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

作者：Dall'Ara Gian Maria^*

来源：Journal of Functional Analysis, 2015, 268(12): 3649-3679.

DOI：10.1016/j.jfa.2014.10.007

• Schrodinger operators
• Matrix-valued potentials
• Discrete spectrum

We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schrodinger operators with non-negative matrix-valued potentials, i.e., operators acting on psi is an element of L-2 (R-n,C-d) by the formula
Hv psi := -Delta psi + V psi
where the potential V takes values in the set of non-negative Hermitian d x d matrices.
The first theorem provides a characterization of discreteness of the spectrum when the potential V is in a matrix-valued A(infinity), 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(infinity) class.
The second result is a sufficient condition for discreteness of the spectrum, which allows certain degenerate potentials, i.e., such that det(V) equivalent to 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 x 2 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).

• 出版日期2015-6-15