One-dimensional Schrödinger Operators and Nonlinear Fourier Analysis

The potential V is a real valued, measurable, locally integrable function. H is then a self-adjoint operator whose domain is an appropriate Sobolev space. Considered as an operator on L2([0,∞)), H is likewise self-adjoint if an appropriate boundary condition at zero, e.g. Dirichlet or Neumann, is specified. The quantum mechanical interpretation is that H0 = − d2 dx2 governs the behavior of a free electron, while HV describes one electron interacting with an external electrical field. The evolution of a particle with initial state ψ0(x) ∈ L2 is described by the time-dependent Schrödinger equation iψt(x, t) = Hψ(x, t), with initial condition ψ(x, 0) = ψ0(x), whose solution is denoted by e−iHtψ0. (1.1) is perhaps the simplest quantum mechanical model describing unbounded motion — though it is by no means simple. The free case V = 0 can be analyzed directly using the Fourier transform