Limit Laws for R-diagonal Variables in a Tracial Probability Space

This paper proves new results on spectral and scattering theory for matrix-valued Schr\"odinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

/pdf/levinson-theorem-for-discrete-schrodinger-operators-on-the-3nlfuawy.pdf

Levinson theorem for discrete Schrodinger operators on the line with matrix potentials having a first moment


 We extend the free convolution of Brown measures of $R$-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.

The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation

Hincin proved that any limit law associated with a triangular array of uniformly infinitesimal random variables is infinitely divisible. Analogous results for the additive and multiplicative free convolution were proved by Bercovici, Belinschi and Pata. We prove an analogous result for the $$\boxplus _{RD}$$
 convolution of measures defined on the positive half-line. This is the convolution arising from the addition of $$*$$
 -free R-diagonal elements of a tracial, noncommutative probability space.

Hinčin’s Theorem for Additive Free Convolutions of Tracial R-Diagonal $$*$$ ∗ -Distributions

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Hinčin’s Theorem for Additive Free Convolutions of Tracial R-Diagonal $$*$$ ∗ -Distributions