Schrödinger Operators with Singular Rank-Two Perturbations and Point Interactions

Schrödinger Operators with Singular Potentials †

We describe classical and recent results on the spectral theory of Schrödinger and Pauli operators with singular electric and magnetic potentials

Operators with Singular Continuous Spectrum, Iv. Hausdorff Dimensions, Rank One Perturbations, and Localization

Although concrete operators with singular continuous spectrum have proliferated recently [7,11,13,17,34,35,37,39], we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and studied recen...

Operators with Singular Continuous Spectrum, Iv. Hausdorff Dimensions, Rank One Perturbations, and Localization

Multilinear Singular Operators with Fractional Rank

We prove bounds for multilinear operators on R given by multipliers which are singular along a k dimensional subspace. The new case of interest is when the rank k/d is not an integer. Connections with the concept of true complexity from Additive Combinatorics are also investigated.

Operators with Singular Continuous Spectrum: Ii. Rank One Operators

For an operator, A, with cyclic vector φ, we study A+ λP where P is the rank one projection onto multiples of φ. If [α,β] ⊂ spec(A) and A has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. §