We continue the program initiated by [J. Éc. Polytech., Math. 12, 1083-1160 (2025)] and show that the Pleijel theorem holds unconditionally on all but four $H$-type groups.
We derive a counting formula for the eigenvalues of Schrödinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue problems into smaller subgraph eigenvalue problems. These methods provide a simple way to calculate the spectra of operators with localized potentials.
We obtain a new upper bound for Neumann eigenvalues of the Laplacian on a bounded convex domain in Euclidean space. As an application of the upper bound we derive universal inequalities for Neumann eigenvalues of the Laplacian.
We consider the space-fractional operator with order $0<α<1$ on the metric star graph. The boundary conditions at the vertices of the metric star graph providing the self-adjointness of the operator are derived. The obtained result is extended to the other topologies of the metric graphs.
We show that for any $ε>0$, $α\in[0,\frac{1}{2})$, as $g\to\infty$ a generic finite-area genus g hyperbolic surface with $n=O\left(g^α\right)$ cusps, sampled with probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below $\frac{1}{4}-\left(\frac{2α+1}{4}\right)^{2}-ε$. For $α=0$ this gives a spectral gap of size $\frac{3}{16}-ε$ and for any $α<\frac{1}{2}$ gives a uniform spectral gap of explicit size.
We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic multiplicity
For a given self-adjoint operator $A$ with discrete spectrum, we completely characterize possible eigenvalues of its rank-one perturbations~$B$ and discuss the inverse problem of reconstructing $B$ from its spectrum.
Given a sequence of regular finite coverings of complete Riemannian manifolds, we consider the covering solenoid associated with the sequence. We study the leaf-wise Laplacian on the covering solenoid. The main result is that the spectrum of the Laplacian on the covering solenoid equals the closure of the union of the spectra of the manifolds in the sequence. We offer an equivalent statement of Selberg's 1/4 conjecture.
We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant $J$-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for $J$-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with $L^p$-potentials. Known bounds on the non-real eigenvalues of such operators are improved.
In this study, the theorem on necessary and sufficient conditions for the solvability of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the algorithm of reconstruction of potential from spectral data (eigenvalues and normalizing numbers) is given.
In this paper, we study asymptotic behavior of the spectrum of the abstract Gurtin-Pipkin integro-differential equation with the kernel, depending on the parameter. The coefficients of this equation are unbounded and the main part is an abstract hyperbolic equation perturbed by terms that include Volterra integral operators.
We prove a Lieb-Thirring type inequality for a complex perturbation of a d-dimensional massive Dirac operator $D_m, m\geq 0$ whose spectrum is $]-\infty , -m]\cup[m , +\infty[$. The difficulty of the study is that the unperturbed operator is not bounded from below in this case, and, to overcome it, we use the methods of complex function theory. The methods of the article also give similar results for complex perturbations of the Klein-Gordon operator.
We obtain uniform, with respect to t asymptotic formulas for the eigenvalues of the operators generated in (0,1) by the Mathieu-Hill equation with a complex-valued potential and by the t-periodic boundary conditions. Then using it we investigate the non-self-adjoint Mathieu-Hill operator H generated in the allreal line by the same equation and establish the necessary and sufficient conditions for the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator.
Let $X$ be a convex co-compact hyperbolic surface and let $δ$ be the Hausdorff dimension of the limit set of the underlying discrete group. We show that the density of the resonances of the Laplacian in strips ${σ\leq \re(s) \leq δ}$ with $|\im(s)| \leq T$ is less than $O(T^{1+δ-ε(σ)})$ with $ε>0$ as long as $σ>δ/2$. This improves the fractal Weyl upper bounds of Zworski and supports numerical results obtained for various models of quantum chaotic scattering.
We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations where the Liouville measure is not (or not known to be) ergodic.
We consider the Schrödinger operator with a periodic potential $p$ on the real line. We assume that $p$ belongs to the Sobolev space $\mH_m$ on the circle for some $m\ge -1$, and we determine the asymptotics of the quasimomentum and the Titchmarsh-Weyl functions, the Bloch functions at high energy.
In this note, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single site potentials decaying at least at a Gaussian speed. We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type $e^{-\log^2|E-2bq|}$.
In this paper it is shown that in case of trace class perturbations the singular part of Pushnitski $μ$-invariant does not depend on the angle variable. This gives an alternative proof of integer-valuedness of the singular part of the spectral shift function. As a consequence, the Birman-Krein formula for trace class perturbations follows.
For certain compactly supported metric and/or potential perturbations of the Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in $\mathbb{H}^{n+1}$, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.