Two-parameter nonresonance condition for the existence of fourth-order boundary value problems
Abstract In this paper, we discuss the existence of the fourth-order boundary value problem { u ( 4 ) = f ( t , u , u ″ ) , 0 t 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : [ 0 , 1 ] × R × R → R is continuous, and partly solve the Del Pino and Manasevich's conjecture on the nonresonance condition involving the two-parameter linear eigenvalue problem. We also present a two-parameter nonresonance condition described by circle.
Abstract: We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schrödinger operators with sparse potentials. We find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model. We also present an improved version of known results on the spectrum of such operators.