The domain of validity of the higher-order Schrödinger equations is analyzed for harmonic-oscillator and Coulomb potentials as typical examples. Then the Cauchy theory for higher-order Hartree–Fock equations with bounded and Coulomb potentials is developed. Finally, the existence of associated ground states for the odd-order equations is proved. This renders these quantum equations relevant for...

The intention of this special session on ”Advances in the numerical solution of nonlinear evolution equations” is to gather mathematicians and theoretical physicists, interconnected through their field of application, the analytical tools, or the numerical methods used. The scope of topics includes but is not limited to Schrödinger type equations, highly oscillatory equations, parabolic problem...

We discuss several aspects of singularities of the solutions of the partial differential equations of Klein–Gordon, Schrödinger and Dirac. In particular we analyze the fold type singularity, of the first and higher orders, and the related characteristic equations. We also consider the field equations as reduction of homogenous equations in higher dimensions, and discuss how singularities of the...

Alternative definitions of the Born approximation and the distorted-wave Born approximation within the framework of the configuration-space Faddeev equations are explored. The most natural definition does not correspond to the Born approximation derived from the Schrödinger equation, even though the exact T-matrices for both formalisms are equivalent. The Schrödinger form is optimal, although i...

A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations

The long-time asymptotics is derived for solutions of the discrete 3dimensional Schrödinger and Klein–Gordon equations. §

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three–manifolds in the case of quadratic nonlinearities. We also discuss the optimali...

The diffusion equation is related to the Schrödinger equation by analytic continuation. The formula E2=p2c2 + m2c4 leads to a relativistic Schrödinger equation, and analytic continuation yields a relativistic diffusion equation that involves fractional calculus. This paper develops stochastic models for relativistic diffusion and equivalent differential equations with no fractional derivatives....