Spectral and scattering theory for Schrödinger operators

Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane

“…Our present goal is to extend the results of [11] to the two-dimensional case and, at the same time, to provide a basis for further study of the planar nonlinear Helmholtz equation. Without loss of generality, we shall focus on the case λ = −1 and therefore deal with the problem (2) − ∆u − u = Q|u| p−2 u in R 2 .…”

“…As in [11], we shall reformulate (2) as an integral equation, involving the resolvent operator R associated to the inhomogeneous Helmholtz equation…”

“…Let 6 ≤ p < ∞, Q ∈ L ∞ (R 2 ) and consider a solution u of (4). Then, u ∈ W 2,q (R 2 ) for all 6 ≤ q < ∞ and it is a strong solution of (2).…”

On the wave equation with a large rough potential

“…Notice that the results of Alsholm and Schmidt [3] do not apply to the potentials like (1.5) since the singularity |x| −2+ is not L 2 loc ; however, in order to apply, e.g. Proposition 1.3, it is sufficient to assume that…”

Finite total cross-sections in nonrelativistic quantum mechanics

“…For rigorous results on Lippman-Schwinger equations, etc. in v-dimensions see Thoe [49] and Alsholm and Schmidt [3].…”