Consecutive Eigenvalues Research Articles

Almost periodic solutions to systems of parabolic equations

In this paper we show that the second‐order differential solution is 𝕃2‐almost periodic, provided it is 𝕃2‐bounded, and the growth of the components of a non‐linear function of a system of parabolic equation is bounded by any pair of con‐secutive eigenvalues of the associated Dirichlet boundary value problems.

Resonance at two consecutive eigenvalues for semilinear elliptic equations

The solvability of the Dirichlet problem for a semilinear elliptic equation is studied in some situations where the classical resonance conditions of Landesman and Lazer may fail.

Nonresonance Conditions on the Potential for a Second-Order Periodic Boundary Value Problem

We consider the periodic problem \[ \begin {array}{*{20}{c}} { - u'' = f(u) + h(t),} \\ {u(0) = u(2\pi ),\qquad u’(0) = u’(2\pi ),} \\ \end {array} \] and prove its solvability for any given $h$, under new assumptions on the asymptotic behaviour of the potential of the nonlinearity $f$, with respect to two consecutive eigenvalues of the associated linear problem.

On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere

AbstractLet 0 = λ0 < λ1 ≤ λ2 ≤ λ3 ≤ … denote the sequence of eigenvalues of the Laplacian of a compact minimal submanifold in a unit sphere. Yang and Yau obtained an upper bound on λn+1 in terms of λn and the sum λ1 + … + λn. In this note we shall prove an improved version of this upper bound by using the method of Hile and Protter.

The shooting method for some nonlinear Sturm-Liouville boundary value problems

The shooting method is used to prove existence and uniqueness of the solution for a semilinear Sturm-Liouville boundary value problem (N).\(\frac{\partial }{{\partial u}}f(x,u)\) “lies between two consecutive eigenvalues” of the related linear problem, the shooting function turns out to be strongly monotone.

On an elliptic boundary value problem at double resonance

Let Ω be a bounded domain in R n ( n ⩾ 2) with smooth boundary ∂Ω. We discuss the existence and multiplicity of solutions of the boundary value problem −Δu = p(x, u) in Ω, u = 0 on ∂Ω , roughly speaking, under the assumption that λ l ⩽ lim t → ± ∞( p(x, t) t ) ⩽ λ l + 1 for two distinct consecutive eigenvalues λ l , λ l + 1 of the first boundary value problem of −Δ.

Asymptotic nonuniform nonresonance conditions in the periodic-Dirichlet problem for semi-linear wave equations

We consider the existence of weak solutions of the periodic-Dirichlet problem on ]0, 2π[×]0, π[for the semi-linear wave equation u tt -uv xx -f(t, x, u=0 when x(t, x)⩽lim in u −1 f(t, x, u)⩽lim supu −1 f(t, x, u) ⩽ Β(t, x ¦u¦→∞ ¦u¦→∞ and α and Β satisfy some nonresonance conditions of non uniform type withr espect to two consecutive nonzero eigenvalues of the associated linear problem. The proof is based upon one generalized continuation theorem for some perturbations of mappings which are not of Fredholm type.

Stability properties of an anisotropic guiding center plasma and relation with the Suydam function

The effect of pressure anisotropy on the equilibrium and stability properties of an unstable guiding center plasma and the dependence of associated stability properties on the Suydam function S are examined. An explicit solution of the guiding center plasma equilibrium equation is obtained as a function of the anitsotropy parameter α≡P∥/P⊥ (assumed constant), and the maximum growth rates for internal kink modes are numerically computed for the entire permissible range of α. For a typical tokamak field configuration with shear in straight cylindrical geometry, it is found that the maximum growth rate is a monotonically increasing function of α. A detailed parameter study of equilibrium and stability properties is presented. The dependence of stability properties on the Suydam function S is investigated by correlating maximum growth rates with the magnitude of S, and by examining the ratio of consecutive eigenvalues for each set of the parameters. The numerical analysis shows that, even though the Suydam function occurs naturally in studies of marginal stability, the maximum growth rate (except for a narrow range of α) is a monotonically decreasing function of S.

Some existence theorems for nonlinear eigenvalue problems associated with elliptic equations

The existence of harmonic functions in a region D of the plane satisfying nonlinear Neumann boundary conditions of the form Ou]On=f(u) where n is the outwardly directed unit normal to the boundary OD of D has been studied by T. CAgLrMAN ([4]), NAKAMORI & SOYAMA ([16]), K. KLINGELH/~FER ([9], [10], [11]), and, in the case when D is the unit disk andf i s of a highly specialized form which also depends on the harmonic conjugate of u, by LEVI-CIVITA ([13]) in his study of periodic progressing water waves. CARLEMAN and NAKAMOm & SUYAMA considered the ease when f ' (u)<0 and found existence results for singular solutions. KLINGELIq61~Im, using the contraction mapping principle, developed existence and uniqueness theorems for regular solutions under assumptions on f (u) which, roughly speaking, require that i f (u) stay away from the (necessarily positive) eigenvalues of the linear problem (the Steklov problem [2], [19]) Ou]On=2u; that is, ;tk < f ' (u) < 2k+ 1 where 2k, 2k+ 1 are consecutive Steldov eigenvalues. These results, however, never yield nontrivial solutions to the problem in the case that f(0) = 0; LEvI-CIVlTA, on the other hand, found (in his special problem) nontrivial solutions even though the problem had the trivial solution u= 0. The problem of finding nontrivial solutions to elliptic equations under such nonlinear boundary conditions (actually a problem in bifurcation theory) is the subject of this paper. Various criteria for uniqueness have been found by many authors including MARTIN, LEVlN, DUNNINGER, and CUSHING (see [5] for bibliography). The problem is the following: to satisfy the equation

Systematic Characterization of mth-Order Energy-Level Spacing Distributions

The mth-order energy-level spacing distributions p(m)(x) for complex spectra are defined in terms of a general joint probability distribution PN(λ1, … λN) for N consecutive eigenvalues. The precise limiting processes involved are explained, and are subsequently used to obtain two formal representations of p(m)(x). Both representations yield p(m)(x) = xm/m! exp (−x) for statistically independent eigenvalues. One of the representations, which is an extension of Dyson's method for m = 0, 1, is applied to the superposition of n independent sequences of levels. General asymptotic results are found for the mth-order distributions for (a) small x, arbitrary n, and (b) arbitrary x with n → ∞.

Class of Ensembles in the Statistical Theory of Energy-Level Spectra

The investigation of a large class of ensembles in the statistical theory of energy-level spectra is initiated. Each member of this class is characterized by a joint probability density for N consecutive eigenvalues of the form PNβ(λ1,…λN)=ΩNβ−1{ ∏ i=1N f(λi)} ∏ k&amp;lt;l |λk−λl|β,where a ≤ λi ≤ b, and β may be 1, 2, or 4. Formal calculations of the nearest-neighbor spacing distribution and the level density are made for β = 2. Results are in terms of asymptotic properties of orthogonal polynomials. It is conjectured that spacing distributions are relatively insensitive to the function f(λ) and the interval [a, b]. When f(λ) = 1 and b = −a = 1, the resulting (Legendre) ensemble has the same spacing distribution as the Gaussian and Dyson ensembles. The level density is concave upward and rapidly increasing for λ ≥ 0, qualitatively resembling actual nuclear and atomic densities. This feature is not present in previously investigated ensembles. Certain invariant matrix ensembles introduced by Dyson, which are of the above type, have the same level density and nearest-neighbor spacing distribution as the Legendre ensemble.