Discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials

Abstract

We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials, i.e., operators acting on ψ∈L2(Rn,Cd) by the formulaHVψ:=−Δψ+Vψ, where the potential V takes values in the set of non-negative Hermitian d×d matrices.The first theorem provides a characterization of discreteness of the spectrum when the potential V is in a matrix-valued A∞ class, thus extending a known result in the scalar case (d=1). We also discuss a subtlety in the definition of the appropriate matrix-valued A∞ class.The second result is a sufficient condition for discreteness of the spectrum, which allows certain degenerate potentials, i.e., such that det⁡(V)≡0. To formulate the condition, we introduce a notion of oscillation for subspace-valued mappings.Our third and last result shows that if V is a 2×2 real polynomial potential, then −Δ+V has discrete spectrum if and only if the scalar operator −Δ+λ has discrete spectrum, where λ(x) is the minimal eigenvalue of V(x).