We address an issue in the proof of Lemma 4.1 in [J. Spectr. Theory 15 (2025), no. 4, 1593–1613].
We characterize locally finite metric trees with finitely many ends of infinite volume for which the Neumann Laplacian has purely discrete spectrum. In particular, for infinite comb graphs this criterion admits a transparent formulation by means of heights and distances between neighboring teeth.
In this paper, we define an analogue of non-interacting quantum fields satisfying (\Delta_{\mathbf{g}}-\lambda^{2})\phi=0 on a Riemannian scattering space (M,\mathbf{g}) with two boundary components, i.e., a manifold with two asymptotically conic ends (meaning asymptotic to the “large end” of a cone). Thus, the Lorentzian spacetime of usual QFT constructions is replaced by a Riemannian manifold with two boundary components, which play a role analogous to the two components of the mass shell for the Klein–Gordon field on Lorentzian spacetimes. Our main result describes a canonical construction of two-point functions satisfying a version of the Hadamard condition.
We study a generalized Steklov problem involving a rough potential on the boundary. We establish sharp L^{p} estimates for the Steklov eigenfunctions on compact manifolds with boundary, controlled by their L^{2} norms on the boundary. We first establish sharp boundary estimates by heat kernel bounds and resolvent estimates for the Dirichlet-to-Neumann operator with a rough potential. And then we combine harmonic extension with the Littlewood–Paley decomposition to obtain sharp interior estimates. These results are new even when there is no potential. As an application, we prove the eigenfunctions are C^{1} if the potential is Lipschitz and refine the previous results by Wang and Zhu (2015) on the lower bound of the size of the boundary nodal sets. A key tool is the commutator estimate for first-order pseudo-differential operators by Calderón (1965), and Coifman and Meyer (1978).
We consider the inverse spectral problem for two classes of Hamiltonians of interacting Fermi particles on an integer lattice of arbitrary dimension subject to a common almost-periodic potential. We show first that the inverse problem can be solved with the help of the KAM (Kolmogorov-Arnold-Moser) techniques from Craig (1983) and P & ouml;schel (1983), initially developed for single-particle models, for a particular class of limit-periodic potentials, and then comment on the adaptations to make in the case of quasi-periodic potentials. The solution is obtained in the class of Hamiltonians with a small amplitude of the kinetic energy operator (i.e., with a low mobility of the particles) featuring a uniform exponential decay of all eigenfunctions which prove to be unimodal.
We study a generalized Steklov problem involving a rough potential on the boundary. We establish sharp L-p estimates for the Steklov eigenfunctions on compact manifolds with boundary, controlled by their L-2 norms on the boundary. We first establish sharp boundary estimates by heat kernel bounds and resolvent estimates for the Dirichlet-to-Neumann operator with a rough potential. And then we combine harmonic extension with the Littlewood-Paley decomposition to obtain sharp interior estimates. These results are new even when there is no potential. As an application, we prove the eigenfunctions are C-1 if the potential is Lipschitz and refine the previous results by Wang and Zhu (2015) on the lower bound of the size of the boundary nodal sets. A key tool is the commutator estimate for first-order pseudo-differential operators by Calder & oacute;n (1965), and Coifman and Meyer (1978).
We consider planar elastic beam Hamiltonians defined on hexagonal lattices. These quantum graphs are constructed from Euler-Bernoulli beams, each governed by the fourth-order Schr & ouml;dinger operator with a real periodic symmetric potential function. In contrast to the second-order Schr & ouml;dinger operator commonly studied in the quantum graph literature, here vertex matching conditions encode the geometry of the underlying graph by their dependence on angles at which the edges meet. We show that on the hexagonal lattice, the dispersion relation has a structure similar to that reported for the periodic second-order Schr & ouml;dinger operator, known as the "graphene Hamiltonian." This property is then utilized to prove the existence of Dirac points (conical singularities). We further discuss the (ir)reducibility of Fermi surfaces. Moreover, we obtain the point spectrum, the absolutely continuous spectrum, and the singular continuous spectrum. Applying perturbation analysis, we derive the dispersion relation for the planar elastic beam Hamiltonians on angle-perturbed irregular hexagonal lattices, defined in a geometric neighborhood of the hexagonal lattice. On these graphs, we find that, unlike the hexagonal lattice, the dispersion relation is not split into purely energy- and quasimomentum-dependent terms; however, Dirac points exist similar to the hexagonal-lattice case.
We prove H & ouml;lder continuity of the Lyapunov exponent L.(omega, E) and of the integrated density of states in neighborhoods of energies that satisfy L(omega, E) > 4 kappa(omega, E).beta(omega)>= 0 for general analytic potentials, with kappa(omega, E) being Avila's acceleration, and beta(omega) measures the closeness of omega to rationals on a logarithmic scale.
We discuss how to generalise a Dirac operator such that the solution of a Dirac equation is of bounded variation rather than continuous. We build the spectral theory for generalised Dirac operators and discuss the connection between them and canonical systems. With the help of de Branges’ theory, we discuss the de Branges space of such an operator and the norm endowed. On the other hand, the Paley–Wiener theorem makes it possible to recover a Dirac operator from a function that plays the same role as the spectral measure, which is known as the Gelfand–Levitan condition.
For a large class of symplectic integer matrices, the action on the torus extends to a symplectic $\mathbb{Z}^r$-action with $r\geq 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the $\mathbb{Z}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\geq 1/2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\geq 1/2$.
This paper is dedicated to the spectral analysis of the semiclassical purely magnetic Laplacian \mathcal{L}_{h} , h>0 , on the plane \mathbb{R}^{2} in the situation where the magnetic field B vanishes uniformly, nondegenerately along an open smooth curve \Gamma . We prove the existence of a discrete spectrum for energy windows of the scale h^{4/3} and give complete asymptotics in the semiclassical parameter h for eigenvalues in such windows. Our strategy relies on the microlocalization of the corresponding eigenfunctions close to the zero locus \Gamma and on the implementation of a Born–Oppenheimer strategy through the use of operator-valued pseudodifferential calculus and superadiabatic projectors. This allows us to reduce our spectral analysis to that of effective semiclassical pseudodifferential operators in dimension 1 and apply the well-known semiclassical techniques à la Helffer–Sjöstrand.
Using recent results on uniform large deviation estimates for random matrix products obtained by Omar Hurtado and Sidhanth Raman (2025), we prove localization for one-dimensional Anderson models with heavy tails.
Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such cocycles and establish a Furstenberg-type formula characterizing the Lyapunov exponent. Using the spectral properties of the corresponding Markov operator and a parameter elimination argument, we prove that Lebesgue almost every cocycle in this space satisfies large deviations estimates and a central limit theorem.
Nonlinear spectral problems arise across a range of fields, including mechanical vibrations, fluid-solid interactions, and photonic crystals. Discretizing infinite-dimensional nonlinear spectral problems often introduces significant computational challenges, particularly spectral pollution and invisibility, which can distort or obscure the true underlying spectrum. We present the first general, convergent computational method for computing the spectra and pseudospectra of nonlinear spectral problems. Our approach uses new results on nonlinear injection moduli and requires only minimal continuity assumptions: specifically, continuity with respect to the gap metric on operator graphs, making it applicable to a broad class of problems. We use the Solvability Complexity Index (SCI) hierarchy, which has recently been used to resolve the classical linear problem, to systematically classify the computational complexity of nonlinear spectral problems. Our results establish the optimality of the method and reveal that Hermiticity does not necessarily simplify the computational complexity of these nonlinear problems. Comprehensive examples - including nonlinear shifts, Klein-Gordon equations, wave equations with acoustic boundary conditions, time-fractional beam equations, and biologically inspired delay differential equations - demonstrate the robustness, accuracy, and broad applicability of our methodology.
In this paper, we consider 2 imes 2 matrix-valued pseudodifferential equations in which the two characteristic sets intersect with finite contact order. We show that the asymptotic behavior of its solution changes dramatically before and after the crossing point, and provide a precise asymptotic formula. The proof relies on a normal form reduction and a detailed analysis of a simple first-order system.
The aim of this article is to present a complete system of Floquet spectral invariants for the discrete Schr & ouml;dinger operators with periodic potentials on periodic graphs. These invariants are polynomials in the potential and determined by cycles in the quotient graph from some specific cycle sets. We discuss some properties of these invariants and give an explicit expression for the linear and quadratic (in the potential) Floquet spectral invariants. The constructed system of spectral invariants can be used to study the sets of isospectral periodic potentials for the Schr & ouml;dinger operators on periodic graphs. In particular, we deduce that under certain assumptions, if a real potential is isospectral to the zero (respectively, "degree") potential, then it must be the zero (respectively, "degree") potential.
We obtain a Szego limit theorem for a family of Toeplitz operators defined on the weighted Bergman space of the unit ball B-n. The symbols of these operators are supported on some isotropic or co-isotropic submanifold Gamma subset of B-n and can be seen, in general, as measures that are singular with respect to the Lebesgue measure on C-n. The theorem given allows to describe the asymptotic behavior of these operators as the parameter of the weighted Bergman space tends to infinity.
If A:D(A)subset of H -> H is an unbounded Fredholm operator of index 0 on a Hilbert space H with a dense domain D(A), then its spectrum is either discrete or the entire complex plane. This spectral dichotomy plays a central role in the study of magic angles in twisted bilayer graphene. This paper proves that if such operators (with certain additional assumptions) are perturbed by certain random trace-class operators, their spectrum is discrete with high probability.
