Calculation of the Spectrum of a Schrodinger Operator

Abstract

Let T denote a selfadjoint linear operator in a complex Hilbert space H, σ (T) the spectrum of T, σ_d (T) the discrete spectrum of T (that is the set of all isolated eigenvalues of T of finite multiplicity) and σ_e (T) = O (T) − σ_d(T) the essential spectrum of T. The operator T is said to be semibounded (below), if σ (T) has a finite lower bound; in this case we denote by ν(T) and μ (T) the greatest lower bound of σ(T) and σ_e (T) respectively, so that ν(T) ≤ µ (T) holds for every semibounded operator T, and v (T) < µ(T) holds if and only if ν(T) belongs to σ_d(T).

Semibounded operators play an important role in the quantum theory of systems with a finite number of degrees of freedom. The operator T of total energy of a N-particle system in the Schrodinger representation is a selfadjoint semibounded differential operator of second order in the Hilbert space L₂(R^m) (with m = 3N). An eigenvalue λ <µ(T) is the total energy corresponding to a stable state of the system; in particular ν(T) (if this number is smaller than ν(T) ) is the enrgy of the ground state. The number µ(T) is the smallest possible energy corresponding to an unstable state of the system and µ(T) - ν(T) is the minimal energy required to lift the system from its ground state to an unstable state.