  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 Schr\"odinger equation for the hydrogen atom. Then,
generalizing the ideas based on Schr\"odinger 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.
