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A note on eigenvalues of ordinary differential operators.

Alan Ho (1997)

Revista Matemática Iberoamericana

In this follow-up on the work of Fefferman-Seco [FS] an improved condition for the discrete eigenvalues of the operator -d2 / dx2 + V(x) is established for V(x) satisfying certain hypotheses. The eigenvalue condition in [FS] establishes eigenvalues of this operator to within a small error. Through an obervation due to C. Fefferman, the order of accuracy can be improved if a certain condition is true. This paper improves on the result obtained in [FS] by showing that this condition does indeed hold....

A Sturm-Liouville problem with spectral and large parameters in boundary conditions and the associated Cauchy problem

Jamel Ben Amara (2011)

Colloquium Mathematicae

We study a Sturm-Liouville problem containing a spectral parameter in the boundary conditions. We associate to this problem a self-adjoint operator in a Pontryagin space Π₁. Using this operator-theoretic formulation and analytic methods, we study the asymptotic behavior of the eigenvalues under the variation of a large physical parameter in the boundary conditions. The spectral analysis is applied to investigate the well-posedness and stability of the wave equation of a string.

Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces

Jean-Marc Delort, Jérémie Szeftel (2005)

Journées Équations aux dérivées partielles

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.

Asymptotic analysis of non-self-adjoint Hill operators

Oktay Veliev (2013)

Open Mathematics

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator...

Asymptotics of eigensections on toric varieties

A. Huckleberry, H. Sebert (2013)

Annales de l’institut Fourier

Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities | ϕ n | 2 = | s N | 2 / | | s N | | L 2 2 for eigensections s N Γ ( X , L N ) approaching a semiclassical ray. Here X is a normal compact toric variety and L is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate and Zelditch....

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

Ziyatkhan Aliyev (2010)

Open Mathematics

We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.

On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev (2011)

Open Mathematics

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator V ϕ : f ( x ) 0 ϕ ( x ) f ( t ) d t be defined on L₂[0,1]. We prove that V ϕ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator V ϕ always equals 1.

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