Spectra of positive and negative energies in the linearized NLS problem

We study the spectrum of the linearized NLS equation in three dimensions in association with the energy spectrum. We prove that unstable eigenvalues of the linearized NLS problem are related to negative eigenvalues of the energy spectrum, while neutrally stable eigenvalues may have both positive and negative energies. The nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum. We derive bounds on the number of unstable eigenvalues of the linearized NLS problem and study bifurcations of embedded eigenvalues of positive and negative energies. We develop the L²‐scattering theory for the linearized NLS operators and recover results of Grillakis [5] with a Fermi golden rule. © 2004 Wiley Periodicals, Inc.     

Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie in forma implicita negli spazi di Banach

Summary This paper deals with the existence of solutions for the implicit Cauchy problem F(t, x, x)=_B, x(t₀)=x₀ in a Banach space B. By using the Kuratowski and the Hausdorff measure of non compactness, we prove an existence theorem for the previous problem (Teorema 1.1) and its extension to non compact intervals (Teorema 2.1). These results generalize the previous ones by R.Conti [1] (in the case B=R), G.Pulvirenti [2] and T. Dominguez Benavides [3], [4] (in the general case). In particular, we relax a Lipschitz condition assumed by all of the abovementioned authors. Some applications of Teorema 2.1 are presented.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

On the eigenvectors and eigenvalues of an operator pair

We establish conditions for orthogonality of eigenvectors and the type of eigenvalues (real or purely imaginary) of an operator pair A, B in an equation Au + Bu = 0which permute (or skew-permute) with a given third operator. We consider equations with differential operators A and B (the differential spectral problem), with a differential operator A and an integro-differential operator B, and also with algebraic operators A and B. Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 14–19.     

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.     

The Analytical (Spectral) Representation of the Solution of Delay Algebraic-Differential Equations

A new approach to finding analytical solutions of linear delay algebraic-differential equations is suggested. The analytical form of the solution is determined in terms of the infinite set of eigenvalues of a parametric matrix whose entries are the delay-time operators exp(–p), where p is the Laplace operator. In order to compute constants in the solution of the homogeneous equations, one must analytically find higher derivatives at the input of the delay operator. The problem of stopping the computation of the infinite spectrum upon determining a certain number of its components is discussed. Bibliography: 5 titles.     

Variational Principles for Eigenvalues of Self-adjoint Operator Functions

Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T() are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T()x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.     

On the uniqueness of solutions of spectral equations

We prove uniqueness of the viscosity solutions of the Dirichlet problem of the spectral equation where is the vector whose components are eigenvalues of a matrix associated with the unknown function u.     

The eigenvalue problem for solitary waves of a boussinesq equation,via a generalization of the rouché theorem

We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane.     

Multiplication and Compact-friendly Operators

During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M commutes with a non-zero finite-rank operator if and only the multiplier function is constant on some non-empty open set.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in [7] ofG. H. Meisters and improves the result [5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result [10] ofG. A. Willis.     

A remark on linearly unstable standing wave solutions to NLS

We obtain an optimal growth estimate of a semigroup generated by a linearized operator around a standing wave solution nonlinear Schrödinger equations in two-dimension. Using the growth estimate of the semigroup, we prove that a linearly unstable standing wave solution is orbitally unstable and that instability of the standing wave solution is mainly caused by a mode of an eigenfunction associated with the rightmost (or the leftmost) eigenvalues of the linearized operator. Our result is obtained by using the method of Yajima and Cuccagna that proved L^p$L^p$-boundedness of the wave operator.     

Gelfand numbers of operators with values in a Hilbert space

In the paper we prove two inequalities involving Gelfand numbers of operators with values in a Hilbert space. The first inequality is a Rademacher version of the main result in [Pa-To-1] which relates the Gelfand numbers of an operator from a Banach spaceX intol ₂ ⁿ with a certain Rademacher average for the dual operator. The second inequality states that the Gelfand numbers of an operatoru froml ₁ ^N into a Hilbert space satisfy the inequality

An algebra of integral operators with fixed singularities in kernels

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue spaceL ₂ (, ), where is a curve with cusps of arbitrary order and is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.Avedanio, N.Krupnik, 1988 and R.Duduchava, N.Krupnik, 1995). Dedicated to Professor Israel Gohberg on the occasion of his 70-th birthday Supported by EPSRC grant GR/K01001     

On a linearized system arising in the study of viscous surface flow down an inclined plane

We consider the linearized problem describing motion of a viscous incompressible fluid flow down an inclined plane under the effect of gravity. We formulate the problem for downward periodic disturbances from the laminar steady flow as an evolution equation in the product of Sobolev spaces \(H^{\frac{3}{2}}_0({\mathbb {T}}) \times {\mathbb {P}}^0H^0(\Omega ) \) . It is crucial to study qualitative behavior of solutions that the operator arising in the linearized problem has compact resolvent operators and generates an analytic semigroup in \(H^{\frac{3}{2}}_0({\mathbb {T}}) \times {\mathbb {P}}^0H^0(\Omega ) \) .     

Pucci eigenvalues on geodesic balls

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng’s eigenvalue comparison theorem for the Laplace–Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng’s bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng’s upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an \({O(n)}\)-invariant hypersurface immersed in \({{\mathbb{R}}^{n+1}}\) with one smooth boundary component are smaller than the eigenvalues of an \({n}\)-dimensional Euclidean ball with the same boundary.     

Invariant differential operators on Hermitian symmetric spaces and their eigenvalues

Let be the invariant Cauchy Riemann operator and the corresponding invariant Laplacians on a bounded symmetric domain. We calculate the eigenvalues ofM _m on spherical functions. In particular we prove that for a symmetric domain of rank two the operatorsM ₁,M ₃ generate all invariant differential operators. We also find the eigenvalues of the generators introduced by Shimura.     

On certain degenerate and singular elliptic PDEs I: Nondivergence form operators with unbounded drifts and applications to subelliptic equations

We prove a Harnack inequality for nonnegative strong solutions to degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts. Our main theorem extends the Harnack inequality for the linearized Monge–Ampère equation due to Caffarelli and Gutiérrez and it is related, although under different hypotheses, to a recent work by N.Q. Le.Since our results are shown to apply to the convex functions $|x{|}^p$ with $p\ge 2$ and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge–Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot–Carathéodory metric in the study of certain subelliptic PDEs.