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We start from applying the general idea of spectral projection (suggested byOlshanski and Borodin and advocated by Tao) to the complex Wishart model.Combining the ideas of spectral projection with the insights from quantummechanics we derive in an effortless way all spectral properties of the complexWishart model: first, the Marcenko-Pastur distribution interpreted as aBohr-Sommerfeld quantization condition for the hydrogen atom; second, hard(Bessel), soft (Airy) and bulk (sine) microscopic kernels from properlyrescaled radial Schr\"odinger equation for the hydrogen atom. Then,generalizing the ideas based on Schr\"odinger equation to the case whenHamiltonian is non-Hermitian, we propose an analogous construction for spectralprojections of universal kernels built from bi-orthogonal ensembles. Inparticular, we demonstrate that the Narain transform is a natural extension ofthe Hankel transform for the products of Wishart matrices, yielding an explicitform of the universal kernel at the hard edge. We also show, how the change ofvariables of the {\it rescaled} kernel allows to make the link to universalkernel of Muttalib-Borodin ensemble. The proposed construction offers a simplealternative to standard methods of derivation of microscopic kernels, basede.g. on Plancherel Rotach limit of orthogonal polynomials of asymptotics in theRiemann-Hilbert problem. Finally, we speculate, that a suitable extension ofthe Bochner theorem for Sturm-Liouville operators may provide an additionalinsight into the classification of microscopic universality classes in randommatrix theory.

Narain transform for spectral deformations of random matrix models