Discreteness of the spectrum of Schr\"odinger operators with non-negative matrix-valued potentials

We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schr\"odinger operators with non-negative matrix-valued potentials, i.e., operators acting on $\psi\in L^2(\mathbb{R}^n,\mathbb{C}^d)$ by the formula $H_V\psi:=-\Delta\psi+V\psi$, where the potential $V$ takes values in the set of non-negative Hermitian $d\times d$ matrices. The first theorem provides a characterization of discreteness of the spectrum when the potential $V$ is in a matrix-valued $A_\infty$ 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 $A_\infty$ class. The second result is a sufficient condition for discreteness of the spectrum, which allows certain degenerate potentials, i.e., such that $\det(V)\equiv0$. 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 $2\times2$ real polynomial potential, then $-\Delta+V$ has discrete spectrum if and only if the scalar operator $-\Delta+\lambda$ has discrete spectrum, where $\lambda(x)$ is the minimal eigenvalue of $V(x)$.