On the shape of the ground state eigenvalue density of a random Hill's equation

Consider the Hill's operator Q = ?d²/dx² + q(x) in which q(x), 0 ≤ x ≤ 1, is a white noise. Denote by f(μ) the probability density function of ?λ₀(q), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics as μ → + ∞. This result is based on a precise Laplace analysis of a functional integral representation for f(μ) established by S. Cambronero and H. P. McKean in 5 . © 2005 Wiley Periodicals, Inc.     

The spectrum of Hill's equation

Letq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=?d ²/dx ²+q(x) in the class of functions of period 2 is a discrete series - ∞<λ₀<λ₁≦λ₂<λ₃≦λ₄<...<λ_2i?1≦λ_2i ↑∞. Let the numer of simple eigenvalues be 2n+1<=∞. Borg [1] proved thatn=0 if and only ifq is constant. Hochstadt [21] proved thatn=1 if and only ifq=c+2p with a constantc and a Weierstrassian elliptic functionp. Lax [29] notes thatn=m if¹ q=4k ² K ² m(m+1)sn ²(2Kx,k). The present paper studies the casen<∞, continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardneret al. [12], Gelfand [13], Gelfand and Levitan [14], Hochstadt [21], and Lax [28–30] in various directions. The content may be summed up in the statement thatq is an abelian function; in fact, from the present standpoint, the whole subject appears as a part of the classical function theory of the hyperelliptic irrationality \(\ell (\lambda ) = \sqrt { - (\lambda - \lambda _0 )(\lambda - \lambda _1 )...(\lambda - \lambda _{2n} )} .\) The casen=∞ requires the development of the theory of abelian and theta functions for infinite genus; this will be reported upon in another place. Some of the results have been obtained independently by Novikov [34], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [22].     

The nonlinear Schrödinger equation with white noise dispersion

Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of this work is to prove that this latter equation is globally well posed in L² or H¹. The main ingredient is the generalization of the classical Strichartz estimates. Additionally, we justify rigorously the formal limit described above.     

On Hill's operator with a matrix potential

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the self‐adjoint operator generated by a system of Sturm–Liouville equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the potential for which the number of gaps in the spectrum of the Hill's operator with matrix potential is finite. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Oscillation and nonoscillation of Hill's equation with periodic damping

We prove new results on the oscillation and nonoscillation of the Hill's equation with periodic damping:

y″+p(t)y′+q(t)y=0,t?0,     

The Cauchy problem for a parabolic equation with coefficients of “white noise” type

Existence and uniqueness conditions are found for a solution possessing a first moment. Sufficient conditions are given for the second moment to be bounded.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 18–28, 1987.     

1D quintic nonlinear Schrödinger equation with white noise dispersion

In this article, we improve the Strichartz estimates obtained in A. de Bouard, A. Debussche (2010) [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.     

Time-space fractional stochastic Ginzburg-Landau equation driven by Gaussian white noise

The current article is devoted to the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations. The dependence of the order of time-fractional derivative, the order of the space-fractional derivative, and the regularity of the initial data are revealed. The global existence and uniqueness of the mild solutions for time-space fractional complex Ginzburg-Landau equation driven by Gaussian white noise are established.     

On the band gap structure of Hill's equation

We revisit the old problem of finding the stability and instability intervals of a second-order elliptic equation on the real line with periodic coefficients (Hill's equation). It is well known that the stability intervals correspond to the spectrum of the Bloch family of operators defined on a single period. Here we propose a characterization of the instability intervals. We introduce a new family of non-self-adjoint operators, formally equivalent to the Bloch ones but with an imaginary Bloch parameter, that we call exponential. We prove that they admit a countable infinite number of eigenvalues which, when they are real, completely characterize the intervals of instability of Hill's equation.     

Two methods of constructing asymptotes for Hill's equation

The asymptotic solution of a particular form of Hill's equation in the neighborhood of the transition curves is found on the basis of a method proposed by Nayfeh. The same differential equation is then solved by the method of averaging, using corrections of up to fourth degree. The second method is shown to give the same asymptote.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 142–145, January, 1992.     

The asymptotic behaviour of the ground state solutions for Hénon equation

The main purpose of this paper is to analyze the asymptotic behaviour of the ground state solution of Hénon equation −Δu=|x|^αu^p−1 in Ω, u=0 on ∂Ω (Ω⊂Rn is a ball centered at the origin). It proved that for p close to 2∗=2n/(n−2)(n?3), the ground state solution u_p has a unique maximum point x_p and as p→2∗. The asymptotic behaviour of u_p is also given, which deduces that the ground state solution is non-radial.     

Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise

Summary The one-dimensional heat equation driven by Fisher-Wright white noise is studied. From initial conditions with compact support, solutions retain this compact support and die out in finite time. There exist interface solutions which change from the value 1 to the value 0 in a finite region. The motion of the interface location is shown to approach that of a Brownian motion under rescaling. Solutions with a finite number of interfaces are approximated by a system of annihilating Brownian motions.