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....

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.

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.

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...

Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities $|{\varphi }_n{|}^2=|s_N{|}^2/\left|\right|s_N{\left|\right|}_{L^2}^2$ for eigensections $s_N\in \Gamma (X,L^N)$ 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....