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We continue the program first initiated by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and develop a modification of the technique introduced in that paper to study the spectral asymptotics, namely the Riesz means and eigenvalue counting functions, of functional difference operators H_0 = \mathcal {F}^{-1} M_{\cosh (\xi)} \mathcal {F} with potentials of the form W(x) = \left \lvert {x} \right \rvert ^pe^{\left \lvert {x} \right \rvert ^\beta } for either \beta = 0 and p > 0 or \beta \in (0, 2] and p \geq 0. We provide a new method for studying general potentials which includes the potentials studied by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and [J. Math. Phys. 60 (2019), p. 103505]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics. [ABSTRACT FROM AUTHOR]

In the present paper, we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Léna on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in \mathbb {R}^2. We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple. [ABSTRACT FROM AUTHOR]

In this paper we consider 0-th order pseudodifferential operators on the circle. We show that inside any interval disjoint from the critical values of the principal symbol, the spectrum is absolutely continuous with possibly finitely many embedded eigenvalues. We also give an example of embedded eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the nonlocal dispersal monostable equation with seasonal succession, which can be used to describe the dynamics of species in an environment alternating between bad and good seasons. We first prove the existence and uniqueness of global positive solution, and then discuss the long time behaviors of solution. It is shown that its dynamics is completely determined by the sign of the principal eigenvalue, i.e., the time periodic problem has no positive solution and the solution of the initial value problem tends to zero when principal eigenvalue is non-negative, while the time periodic positive solution exists uniquely and is globally asymptotically stable when principal eigenvalue is negative. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Schrödinger-Poisson system with nonhomogeneous boundary conditions \begin{equation*} \begin {cases} -\frac {1}{2}\triangle {u}+\phi u =\omega u, & \text {in }\Omega, \\ -\triangle {\phi }=4\pi u^2, & \text {in } \Omega,\\ \phi =h,u=0, & \text {on } \partial \Omega, \end{cases} \end{equation*} where \Omega is a smooth and bounded domain in \mathbb {R}^3, h is a given nonnegative regular function on \partial \Omega and \omega \in \mathbb {R}. By using variational method and bifurcation theory, we obtain the existence of positive solutions to the above system for \omega larger than some positive constant. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the smallest eigenvalue \lambda _N, of the (N+1)\times (N+1) Hankel matrix \mathcal {M}_N=(\mu _{j+k})_{0\le j,k\le N} generated by the semi-classical Hermite weight w(z,t)=|z|^\lambda \exp \left (-z^2+tz\right), z, t \in \mathbb {R}, \lambda >-1. An asymptotic expression of the orthonormal polynomials \mathcal {P}_N(z) with the semi-classical Hermite weight w(z,t) is established as N tends to infinity. Based on the orthonormal polynomials \mathcal {P}_N(z), we obtain the specific asymptotic formulas of \lambda _{N}. [ABSTRACT FROM AUTHOR]

In this paper a class of differential operators with retarded arguments on a lasso graph is studied. We derive the asymptotic expressions of its large eigenvalues and obtain a new regularized trace formula for this class of differential operators. [ABSTRACT FROM AUTHOR]

We prove that the (k+d)-th Neumann eigenvalue of the biharmonic operator on a bounded connected d-dimensional (d\ge 2) Lipschitz domain is not larger than its k-th Dirichlet eigenvalue for all k\in \mathbb {N}. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the (k+d+1)-th Neumann eigenvalue of the biharmonic operator does not exceed its k-th Dirichlet eigenvalue for all k\in \mathbb {N}. In particular, in two dimensions, this special class consists of domains having an axis of symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we give a mild solution of our problem. Our main tool here is using some Sobolev embeddings. [ABSTRACT FROM AUTHOR]

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction diffusion equations with skew-gradient structure has been established in a previous paper (see Paul Cornwell [Indiana Univ. Math. J. 68 (2019), pp. 1801-1832]), but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the first eigenvalue of the stability operator on an integral n-varifold with constant mean curvature in the unit sphere \mathbb {S}^{n+1}. We find the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori. [ABSTRACT FROM AUTHOR]

In this paper, we establish lower bounds for the eigenvalues of even ordered quasilinear eigenvalue problems with indefinite weights. We first consider the case when the weight function is identically one and later generalize the results to any weight function within the p-th Muckenhopt class. In the process, we obtain new Lyapunov-type inequalities for the corresponding problems and utilize them to obtain the lower bound of the k-th eigenvalues. In particular, we show that the first eigenvalue is bounded below in terms of the integral of the weight instead of the integral of its positive part. Our results provide a much sharper bound than existing results. [ABSTRACT FROM AUTHOR]

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of q-difference operators. The q-difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of q-difference operators is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial estimates for Steklov eigenvalues on graphs. [ABSTRACT FROM AUTHOR]

The current paper establishes criteria for the existence of principal eigenvalues of time periodic cooperative linear nonlocal dispersal systems with Dirichlet type, Neumann type or periodic type boundary conditions. It is shown that such a nonlocal dispersal system has a principal eigenvalue in the following cases: the nonlocal dispersal distance is sufficiently small; the spatial inhomogeneity satisfies a so-called vanishing condition; or the spatial inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the principal eigenvalue of a time periodic cooperative linear nonlocal dispersal system (if it exists) is algebraically simple. A linear nonlocal dispersal system may not have a principal eigenvalue. The results established in the current paper extend those in literature for time independent or periodic nonlocal dispersal equations to time periodic cooperative nonlocal dispersal systems and will serve as a basic tool for the study of cooperative nonlinear systems with nonlocal dispersal. [ABSTRACT FROM AUTHOR]

In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger operator-div(Pr ∇ ·)-Wr² on a closed hypersurface Σn of ℝn+1, where Wr is a power of the (r + 1)-th mean curvature of Σn, which we will ask to be positive. If this eigenvalue is null, we will have a characterization of the sphere. This theorem generalizes the result of Harrell and Loss proved to the Laplace-Beltrame operator penalized by the square of the mean curvature. [ABSTRACT FROM AUTHOR]

We study first nonzero eigenvalues for the p-Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the p-Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for p\in (1, 2]. Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the p-Laplacian on compact Kähler manifolds with smooth boundary for p\in (1, \infty). Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved by Li and Wang [Trans. Amer. Math. Soc. 374 (2021), pp. 8081–8099]. [ABSTRACT FROM AUTHOR]

In this paper, we study the Crisanti-Sommers variational problem, which is a variational formula for the free energy of spherical mixed p-spin glasses. We begin by computing the dual of this problem using a min-max argument. We find that the dual is a 1D problem of obstacle type, where the obstacle is related to the covariance structure of the underlying process. This approach yields an alternative way to understand Replica Symmetry Breaking at the level of the variational problem through topological properties of the coincidence set of the optimal dual variable. Using this duality, we give an algorithm to reduce this a priori infinite dimensional variational problem to a finite dimensional one, thereby confining all possible forms of Replica Symmetry Breaking in these models to a finite parameter family. These results complement the authors’ related results for the low temperature Γ-limit of this variational problem. We briefly discuss the analysis of the Replica Symmetric phase using this approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the family {Lj(s)}j=1∞ of L-functions associated to an orthonormal basis {uj}j=1∞ of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κj2 + 1/4}j=1∞. We prove the following effective non-vanishing result: At least 50 % of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals. [ABSTRACT FROM AUTHOR]

This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors' earlier work for the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend our earlier results to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ RN be an open bounded connected set. We consider the eigenvalue problem -Δpu = λρ∣u∣p-2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(∣Ω∣-γ) for a fixed 0 < γ < ∣Ω∣. The optimization of the map ρ ↔ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry. [ABSTRACT FROM AUTHOR]

Let Q be a probability measure on a finite group G, and let H be a subgroup of G. We show that a necessary and sufficient condition for the random walk driven by Q on G to induce a Markov chain on the double coset space H\backslash G/H is that Q(gH) is constant as g ranges over any double coset of H in G. We obtain this result as a corollary of a more general theorem on the double cosets H \backslash G / K for K an arbitrary subgroup of G. As an application we study a variation on the r-top to random shuffle which we show induces an irreducible, recurrent, reversible and ergodic Markov chain on the double cosets of \mathrm {Sym}_r \times \mathrm {Sym}_{n-r} in \mathrm {Sym}_n. The transition matrix of the induced walk has remarkable spectral properties: we find its invariant distribution and its eigenvalues and hence determine its rate of convergence. [ABSTRACT FROM AUTHOR]

We show that, given any static spacetime whose spatial slices are asymptotically Euclidean (or, more generally, asymptotically conic) manifolds modeled on the large end of the Schwarzschild exterior, there exist stationary solutions to the Klein–Gordon equation having Schwartz initial data. In fact, there exist infinitely many independent such solutions. The proof is a variational argument based on the long range nature of the effective potential. We give two sets of test functions which serve to verify the hypothesis of the variational argument. One set consists of cutoff versions of the hydrogen bound states and is used to prove the existence of eigenvalues near the hydrogen spectrum. [ABSTRACT FROM AUTHOR]

Let M be a closed differentiable manifold of dimension at least 3. Let \Lambda _0 (M) be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on M can have. We prove that for any k greater than or equal to \Lambda _0 (M), there exists a Riemannian metric on M such that its conformal Laplacian has exactly k negative eigenvalues. Also, we discuss upper bounds for \Lambda _0 (M). [ABSTRACT FROM AUTHOR]

We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on S^m. For S^{2}, the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides' work and recent developments on the hyperbolic center of mass and provides also a simpler proof for S^2. [ABSTRACT FROM AUTHOR]

Measure differential equations can model non-classical problems like the quantum effects. In this paper we will solve extremal problems for eigenvalues of measure differential equations by exploiting the approximation of general measures by smooth measures and the continuity results of eigenvalues in weak* topology of measures. [ABSTRACT FROM AUTHOR]

In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton-Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton-Jacobi equations is also obtained. [ABSTRACT FROM AUTHOR]

The spectrum of the singular indefinite Sturm-Liouville operator ... with a real potential q ∊ L¹(ℝ) covers the whole real line, and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction have been obtained. In this paper the bound ... on the absolute values of the non-real eigenvalues λ of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L¹-norm of the negative part of q. [ABSTRACT FROM AUTHOR]

In this paper we establish an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases. [ABSTRACT FROM AUTHOR]

In a paper by Z. Lu and J. Rowlett, it is shown that the eigenvalues of the weighted Laplacian can be approximated by eigenvalues of a naturally associated family of narrow graphs. In this paper, we generalize this result to the p-Laplacian. Our approach features overcoming the nonlinearity of the p-Laplacian when p ≠ 2, which is different from the Laplacian case. [ABSTRACT FROM AUTHOR]

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p-Laplacian, an indefinite nonlinearity, and depend on a real parameter λ. A special focus is given to the extreme value λ∗ of the Nehari manifold that determines the threshold of applicability of the Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where the parameter λ lies above the threshold λ∗ is obtained. [ABSTRACT FROM AUTHOR]

In this paper we obtain lower bound estimates of the spectrum of the Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative sectional curvature. Our main theorem generalizes many previously known estimates and applies for both immersions and submersions. [ABSTRACT FROM AUTHOR]

In this paper we establish an upper bound for the first eigenvalue of a Shrödinger operator on compact Riemann serfaces in terms of its diameter. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any Hs (Ω) solution u of the problem s (0.1) (-Δ)su = f(u) in Ω,u = 0 on ∂ύ, belongs to L∞ (Ω) for the nonlinearity of f(t) being subcritical and critical. This implies that the solution u is classical if f(t) is C1, γ for some 0 < γ < 1. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of almost periodic mild solutions to some second-order equations. Using dichotomy tools and the Schauder fixed point theorem, the existence of almost periodic mild solutions to those second-order evolution equations is established. To illustrate our abstract results, the existence of almost periodic solutions to a damped second-order boundary value problem is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we show sharp estimates for the first eigenvalue λ1,p of the weighted p-Laplacian on smooth metric measure spaces (M, g, e−f dv). When the Bakry-Émery curvature Ricf is bounded from below and the weighted function f is of sublinear growth, we prove some rigidity properties provided that the first eigenvalue λ1,p obtains its optimal value. [ABSTRACT FROM AUTHOR]

In this paper we consider a real analytic linear quasi-periodic system of 2-dimension, whose coefficient matrix depends on a small parameter Cm-smoothly and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption with respect to the small parameter, we prove that the system is reducible for many of the sufficiently small parameters. [ABSTRACT FROM AUTHOR]

Let P(h), h ∈]0, 1] be a semiclassical scalar differential operator of order 2. The existence of a supersymmetric structure given by a matrix G(x; h) was exhibited by H'erau, Hitrik, and Sj¨ostrand (2011) under rather general assumptions. In this paper we give a sufficient condition on the coefficients of P(h) so that the matrix G(x; h) enjoys some nice estimates with respect to the semiclassical parameter. [ABSTRACT FROM AUTHOR]

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus (ℝ/ℤ)³, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant- sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. [ABSTRACT FROM AUTHOR]

We derive a sharp inequality relating the second and fourth elementary symmetric functions of the eigenvalues of a trace-free matrix and give two applications. First, we give a new proof of the classification of conformally flat hypersurfaces in spaceforms. Second, we construct a functional which characterizes rotational hypersurfaces and catenoids. [ABSTRACT FROM AUTHOR]

Let \phi be an L^2-normalized Hecke–Maaß cusp form on the locally symmetric space X≔\mathrm {PGL}_n(\mathbb {Z}[i])\backslash \mathrm {PGL}_n(\mathbb {C}) / \mathrm {PU}_n. If \Omega is a compact subset of X, then we prove the bound \|\phi |_{\Omega }\|_{\infty }\ll _{\Omega } \lambda _{\phi }^{n(n-1)/4-\delta } for some \delta >0 depending only on n, where \lambda _{\phi } is the Laplace eigenvalue of \phi. [ABSTRACT FROM AUTHOR]

We construct nonminimal and irreducible solutions to the Ginzburg–Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles. [ABSTRACT FROM AUTHOR]

We study Schrödinger operators on compact finite metric graphs subject to \delta-coupling and standard boundary conditions. We compare the n-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. [Comm. Math. Phys. 388 (2021), pp. 1603–1635] obtained for domains in \mathbb {R}^2 to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained by Sofer [ Spectral curves of quantum graphs with \delta _s type vertex conditions , arXiv:2212.09143, 2022] and Band et al. [ Differences between Robin and Neumann eigenvalues on metric graphs , arXiv:2212.12531, 2022] for metric graphs. In addition, based on our main result, we introduce some surface measures for a (quantum) graph which might prove useful in the future. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the second eigenfunctions of the p- Laplacian, p > 1, are not radial on the unit ball in RN, for any N ⩾ 2. Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs {τn,ψn} such that ψn is nonradial and has exactly 2n nodal domains. A few related open problems are also stated. [ABSTRACT FROM AUTHOR]

In this paper we prove a Lyapunov type inequality for quasilinear problems with indefinite weights. We show that the first eigenvalue is bounded below in terms of the integral of the weight, instead of the integral of its positive part. We apply this inequality to some eigenvalue homogenization problems with indefinite weights. [ABSTRACT FROM AUTHOR]

The present paper gives a priori upper and lower bounds on nonreal eigenvalues of regular indefinite Sturm-Liouville problems only under the integrability conditions. More generally, a lower bound on non-real eigenvalues of the self-adjoint operator in Krein space is obtained. [ABSTRACT FROM AUTHOR]

We consider the decay property of the eigenvalues of the Neumann-Poincaré operator in two dimensions. As is well known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having C1,\α boundary with α ∈ (0,1). We show that the eigenvalues λk of the Neumann-Poincaré operator ordered by size satisfy that |λk| = O(k-p-α +1/2) for an arbitrary simply connected domain having C1+p,α boundary with p ≥ 0,~ α ∈ (0,1), and p+α > 1/2. [ABSTRACT FROM AUTHOR]