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In a recent paper, Rodrigues and Sola-Morales construct an example of a continuously Frechet differentiable discrete dynamical system in a separable Hilbert space for which the origin is an exponentially asymptotically stable fixed point, although its derivative at 0 has spectral radius greater than one. For maps on general Banach spaces we demonstrate that the slightly stronger, but also widely used concept of exponential stability allows a complete characterization in terms of the spectral radius. Moreover, under a spectral gap condition valid for compact and finite-dimensional linearizations these two stability notions are shown to be equivalent.

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained., Comment: 38 pages

Consider the semilinear Schrodinger equation { − △ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where both V ( x ) and f ( x , u ) are periodic in x, 0 belongs to a spectral gap of − △ + V , and f ( x , u ) is subcritical and allowed to be super-linear at some x ∈ R N and asymptotically linear at the other x ∈ R N . In the existing works in the literature, it is commonly assumed that lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ uniformly in x ∈ R N , to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ , a.e. x ∈ G just for some domain G ⊂ R N .

This paper deals with a non-self-adjoint eigenvalue problem { a ( x ) y ″ ( x ) + b ( x ) y ′ ( x ) = λ y ( x ) , y ( 0 ) = ∫ 0 1 y ( x ) d ν 0 ( x ) , y ( 1 ) = ∫ 0 1 y ( x ) d ν 1 ( x ) , which is associated with the generator of one dimensional diffusions with random jumps from the boundary. We focus on the dependence of spectral gap, eigenvalues and eigenfunctions on the coefficients a, b and the probability distributions ν 0 , ν 1 . To prove this, we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis. Moreover, we also prove that zero is an algebraically simple eigenvalue of the problem.

We prove the existence of inertial manifolds for the incompressible hyperviscous Navier–Stokes equations on the two or three-dimensional torus: { u t + ν ( − Δ ) β u + ( u ⋅ ∇ ) u + ∇ p = f , ( t , x ) ∈ R + × T d , div u = 0 , where d = 2 or 3 and β ≥ 3 / 2 . Since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions, we employ the spatial averaging method introduced by Mallet-Paret and Sell in [26] .

In a seminal work, B. Simon provided a classification of nonnegative Schr��dinger operators $-��+V$ into subcritical and critical operators based on the long-term behaviour of the associated heat kernel. Later works by others developed an alternative subcritical/critical classification based on whether or not the operator admits a weighted spectral gap. All these works dealt only with potentials that ensured the validity of Harnack's inequality, typically potentials in $L^p_{loc}$ for some $p > \frac{N}{2}$. This paper extends such a weighted spectral gap classification to a large class of locally integrable potentials $V$ called balanced potentials here. Harnack's inequality will not hold in general for such potentials. In addition to the standard potentials in the space $L^p_{loc}$ for some $p > \frac{N}{2}$, this class contains potentials which are locally bounded above or more generally, those for which the positive and negative singularities are separated as well as those for which the interaction of the positive and negative singularities is not "too strong". We also establish for such potentials, under the additional assumption that $V^+ \in L^p_{loc}$ for some $p > \frac{N}{2}$, versions of the Agmon-Allegretto-Piepenbrink Principle characterising the subcritical/critical operators in terms of the cone of positive distributional super-solutions (or solutions). A compressed version of this work has appeared in the paper : M. Lucia and S. Prashanth, Criticality theory for Schrodinger operators with singular potential, J. Differential Equations 265(2018), 3400-3440. We also remove here some restrictions imposed in section 10.3 of this paper., 63 pages, 1 figure

We perturb a linear Schrodinger equation with Lame potential with a small positive or negative potential. The new perturbed operator has one or more eigenvalues, at most one in each spectral gap. We then add a nonlinear term and study the stability of the corresponding nonlinear stationary waves.

In this paper we study the spectral gap of the Dirichlet Laplacian on a periodically curved strip. We begin with the definition of the planar strip and the Dirichlet Laplacian. For γ ∈ C(R; R), let $$ {k_\gamma }:R \ni s \mapsto \left( {{a_\gamma }\left( s \right),{b_\gamma }\left( s \right)} \right) \ni {R^2}$$ be a curve parameterized by its arc length whose curvature at s is γ(s). For d > 0, we define $${\Omega _{d,\gamma }} \equiv \left\{ {\left( {{a_\gamma }\left( s \right) - u\frac{d}{{ds}}{b_\gamma }\left( s \right),{b_\gamma }\left( s \right) + u\frac{d}{{ds}}{a_\gamma }\left( s \right)} \right) \in {R^2};s \in R, - d < u < d} \right\}.$$

We offer two methods of inserting eigenvalues into spectral gaps of a given background Jacobi operator: The single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi-bounded background Jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps. Moreover, we prove unitary equivalence of the commuted operators, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. In addition we compute the (matrix-valued) Weyl m -functions of the commuted operator in terms of the background Weyl m -functions. Finally we show how to iterate the above methods and give explicit formulas for various quantities (such as eigenfunctions and spectra) of the iterated operators in terms of the corresponding background quantities and scattering matrix. Concrete applications include an explicit realization of the isospectral torus for algebro-geometric finite-gap Jacobi operators and the N -soliton solutions of the Toda and Kac–van Moerbeke lattice equations with respect to arbitrary background solutions.

We study the Floquet solutions of quasi-periodic Schrodinger operators on flows which satisfy a Diophantine condition. Using these solutions, we show that the resolvent of such an operator is smooth with respect to certain derivations in the C∗-algebra associated with the flow, and that every projection corresponding to a spectral gap has a finite localization length. Based on these results, we derive a formula which quantizes the integral components of the integrated density of states for the operator on each spectral gap.

We prove existence results for nonlinear two-point boundary value problems. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities. Our solvability conditions allow the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense.

We present a streamlined approach to relative oscillation criteria based on effective Pruefer angles adapted to the use at the edges of the essential spectrum. Based on this we provided a new scale of oscillation criteria for general Sturm-Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover the Gesztesy-Uenal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt., 22 pages