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On the analytic side, we prove the quantum ergodicity (QE) of Hamiltonian operators on certain series of unitary flat bundles, using mixed quantization techniques. On the dynamical side, we introduce a new family of partially hyperbolic flow associated with QE and establish its ergodicity.

In this work, we study the eccentricity spectra of zero divisor graphs (ZDGs) associated with the ring $\mathbb{Z}_n.$ While previous studies have examined the Laplacian and distance Laplacian spectra of ZDGs, the eccentricity spectra have remained largely unknown due to the unique features of the eccentricity matrix. More specifically, we prove that for a prime $p$, the ZDG and extended ZDG of $\mathbb{Z}_{p^t}$ have integral eccentricity eigenvalues for $t \geq 3$ and $t \geq 2$, respectively. We also find the eccentricity spectra for specific classes of ZDGs and the relationship between the eccentricity matrix of these ZDGs and their tree structures using matrix analysis tools. In addition, for the usefulness of the energy gap in applications, we have calculated the eccentricity energy gap of ZDGs. These findings reveal interesting behaviours of the eccentricity matrix and may contribute to a more profound understanding of the structural properties of ZDGs., Comment: 30 pages, 6 figs. We welcome comments

The recent discovery of ``magic angles" in twisted bilayer graphene (TBG) has spurred extensive research into its electronic properties. The primary tool for studying this thus far has been the famous Bistritzer-MacDonald model, which relies on several approximations. This work aims to build the first steps in studying magic angles without using this model. Thus, we study a model for TBG in AA stacking \emph{without} the approximations mentioned above in the continuum setting, using two copies of a potential with the symmetries of graphene, sharing a common origin and twisted with respect to each other. We describe the angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones in the vertices of the Brillouin zone for such angles. Furthermore, we show that for small potentials, the slope of the Dirac cones is small for commensurate angles that are close to incommensurate angles. This work is the first to establish the existence of Dirac cones for twisted bilayer graphene in the continuum setting without relying on the above model. This work is the first in a series of works to build a more fundamental understanding of the phenomenon of magic angles.

We establish global bounds for solutions to stationary and time-dependent Schr\"odinger equations associated with the sublaplacian $\mathcal L$ on the Heisenberg group, as well as its pure fractional power $\mathcal L^s$ and conformally invariant fractional power $\mathcal L_s$. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schr\"odinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case $s=1$, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results., Comment: 28 pages

We explain in which sense Schr\"odinger operators with complex potentials appear to violate locality (or Weyl's asymptotics), and we pose three open problems related to this phenomenon.

The class of stochastic matrices that have a stochastic $c$-th root for infinitely many natural numbers $c$ is introduced and studied. Such matrices are called arbitrarily finely divisible, and generalise the class of infinitely divisible matrices. In particular, if $A$ is a transition matrix for a Markov process over some time period, then arbitrarily finely divisibility of $A$ is the necessary and sufficient condition for the existence of transition matrices corresponding to this Markov process over arbitrarily short periods. In this paper, we lay the foundation for research into arbitrarily finely divisible matrices and demonstrate the concepts using specific examples of $2 \times 2$ matrices, $3 \times 3$ circulant matrices, and rank-two matrices.

We present a review of the work L. Raymond from 1995. The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian Hamiltonian is a one-dimensional Schr\"odinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, $V\in\mathbb{R}$. The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If $V>4$, then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if $V>4$, then all the gaps, as predicted by the gap labelling theorem, are there., Comment: 63 pages, 5 figures

This paper investigates uniqueness results for perturbed periodic Schr\"odinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -\Delta + V + v$, where $\Delta$ is the discrete Laplacian, $V: \mathbb{Z}^d \rightarrow \mathbb{R}$ is a periodic potential, and $v: \mathbb{Z}^d \rightarrow \mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-\Delta u + V u + v u = \lambda u$, for $\lambda \in \mathbb{C}$, admits only the trivial solution $u \equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of $\lambda$., Comment: 21 pages

Let $H = -d^2/dx^2 + q(x)$, $x \in \mathbb{R}$, where $q(x)$ is a periodic potential, and suppose that the spectrum $\sigma(H)$ of $H$ is the positive semi-axis $[0, \infty)$. In the case where $q(x)$ is real-valued (and locally square-integrable) a well-known result of G. Borg states that $q(x)$ must vanish almost everywhere. However, as it was first observed by M.G. Gasymov, there is an abundance of complex-valued potentials for which $\sigma(H) = [0, \infty)$. In this article we conjecture a characterization of all complex-valued potentials whose spectrum is $[0, \infty)$. We also present an analog of Borg's result for complex potentials., Comment: 22 pages

Two graphs $G$ and $H$ are \emph{cospectral} if the adjacency matrices share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature, e.g. the famous GM-switching method. In this paper, we shall construct cospectral graphs via regular rational orthogonal matrix $Q$ with level two and three. We provide two straightforward algorithms to characterize with adjacency matrix $A$ of graph $G$ such that $Q^TAQ$ is again a (0,1)-matrix, and introduce two new switching methods to construct families of cospectral graphs which generalized the GM-switching to some extent., Comment: 19 pages,16 figures

The Newtonian potential operator for the Helmholtz equation, which is represented by the volume integral with fundamental solution as kernel function, is of great importance for direct and inverse scattering of acoustic waves. In this paper, the eigensystem for the Newtonian potential operator is firstly shown to be equivalent to that for the Helmholtz equation with nonlocal boundary condition for a bounded and simply connected Lipschitz-regular domain. Then, we compute explicitly the eigenvalues and eigenfunctions of the Newtonian potential operator when it is defined in a 3-dimensional ball. Furthermore, the eigenvalues' asymptotic behavior is demonstrated. To illustrate the behavior of certain eigenfunctions, some numerical simulations are included., Comment: 19 pages,4 figures

We provide the first counter-example showing that the ground state energy of electrons in an external Coulomb potential is not always a convex function of the number of electrons. This property had been conjectured to hold for decades and it plays an important role in quantum chemistry. Our counter-example involves an external potential generated by six nuclei of small fractional charges, placed far away from each other. The ground state energy of 3 electrons is proved to be higher than the average of the energies for 2 and 4 electrons. In addition, we show that the nuclei can bind 2 or 4 electrons, but not 3. Although the conjecture remains open for real nuclei (of integer charges), our work sets some doubt on the validity of the energy convexity for general atoms and molecules., Comment: This paper is dedicated to Trygve Helgaker on the occasion of his 70th birthday

We prove that mean decay of the coefficient of Krein system is equivalent to the mean decay of the Fourier transform of its Szeg\H{o} function.

In this article, we consider the Dirac operator with constant magnetic field in $\mathbb R^2$. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similarly to the Landau (Schr\"odinger) operator, we demonstrate that a three-terms asymptotic formula holds for the eigenvalue counting function. One of the main novelties of this work is the treatment of some perturbations of variable sign. In this context we explore some remarkable phenomena related to the finiteness or infiniteness of the discrete eigenvalues, which depend on the interplay of the different terms in the matrix perturbation.

The longstanding \emph{nonnegative inverse eigenvalue problem} (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a \emph{Perron similarity} if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevich arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of $4 \times 4$ stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. \emph{Extremal} elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the \emph{discrete Fourier transform (DFT) matrix} corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices.

In the context of irreducible ultracontractive Dirichlet metric measure spaces, we demonstrate the discreteness of the Laplacian spectrum and the corresponding diffusion's irreducibility in connected open sets, without assuming regularity of the boundary. This general result can be applied to study various questions, including those related to small deviations of the diffusion and generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.

We consider a natural eigenvalue problem for the vector Laplacian related to stationary Maxwell's equations in a cavity and we prove that an analog of the celebrated Faber-Krahn inequality doesn't hold., Comment: 8 pages, 2 figures

As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant in forms of $C(k)\sqrt{\lambda_k(G)}$, where $\lambda_k(G)$ is the $k$-th eigenvalue of the graph Laplacian and $C(k)$ is a constant depending only on $k$. In this article, we prove some new bounds for multi-way Cheeger constants. By shifting the index of the eigenvalue via cyclomatic number, we establish upper bound estimates with an absolute constant instead of $C(k)$. This, in particular, gives a more direct proof of Miclo's higher order Cheeger inequalities on trees. We also show a new lower bound of the multi-way Cheeger constants in terms of the spectral radius of the graph. The proofs involve the concept of discrete nodal domains and a probability argument showing generic properties of eigenfunctions., Comment: 13 pages, 1 figure

In this work we analyze the spectral $\zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we find the characteristic function which implicitly provides the eigenvalues associated with a given self-adjoint extension $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $\zeta$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $\zeta$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $\zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $\zeta$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$., Comment: 39 pages

In this paper we present a complete spectral analysis of Dirac operators with non-Hermitian matrix potentials of the form $i\operatorname{sgn}(x)+V(x)$ where $V\in L^1$. For $V=0$ we compute explicitly the matrix Green function. This allows us to determine the spectrum, which is purely essential, and its different types. It also allows us to find sharp enclosures for the pseudospectrum and its complement, in all parts of the complex plane. Notably, this includes the instability region, corresponding to the interior of the band that forms the numerical range. Then, with the help of a Birman-Schwinger principle, we establish in precise manner how the spectrum and pseudospectrum change when $V\not=0$, assuming the hypotheses $\|V\|_{L^1}<1$ or $V\in L^1\cap L^p$ where $p>1$. We show that the essential spectra remain unchanged and that the $\varepsilon$-pseudospectrum stays close to the instability region for small $\varepsilon$. We determine sharp asymptotic for the discrete spectrum, whenever $V$ satisfies further conditions of decay at infinity. Finally, in one of our main findings, we give a complete description of the weakly-coupled model., Comment: 43 pages, 4 figures, 1 table

We consider the magnetic Dirac operator on a curved strip whose boundary carries the infinite mass boundary condition. When the magnetic field is large, we provide the reader with accurate estimates of the essential and discrete spectra. In particular, we give sufficient conditions ensuring that the discrete spectrum is non-empty.

We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs. Their spectra consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain periodic graphs of smaller dimensions called subcovering graphs. For example, rolling up a planar hexagonal lattice along different directions will lead to nanotubes with various chiralities. We show that the subcovering graph is asymptotically isospectral to the original periodic graph as the length of the "chiral" (roll up) vectors tends to infinity and get asymptotics of the band edges of the Schr\"odinger operator on the subcovering graph. We also obtain a criterion for the subcovering graph to be just isospectral to the original periodic graph. By isospectrality of periodic graphs we mean that the spectra of the Schr\"odinger operators on the graphs consist of the same number of bands and the corresponding bands coincide as sets., Comment: 27 pages, 6 figures. arXiv admin note: text overlap with arXiv:2402.10780

We present a novel theoretical framework connecting k-component edge connectivity with spectral graph theory and homology theory to pro vide new insights into the resilience of real-world networks. By extending classical edge connectivity to higher-dimensional simplicial complexes, we derive tight spectral-homological bounds on the minimum number of edges that must be removed to ensure that all remaining components in the graph have size less than k. These bounds relate the spectra of graph and simplicial Laplacians to topological invariants from homology, establishing a multi-dimensional measure of network robustness. Our framework improves the understanding of network resilience in critical systems such as the Western U.S. power grid and European rail network, and we extend our analysis to random graphs and expander graphs to demonstrate the broad applicability of the method. Keywords: k-component edge connectivity, spectral graph theory, homology, simplicial complexes, network resilience, Betti numbers, algebraic connectivity, random graphs, expander graphs, infrastructure systems

We study the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length $t$ of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial. The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus, in the spirit of the celebrated b-algebra of Melrose, which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents. We obtain smoothness of the spectral projectors, and $t^2 \log t$ regularity for the cusp-surgery trace of the relative resolvent in the degeneracy process as $t \searrow 0$., Comment: 50 pages, 8 figures

We prove Weyl laws for Schr\"odinger operators with critically singular potentials on compact manifolds with boundary. We also improve the Weyl remainder estimates under the condition that the set of all periodic geodesic billiards has measure 0. These extend the classical results by Seeley, Ivrii and Melrose. The proof uses the Gaussian heat kernel bounds for short times and a perturbation argument involving the wave equation., Comment: 19 pages

We study the topological properties of spaces of reflectionless canonical systems. In this analysis, a key role is played by a natural action of the group $\operatorname{PSL}(2,\mathbb R)$ on these spaces.

This paper studies moving 180-degree N\'eel walls in ferromagnetic thin films under the reduced model for the in-plane magnetization proposed by Capella, Melcher and Otto [5], in the case when a sufficiently weak external magnetic field is applied. It is shown that the linearization around the moving N\'eel wall's phase determines a spectral problem that is a relatively bounded perturbation of the linearization around the static N\'eel wall, which is the solution when the external magnetic field is set to zero and which is spectrally stable. Uniform resolvent-type estimates for the linearized operator around the static wall are established in order to prove the spectral stability of the moving wall upon application of perturbation theory for linear operators. The spectral analysis is the basis to prove, in turn, both the decaying properties of the generated semigroup and the nonlinear stability of the moving N\'eel wall under small perturbations, in the case of a sufficiently weak external magnetic field. The stability of the static N\'eel wall, which was established in a companion paper [4], plays a key role to obtain the main result., Comment: 40 pages, 4 figures

We prove that on convex domains, first mixed Laplace eigenfunctions have no interior critical points if the Dirichlet region is connected and sufficiently small. We use this result to construct a new family of polygonal domains for which Rauch's hot spots conjecture holds and to prove a new general theorem regarding the hot spots conjecture., Comment: 6 pages, 1 figure. Comments welcome!

We analyze the spectrum of the hexagonal lattice graph with a vertex coupling which manifestly violates the time reversal invariance and at high energies it asymptotically decouples edges at even degree vertices; a comparison is made to the case when such a decoupling occurs at odd degree vertices. We also show that the spectral character does not change if the equilateral elementary cell of the lattice is dilated to have three different edge lengths, except that flat bands are absent if those are incommensurate., Comment: 12 pages, 4 figures

For the Laplacian in spherical and hyperbolic spaces, Robin eigenvalues in two dimensions and Dirichlet eigenvalues in higher dimensions are shown to satisfy scaling inequalities analogous to the standard scale invariance of the Euclidean Laplacian. These results extend work of Langford and Laugesen to Robin problems and to Dirichlet problems in higher dimensions. In addition, scaled Robin eigenvalues behave exotically as the domain expands to a 2-sphere, tending to the spectrum of an exterior Robin problem., Comment: 38 pages, 14 figures

In this work, we consider a general time-periodic linear transport equation with integral source term. We prove the existence of a Floquet principal eigenvalue, namely a real number such that the equation rescaled by this number admits nonnegative periodic solutions. We also prove the exponential attractiveness of these solutions. The method relies on general spectral results about positive operators.

This paper deals with the Sturm-Liouville problem that feature distribution potential, polynomial dependence on the spectral parameter in the first boundary condition, and analytical dependence, in the second one. We study an inverse spectral problem that consists in the recovery of the potential and the polynomials from some part of the spectrum. We for the first time prove local solvability and stability for this type of inverse problems. Furthermore, the necessary and sufficient conditions on the given subspectrum for the uniqueness of solution are found, and a reconstruction procedure is developed. Our main results can be applied to a variety of partial inverse problems. This is illustrated by an example of the Hochstadt-Lieberman-type problem with polynomial dependence on the spectral parameter in the both boundary conditions.

In this paper we prove sharp Lieb-Thirring (LT) inequalities for the family of shifted Coulomb Hamiltonians. More precisely, we prove the classical LT inequalities with the semi-classical constant for this family of operators in any dimension $d\geqslant 3$ and any $\gamma \geqslant 1$. We also prove that the semi-classical constant is never optimal for the Cwikel-Lieb-Rozenblum (CLR) inequalities for this family of operators in any dimension. In this case, we characterize the optimal constant as the minimum of a finite set and provide an asymptotic expansion as the dimension grows. Using the same method to prove the CLR inequalities for Coulomb, we obtain more information about the conjectured optimal constant in the CLR inequality for arbitrary potentials., Comment: Added a remark regarding the optimal LT inequality in d=3 and fixed some typos

This paper presents significant advancements in tensor analysis and the study of random walks on manifolds. It introduces new tensor inequalities derived using the Mond-Pecaric method, which enriches the existing mathematical tools for tensor analysis. This method, developed by mathematicians Mond and Pecaric, is a powerful technique for establishing inequalities in linear operators and matrices, using functional analysis and operator theory principles. The paper also proposes novel lower and upper bounds for estimating column sums of transition matrices based on their spectral information, which is critical for understanding random walk behavior. Additionally, it derives bounds for the right tail of weighted tensor sums derived from random walks on manifolds, utilizing the spectrum of the Laplace-Beltrami operator over the underlying manifolds and new tensor inequalities to enhance the understanding of these complex mathematical structures.

We show that every ergodic dynamical system induces a system with pure Lebesgue spectrum of infinite multiplicity.

Many first-passage processes in complex media and related diffusion-controlled reactions can be described by means of eigenfunctions of the mixed Steklov-Neumann problem. In this paper, we investigate this spectral problem in a common setting when a small target or escape window (with Steklov condition) is located on the reflecting boundary (with Neumann condition). We start by inspecting two basic settings: an arc-shaped target on the boundary of a disk and a spherical-cap-shaped target on the boundary of a ball. We construct the explicit kernel of an integral operator that determines the eigenvalues and eigenfunctions and deduce their asymptotic behavior in the small-target limit. By relating the limiting kernel to an appropriate Dirichlet-to-Neumann operator, we extend these asymptotic results to other bounded domains with smooth boundaries. A straightforward application to first-passage processes is presented; in particular, we revisit the small-target behavior of the mean first-reaction time on perfectly or partially reactive targets, as well as for more sophisticated surface reactions that extend the conventional narrow escape problem.

We construct a unitary operator between Hilbert spaces of generalized eigenfunctions of Coulomb operators and the Laplace-Beltrami operator of hyperbolic space that intertwines their respective Poisson operators on $L^2(\mathbb{S}^{d-1})$. The constructed operator generalizes Fock's unitary transformation, originally defined between the discrete spectra of the attractive Coulomb operator and the Laplace-Beltrami operator on the sphere, to the setting of continuous spectra. Among other connections, this map explains why the scattering matrices are the same in these two different settings, and it also provides an explicit formula for the Poisson operator of the Coulomb Hamiltonian., Comment: 21 pages, 5 figures

We obtain the sharp arithmetic Gordon's theorem: that is, absence of eigenvalues on the set of energies with Lyapunov exponent bounded by the exponential rate of approximation of frequency by the rationals, for a large class of one-dimensional quasiperiodic Schr\"odinger operators, with no (modulus of) continuity required. The class includes all unbounded monotone potentials with finite Lyapunov exponents and all potentials of bounded variation. The main tool is a new uniform upper bound on iterates of cocycles of bounded variation.

This work is concerned with the spectral analysis of a piezoelectric energy harvesting model based on a coupled bending-torsion beam. After building the problem's operator setting and showing that the governing operator is nonselfadjoint with a purely discrete spectrum, we derive an asymptotic approximation of its spectrum. In doing so, we also prove that the addition of energy harvesting can be viewed as a weak perturbation of the underlying beam dynamics, in the sense that no piezoelectric parameters appear in the spectral approximation's first two orders of magnitude. We conclude by outlining future work based on numerical simulations., Comment: 21 pages

We prove H\"older continuity of the Lyapunov exponent $L(\omega,E)$ and the integrated density of states at energies that satisfy $L(\omega,E)>4\kappa(\omega,E)\cdot \beta(\omega)\geq 0$ for general analytic potentials, with $\kappa(\omega,E)$ being Avila's acceleration.

We consider eigenvalue sums of Schr\"odinger operators $-\Delta+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q

The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in \textit{graphene} and \textit{carbon $(p,q)$--nanotubes}. The chemical and physical attributes of these two carbon allotropes are closely linked to the average number of closed paths of different lengths $k\in\mathbb{N}_0$ on their respective graph representations. Considering that a carbon $(p,q)$--nanotube can be thought of as a graphene sheet rolled up in a matter determined by the \textit{chiral vector} $(p,q)$, our findings are based on the study of \textit{random eigenvalues} of both the hexagonal and triangular lattices presented in \cite{bille2023random}. This study reveals that for any given \textit{chiral vector} $(p,q)$, the sequence of counts of closed paths forms a moment sequence derived from a functional of two independent uniform distributions. Explicit formulas for key characteristics of these distributions, including probability density function (PDF) and moment generating function (MGF), are presented for specific choices of the chiral vector. Moreover, we demonstrate that as the \textit{circumference} of a $(p,q)$--nanotube approaches infinity, i.e., $p+q\rightarrow \infty$, the $(p,q)$--nanotube tends to converge to the hexagonal lattice with respect to the number of closed paths for any given length $k$, indicating weak convergence of the underlying distributions.

We consider a dilute Bose gas in the thermodynamic limit and prove a lower bound on the free energy for low temperatures which is in agreement with the conjecture of Lee-Huang-Yang on the excitation spectrum of the system. Combining techniques of \cite{FS2} and \cite{HHNST}, we give a simpler and shorter proof resolving the case of strong interactions, including the hard-core potential., Comment: 33 pages

In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for capillary convex bodies in the half-space and prove a general Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The conjecture [16, Conjecture 1.5] follows as its consequence., Comment: 18 pages

In this note, we show that for a family of quantum walk models with electric fields, the spectrum is the unit circle for any irrational field. The result also holds for the associated CMV matrices defined by skew-shifts. Generalizations to CMV matrices with skew-shifts on higher dimensional torus are also obtained.

The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.

We provide inequalities enabling to bound the error between the exact solution and an approximated solution of an eigenvalue problem, obtained by subspace projection, as in the reduced basis method. We treat self-adjoint operators and degenerate cases. We apply the bounds to the eigenvector continuation method, which consists in creating the reduced space by using basis vectors extracted from perturbation theory.

In this note we consider achieving the largest principle eigenvalue of a Robin Laplacian on a bounded domain $\Omega$ by optimizing the Robin parameter function under an integral constraint. The main novelty of our approach lies in establishing a close relation between the problem under consideration and the asymptotic behavior of the Dirichlet heat content of $\Omega$. By using this relation we deduce a two-term asymptotic expansion of the principle eigenvalue and discuss several applications., Comment: 10 -ages, no figures