Narain transform for spectral deformations of random matrix models

Abstract

We start from applying the general idea of spectral projection (suggested by Olshanski and Borodin and advocated by Tao) to the complex Wishart model. Combining the ideas of spectral projection with the insights from quantum mechanics we derive in an effortless way all spectral properties of the complex Wishart model: first, the Marcenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) microscopic kernels from properly rescaled radial Schrödinger equation for the hydrogen atom. Then, generalizing the ideas based on Schrödinger equation to the case when Hamiltonian is non-Hermitian, we propose an analogous construction for spectral projections of universal kernels built from bi-orthogonal ensembles. In particular, we demonstrate that the Narain transform is a natural extension of the Hankel transform for the products of Wishart matrices, yielding an explicit form of the universal kernel at the hard edge. We also show, how the change of variables of the {\it rescaled} kernel allows to make the link to universal kernel of Muttalib-Borodin ensemble. The proposed construction offers a simple alternative to standard methods of derivation of microscopic kernels, based e.g. on Plancherel Rotach limit of orthogonal polynomials of asymptotics in the Riemann-Hilbert problem. Finally, we speculate, that a suitable extension of the Bochner theorem for Sturm-Liouville operators may provide an additional insight into the classification of microscopic universality classes in random matrix theory.