)|\Omega|+1}. \] Equality holds if and only if $\Omega$ is a ball. Since \[ |\delta\Omega|=(n-2)|\Omega|+2, \] the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.

Let $\Omega\subset\mathbb R^2$ be a bounded connected Lipschitz domain, and let $\{\mu_j(\Omega)\}_{j\geq1}$ and $\{\lambda_j(\Omega)\}_{j\geq1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that $$ \mu_3(\Omega)<\lambda_1(\Omega), $$ thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.

The resistance Laplacian is a graph matrix associated with the effective resistance metric and provides a global counterpart of the classical graph Laplacian. Although it inherits several fundamental properties of the ordinary Laplacian, including a connected graph partitioning theorem analogous to that of Fiedler, its intrinsic spectral structure has remained largely unexplored. In this paper, we develop a structural theory of the resistance Laplacian. We derive a canonical decomposition that separates its intrinsic, average, and deviation components, thereby revealing how the global geometry induced by effective resistance differs from the local geometry encoded by the ordinary Laplacian. Building upon this decomposition, we establish several structural and spectral properties of the associated deviation operator, obtain variational characterizations of the largest eigenvalue and its corresponding eigenspace, and express the resistance Laplacian in Laplacian coordinates, thereby elucidating the relationship between the eigenspaces of the two operators. Finally, we formulate resistance-based graph partitioning objectives whose spectral relaxations recover the dominant eigenvector of the resistance Laplacian, providing a variational interpretation of the connected partition theorem. Experimental results on synthetic and real-life datasets demonstrate the effectiveness of the proposed framework for graph partitioning, data clustering, and exploratory anomaly detection.

We study uniform observability for Laplace eigenfunctions with mixed boundary conditions whose reflection signs do not define a scalar character. For the DDN/NND sectors of the equilateral rhombus, the resulting $\mathbb Z_2$ monodromy is resolved by a degree-two branched arithmetic translation surface of genus two. On an explicit class of admissible open sets, every exact mixed-sector eigenfunction satisfies a local $L^2$ lower bound uniform in the eigenvalue, multiplicity, and choice within the eigenspace. The proof combines exact unfolding, semiclassical defect measures, and the Veech dichotomy to exclude concentration near the saddle/conic network and inside periodic cylinders. We also obtain a reflection-overlap observability result for the full rhombus and isolate the remaining saddle-network obstruction for arbitrary open observation sets.