Schrödingeroperatoren

Schrödinger operators are a class of linear operators that play a crucial role in quantum mechanics and mathematical physics. They are named after Erwin Schrödinger, who introduced them in the context of his wave equation. Schrödinger operators are typically defined on a Hilbert space, which is a complete vector space equipped with an inner product. The most general form of a Schrödinger operator is given by:

H = -Δ + V

where H is the Hamiltonian operator, Δ is the Laplacian operator, and V is a potential function. The Laplacian operator represents the kinetic energy of a particle, while the potential function V represents the potential energy. The Schrödinger equation, which describes the time evolution of a quantum system, is given by:

iħ ∂ψ/∂t = Hψ

where i is the imaginary unit, ħ is the reduced Planck constant, ψ is the wave function, and t is time. Schrödinger operators are self-adjoint if the potential function V is real-valued and satisfies certain conditions. Self-adjointness is a crucial property that ensures the existence of a unique solution to the Schrödinger equation. Schrödinger operators are also used in the study of spectral theory, which deals with the eigenvalues and eigenfunctions of linear operators. The spectral properties of Schrödinger operators are closely related to the scattering theory of quantum mechanical systems.