Schrödinger Operator

A Schrödinger operator is a differential operator consisting of a kinetic-energy term and a multiplication operator given by a potential. For a particle of mass moving in , it has the form

(1)

on a suitable dense domain in the Hilbert space . In mathematical treatments, units are often chosen so that the same operator is written .

The stationary Schrödinger equation is the spectral problem

(2)

The domain and boundary conditions are part of the definition of . Conditions on the potential are used to ensure that is self-adjoint. Self-adjointness implies that its spectrum is real and that it generates unitary time evolution. Bound states correspond to square-integrable eigenfunctions, while scattering states are associated with the continuous spectrum.

The free-particle operator has . Other important examples include the harmonic oscillator, the quartic oscillator, Coulomb potentials, periodic potentials, and random potentials.

Exact eigenfunctions of singular radial operators can sometimes be found by matching coefficients. For example, consider

(3)

where , , , and is a nonnegative integer. Put . The function

(4)

satisfies with iff

(5)

Under this constraint, is positive and square-integrable on and represents the ground state of the self-adjoint radial operator (Dong et al. 1998).

More general trial functions multiply by a finite Laurent polynomial in . Substitution gives algebraic conditions on its coefficients, the eigenvalue, and the potential parameters. Trott (2006) uses a Gröbner basis to eliminate the unknowns for the trial factor . The resulting conditions apply only at the constrained parameter values. Quasi-exact solvability describes the related reduction obtained from an explicitly known finite-dimensional invariant subspace.