Abstract. We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schrödinger operators with nonnegative matrix-valued potentials, i.e., operators acting on ψ ∈ L 2 (R n , C d ) by the formula H V ψ := −∆ψ + V ψ, 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 subspacevalued mappings.Our third and last result shows that if V is a 2×2 real polynomial potential, then −∆ + V has discrete spectrum if and only if the scalar operator −∆ + λ has discrete spectrum, where λ(x) is the minimal eigenvalue of V (x).
Abstract. We prove new pointwise bounds for weighted Bergman kernels in C n , whenever a coercivity condition is satisfied by the associated weighted Kohn Laplacian on (0, 1)-forms. Our results extend the ones obtained by Christ in [Chr91].Our main idea is to develop a version of Agmon theory (originally introduced in [Agm82] to deal with Schrödinger operators) for weighted Kohn Laplacians on (0, 1)-forms, inspired by the fact that these are unitarily equivalent to certain generalized Schrödinger operators.
, we prove a spectral multiplier theorem of Mihlin-Hörmander type for L, whose smoothness requirement is optimal and independent of V . The assumption on the second derivative V ′′ can actually be weakened to a Hölder-type condition on V ′ . The proof hinges on the spectral analysis of one-dimensional Schrödinger operators, including universal estimates of eigenvalue gaps and matrix coefficients of the potential.
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