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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]

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 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 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]

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 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]

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 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]

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 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]

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 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]

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]

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 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 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]

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 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 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]

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]

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]

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]

In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an $ n$-dimensional Euclidean space, Payne, Pólya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yang-type universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the $ n$-dimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing ``nice'' trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to $ \tfrac{(n-1)^4}{16}$ $ \tfrac{(n-1)^4}{16}$ [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]

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 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]

This paper addresses a gap in the classification of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the eigenspaces is n - 1, then the metric is a warped product where the base is an open interval, a conclusion we will show to be true under a milder trace condition. Furthermore, we construct examples of Codazzi tensors having two eigenvalue functions, one of which has eigenspace dimension n-1, where the metric is not a warped product with interval base, refuting a claim by A. L. Besse that the warped product conclusion holds without any restriction on the trace. [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 concerns the question whether the cone spectral radius rC(f) of a continuous compact order-preserving homogenous map f : C → C on a closed cone C in Banach space X depends continuously on the map. Using the fixed point index we show that if there exists 0 < a1 < a2 < a3 < ... not in the cone spectrum, σC(f), and limk→∞ ak = rC(f), then the cone spectral radius is continuous. An example is presented showing that if such a sequence (ak)k does not exist, continuity may fail. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that if C is a polyhedral cone with m faces, then σC(f) contains at most m - 1 elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each nonpolyhedral cone there exist maps whose cone spectrum is infinite. [ABSTRACT FROM AUTHOR]

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables and eigenfunction expansions in time and space are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator. [ABSTRACT FROM AUTHOR]

Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension ≤ 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

Let χ : M ↠ Sn+1(1) be an n-dimensional compact hypersurface with constant scalar curvature n(n - 1)r, r ≥ 1, in a unit sphere Sn+1(1), n ≥ 5, and let Js be the Jacobi operator of M. In 2004, L. J. Alías, A. Brasil and L. A. M. Sousa studied the first eigenvalue of Js of the hypersurface with constant scalar curvature n(n - 1) in Sn+1(1), n ≥ 3. In 2008, Q.-M. Cheng studied the first eigenvalue of the Jacobi operator Js of the hypersurface with constant scalar curvature n(n-1)r, r > 1, in Sn+1(1). In this paper, we study the second eigenvalue of the Jacobi operator Js of M and give an optimal upper bound for the second eigenvalue of Js. [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 study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is nonnegative-definite. [ABSTRACT FROM AUTHOR]

This paper concerns minimization and maximization of the first eigenvalue in problems involving the $p$-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of $N=2$ and $p$ close to $2$, our equation models the vibration of a nonhomogeneous membrane $Omega $ which is fixed along the boundary. Given several materials (with different densities) of total extension $|Omega |$, we investigate the location of these material inside $Omega $ so as to minimize or maximize the first mode in the vibration of the membrane. [ABSTRACT FROM AUTHOR]

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if $A$ is an $(n+1)times (n+1)$ nonnegative matrix whose {bf nonzero} eigenvalues are: $lambda _0 geq |lambda _i|$, $i=1,ldots ,r$, $r leq n$, then for all $x geq lambda _0$, begin {equation} prod _{i=0}^{r} (x-lambda _i) leq x^{r+1}-lambda _0^{r+1}.tag *{$(ast )$} endequation }}}par To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when $2(r+1)geq (n+1)$, while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when $nleq 4$ and when the spectrum of $A$ is real. They also showed that the conjecture is asymptotically true with the dimension. par Here we prove a slightly stronger inequality than in $(ast )$, from which it follows that the Boyle--Handelman conjecture is true. Actually, we do not start from the assumption that the $lambda _i$'s are eigenvalues of a nonnegative matrix, but that $lambda _1,ldots , lambda _{r+1}$ satisfy $lambda _0geq |lambda _i|$, $i=1,ldots , r$, and the trace conditions: begin {equation} sum _{i=0}^{r} lambda _i^k geq 0, mbox {for all} k geq 1.tag *{$(ast ast )$} endequation }}} A strong form of the Boyle--Handelman conjecture, conjectured in 2002 by the present authors, says that ($*$) continues to hold if the trace inequalities in ($**$) hold only for $k=1,ldots ,r$. We further improve here on earlier results of the authors concerning this stronger form of the Boyle--Handelman conjecture. [ABSTRACT FROM AUTHOR]

In this paper we derive a trace formula for the Schrödinger operator on the half-line. As a consequence we obtain a Schmincke type inequality with sharp constant. The main tool in our investigation is the inverse spectral Gelfand-Levitan theory, which allows us to compare two Schrödinger operators whose spectra differ by few eigenvalues. [ABSTRACT FROM AUTHOR]