11,440 papers in this slice of arXiv.
Emanuel Milman
We record several elementary relations between spectral and isoperimetric parameters of a flat torus TΛ=Rn/Λ and the covariance structure of a fundamental domain K for the lattice Λ⊂Rn. For every measurable fundamental domain K and nonzero vector ξ in the dual lattice Λ∗, we observe the sharp directional variance estimate ⟨CovKξ,ξ⟩≥121. This yields a lower bound on the torus spectral gap λSG(TΛ) (equivalently, the length of the shortest nonzero dual vector λ1(Λ∗)) in terms of the maximal covariance of K: λSG(TΛ)=4π2λ1(Λ∗)2≥3∥CovK∥opπ2. Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant DChe(TΛ) using an old argument of Hadwiger. In particular, when the Voronoi cell KΛ of a lattice with detΛ=1 is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that DChe(TΛ),λSG(TΛ),λ1(Λ∗)≥c>0, where c>0 is a universal constant independent of dimension n; this may be thought of as a positive resolution of the Kannan--Lovász--Simonovits conjecture for all flat tori. While there are lattices Λ and corresponding Voronoi cells K=KΛ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).
Habib Ammari, Yat Tin Chow, Fuqun Han
We develop a topological framework for wave scattering by a medium with self-similar or prefractal geometry. Physical scaling identities motivate an auxiliary boundary-integral model that is periodic in logarithmic scale, and the Bloch-Floquet-Zak transform fiberizes its interscale coupling. For sufficiently small, well-separated components, equilibrium densities define a computable finite-dimensional projected matrix, while Riesz projections select the corresponding invariant spectral subspace of the full boundary symbol. Under the stated spectral-isolation and point-gap conditions, we prove that the determinant winding of the exact Riesz-reduced family agrees with that of the projected matrix. For the regular-simplex configurations, we obtain nonzero winding formulas and track the resulting local winding data across finite prefractal levels and along a geometric sequence of wavenumbers. We also obtain conditional results for different dilation centers and show that the Zak phase of the chiral Hermitianization is equal to π times the point-gap winding modulo 2π. Thus, the scale-periodic boundary model admits a rigorous topological reduction.
Xing Liang
In this paper, we establish variational formulas and a rearrangement inequality for principal eigenvalues of asymmetric second-order difference operators on periodic lattices. For a general irreducible nearest-neighbor operator, positive right and left eigenvectors give an explicit saddle point and hence equal minimum--maximum and maximum--minimum formulas over positive profiles and periodic logarithmic correctors. The corrector is unique and satisfies a nonlinear discrete flux-conservation law. For exponentially tilted symmetric diffusion, the formula separates the discrete Dirichlet and potential terms from an exact nonlinear periodic correction. In one dimension the flux is constant, and an elementary scalar-flux representation yields a bell-shaped cyclic rearrangement that maximizes the tilted principal eigenvalue for every tilt.
Zuoqin Wang, Hanzhang Yun
Let Λ be the Dirichlet-to-Neumann operator on the boundary of a compact three-dimensional Riemannian manifold. Using the Lee--Uhlmann full-symbol formula and Weyl's invariant theory, we compute the Guillemin--Wodzicki residues of Λ and Λ2 explicitly, and hence the first two logarithmic coefficients in the Steklov heat trace. As an application, we prove that geodesic balls in simply connected three-dimensional space forms are determined by their Steklov spectra among smooth domains in the same space form. More generally, we obtain a rigidity theorem in the class of compact constant-curvature manifolds with smooth, not necessarily connected, boundary. In the Euclidean case, we also prove spectral uniqueness for concentric spherical shell regions.
Pingxin Gu, Haizhong Li, Yao Wan
In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space Hn for n≥3. We note that when n≥4, the result was obtained in our previous paper [23]. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [13] for the hyperbolic case.
Mikhail Dolgopolov
We study the strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant at a distinguished flat momentum, valid for every quasimomentum K, obtained via a uniform elliptic reduction of the two-dimensional lattice integral. Second, we formulate and prove a general order-matching criterion. This criterion decides whether a leading-order Fredholm determinant asymptotic, with relative accuracy of order 1/mu, suffices to determine the constant-order additive correction in the strong-coupling energy, or if the next-order refinement is required. Applying the criterion at K = pi, we derive the complete strong-coupling ground-state asymptotics, including the coefficient of the order 1/mu term. The results are cross-validated against the exact closed-form benchmark and two independent high-precision numerical schemes. Furthermore, we determine the corresponding spectral gap and prove an exact Fredholm-determinant underestimation factor equal to 2. Additionally, we independently confirm that the previously established K = 0 constant is correctly derived and requires no analogous refinement. Finally, we compare the orders of asymptotic precision controlled across recent lattice few-body models and draw a structural parallel with the parity classification of topological band structures.
Kai Hippi, Søren Mikkelsen
The Shnirelman-Zelditch-Colin de Verdière theorem and its weak mixing extension relate quantum ergodicity and quantum mixing with ergodicity and weak mixing of the geodesic flow. Integrable systems, such as the flat torus, do not satisfy either in general. Restricting to position-dependent observables, Marklof and Rudnick established equidistribution for almost all eigenfunctions of rational polygons. In this note, we extend this result to off-diagonal elements for a subset of rational polygons not including the torus using weak mixing of the directional billiard flow for almost all directions. Additionally, we provide a different proof that establishes the same result for 2-tori.
Jochen Glück, Phillip Krokor
Every bounded linear operator T on a complex Banach space E is known to extend to a lattice homomorphism T that acts on the so-called complex free Banach lattice over E. We show the following three results about the spectrum σ(T) of T: (i) σ(T) always contains the spectrum σ(T); this answers a recent question of de Hevia and Tradacete. (ii) It can happen that σ(T) is a singleton while σ(T) is the entire unit circle; this shows that σ(T) is not the closure of the cyclic hull of σ(T) in general. (iii) Finally, we give a full characterization of σ(T) in terms of the spectral properties of T. Some of our arguments also give new results about the spectral properties of general lattice homomorphisms. As a main tool we make extensive use of the Banach lattice functional calculus for continuous positively homogeneous functions.
Alexander Van Werde
We investigate a notion of cospectrality for integer matrices that is parameterized by algebraic number fields. Given a number field and a symmetric integer matrix, we wonder when conjugating the integer matrix by an orthogonal matrix with entries in the given field can produce new integer matrices. Our results concern sufficient conditions for the associated notion of spectral determination, and we give constraints on the orthogonal matrices when the conditions are not applicable. The results use the discriminant of the characteristic polynomial and properties of Krylov subspaces. We leverage the theory to develop an algorithm to find all cospectral mates over a given (small) field. An implementation of the algorithm is made available.
Jin Sun, Lili Wang, Tao Wang
We prove the Weinstock inequality for the first Steklov eigenvalue of convex domains in hyperbolic space Hn, resolving Open Question 4.27 of Colbois-Girouard-Gordon-Sher(2024) for the remaining case n=3. Our argument replaces the global monotonicity required in earlier work Gu-Li-Wan(2025) with a one-crossing property, which is established via an explicit slope comparison. The proof works uniformly for all n≥3.
Tao Hu, Jiachen Shi, Quanyu Tang
We construct pairs of noncongruent bounded Euclidean plane domains with identical Steklov spectra, including multiplicities. This gives a negative answer to the planar Steklov analogue of Kac's question ``Can one hear the shape of a drum?'' The domains are simply connected and strictly convex, have real-analytic boundaries, and may be chosen arbitrarily close to a disk in the C∞ topology.
Luigi Provenzano
We prove that, for n≥3, there is no constant Cn depending only on n such that the Steklov eigenvalues σk(Ω) satisfy σk(Ω)≤Cn(∣∂Ω∣k)n−11 for every smooth bounded domain Ω⊂Rn and every k≥1.
Dong Zhang
We propose Choquet extension for set-pair functions, Lp integration of Choquet extensions, and global extension constants, and apply these to investigate optimization problems and bound many combinatorial and geometric quantities. Our research line is also applicable to the study of original Choquet extension; within this framework, parallel results for the original version are obtained. Specifically, we use Choquet-type extensions to build an equivalent functional representation of set-based fractional optimization, which finds applications in various settings, such as maxcut, bipartiteness ratio, and conductance on graph limits or Riemannian manifolds. We further establish the Lp integration of a family of Choquet extensions, and apply it to derive spectral bounds for conductance and other combinatorial quantities on measure spaces. A monotonicity inequality on global extension constants is proposed, which unifies classical estimates and uncovers new bounds for a lot of geometric and combinatorial quantities, such as torsional rigidity, Cheeger constants, Dirichlet p-isoperimetric constant, and p-Laplacian eigenvalues, in totally distinct underlying structures----including hypergraphs, graph limits, Riemannian manifolds and metric measure spaces.
Søren Fournais, Alexander V. Sobolev
Let ψ(x), x∈R3N, be an eigenfunction of the N-particle atomic Schrödinger operator. We consider the one-particle density matrix γ(x,y) and one-particle kinetic energy density ϰ(x,y), x,y∈R3, associated with the eigenfunction ψ. Both functions play a central role in quantum chemistry computations of atomic and molecular bound states: the knowledge of the eigenvalue behaviour of the integral operators Γ and K with kernels γ(x,y) and ϰ(x,y) serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues λk(Γ)>0 and λk(K)>0: k→∞limk38λk(Γ)=A38,k→∞limk2λk(K)=B2, where A and B are non-negative constants given explicitly in terms of the eigenfunction ψ. These asymptotics are determined by the singularities of the function ψ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for ψ. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction ψ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues λk(Γ) and λk(K). The asymptotic formulas take the form k→∞limk310λk(Γ)=(Aasym)310,k→∞limk38λk(K)=(Basym)38, where Aasym and Basym are non-negative constants given explicitly in terms of the gradient of ψ.
Pedro Freitas, Isabel Salavessa
We characterise the periodic orbits of geodesic billiards on spherical lunes on Sn. In the case of angle openings of the form π/p for positive integer p we fully determine their Dirichlet and Neumann spectra. We then show that lunes with an angle opening smaller than π which is not a rational multiple of π, or those with an angle opening of the form π/p for p larger than one satisfy Pólya's conjecture eventually, independently of whether the corresponding geodesic billiards satisfy the nonperiodicity condition or not. For lunes with an angle opening π/p we further provide a two-term asymptotic formula for the eigenvalues based on sharp upper and lower bounds, together with a corresponding two-term counting function established using the geoesic billiards approach. Finally,we give an explicit bound on p in terms of the dimension ensuring the corresponding lunes satisfy Pólya's conjecture for all eigenvalues.
Noah Koerner
We consider the magnetic Neumann Laplacian on bounded domains in R2 with outward peaks. The operator is associated with a large magnetic field depending on a parameter λ, and we investigate the behaviour of the eigenvalues as λ tends to +∞. We show that their asymptotic expansion is influenced by the sharpness q and geometry of the peak, and that its main term is of order λq+12. This is an extension of previous works on magnetic Neumann Laplacians in smooth domains and domains with corners.
Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants
Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain Ω with real-analytic Jordan boundary, which is not a disc and for which there exist k∈(27.4381178838,27.4381198839) and a nonzero real-valued function u∈Cω(Ω) satisfying (Δ+k2)u=0 in Ω, with u=0,∂νu=constant=0 on ∂Ω. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on u. The domain has dihedral symmetry of order 26, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies σ∂Ω(kω)=0 for ω∈S1. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.
Gregory Berkolaiko, Denis Borisov, Marshall King +1
We study the negative spectrum of the Laplacian on a metric graph with general vertex matching conditions and with two length scales: a compact core whose edges have length of order a small parameter ε, together with finitely many edges of infinite length. As ε→0, some negative eigenvalues may escape to −∞, and we describe precisely how. There are exactly two rates of escape, ε−1 and the fractional rate ε−2/3. We determine the number of eigenvalues of each rate, together with their leading coefficients, explicitly from the vertex conditions. The analysis rests on the Dirichlet-to-Neumann map of the graph and on an implicit Rellich-type theorem, that identifies the power-law rates of the solution branches of a nonlinear 2-parameter matrix pencil with the leading orders of a one-parameter Hermitian family.
Sebastián Muñoz-Thon
We study the long-time asymptotics of heat kernels on Abelian covers of compact manifolds and on Abelian covers of principal bundles. For Abelian covers, we establish three complementary asymptotic expansions: a distributional expansion describing correlations of the heat semigroup, a local pointwise expansion of the heat kernel on compact subsets, and a global expansion valid on the natural diffusive scale, revealing the Gaussian profile governing heat propagation. We then extend these results to horizontal heat kernels on principal bundles under natural holonomy and curvature assumptions. In this setting, we show that the leading asymptotics are entirely determined by the geometry of the underlying Abelian cover, while the nontrivial fiber modes decay exponentially fast due to a uniform spectral gap. The proof combines Floquet theory on Abelian covers with the Borel--Weil decomposition on principal bundles.