Narain transform for spectral deformations of random matrix models

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.

form a bi-orthogonal basis, it is possible to infer the microscopic limit of the kernels using the spectral projection method. We demonstrate this on two examples -singular values of products of Gaussian matrices [22] and the Muttalib-Borodin ensemble [23,24]. In both cases the Narain transform [25][26][27] allows one to recover the Meijer-G hard edge universality, generalizing Bessel kernel. Again, the spectral projection translates to the truncation of the phase space of the associated transform. Section 4 concludes the paper. In appendix A, we show an alternative mapping of the Wishart ensemble to the 2-dimensional hydrogen atom problem [28]. In appendix B we recover the Marchenko-Pastur distribution from the WKB approximation. In appendix C we recall some properties of the Meijer-G functions.

II. SPECTRAL PROJECTIONS FROM HYDROGEN ATOM PROBLEM
A. Complex Wishart ensemble Let us consider Hermitian matrix M = XX † , where X is the complex N × T matrix with entries given by the probability density function P (X)dX = Z −1 N T e − 1 σ 2 N,T α,j |Xα,j | 2 N,T αj d X αj d X αj . Here Z −1 N T provides the normalization and σ 2 is the variance of the complex Gaussian distribution, which we set to 1, to simplify the expressions. This defines complex Wishart matrix [29]. Switching to eigenvalues, we arrive, using standard methods [30], at their joint probability density with α = T − N , and the Vandermonde determinant (last term) is the price for switching from elements of X to eigenvalues λ i of matrix M . Standard orthogonal polynomials trick [12] allows one to rewrite the probability distribution as with the correlation kernel where ψ l (λ) = e −λ/2 λ α/2 P l (λ) and P l are monic polynomials. This form already suggests links to quantum mechanics. The first equality in (2) represents the joint probability of eigenvalues as the square of the Slater determinant, therefore can be interpreted as the quantum probability density of non-interacting spinless fermions (see [31] for a review). This also explains why the eigenvalue density is expressed solely in terms of a two-point function (second expression on the r.h.s. of (2)). Next, we see that the most natural choice of polynomials is dictated by the weight w α (λ) = λ α e −λ . Such polynomials, orthonormal on the positive part of the real axis, are the associated Laguerre polynomials and appear in the radial part of the Schrödinger equation. Indeed, upon standard separation of variables in the wavefunction, ϕ( r) = R(r)Y m l (θ, ψ), it reads where y(r) = rR(r). Switching to dimensionless variable x = r , where ( /2) 2 = −2µE/ 2 , putting 2µ = 1 and all other physical constants to 1, we recover [32] where y = y l n = e −x/2 x (k+1)/2 L k j (x). Here k = 2l + 1 and the principal quantum number is related to the order of Laguerre polynomial as n = j + l + 1. Note, that = 1/n, or, equivalently, E n = −1/4n 2 , since in our units Bohr's radius equals to 2. To map this random matrix problem to the hydrogen atom we associate ψ l (λ) = √ xy(x). This completes the dictionary between hydrogen atom problem and the Wishart kernel. In Appendix A we also present a mapping into 2D hydrogen atom with 1/r potential [28], in which the relation between eigenfunctions of the radial part of the Schrödinger equation and ψ is even more explicit.
The Schrödinger equation for ψ expressed in terms of the parameters of the Wishart ensemble reads Finally, let us note that in the bra-ket notations the kernel readsK N = N −1 k=0 |ψ k ψ k | thus it is the operator projecting onto the set of N lowest eigenstates. Indeed, due to the orthonormality of eigenfunctionsK 2 N =K N . Last equation, when calculated in coordinate representation, yields well-known reproducing property x|K N |y ≡ K N (x, y) = K N (x, z)K N (z, y)dz.

B. Macroscopic density from the semiclassical approximation
To have the finite support of the spectral density in the large N limit, we rescale x → T x. Upon this scaling and identifying momentum 1 as p = − i T d dx (in analogy to ↔ 1/T ) in the limit N, T → ∞ with c = N/T fixed we obtain the Schrödeinger equation (p 2 + V ef f )ψ = − 1 4 ψ with the effective potential Thanks to the fermionic analogy, the mean spectral density is the same as the probability density of non-interacting fermions. The latter is obtained by integrating the Wigner function over the set of momenta. In our case the Wigner function is constant on the region of the phase space p 2 + V (x) ≤ 1 4 and zero outside [33] (see Fig. 1), therefore the density of eigenvalues is proportional to the momentum and the Bohr-Sommerfeld quantization condition on the RMT side corresponds to the normalization of density This allows us to obtain the density of eigenvalues Here r ± = (1± √ c) 2 are classical turning points in WKB approximation. In Appendix B we provide another derivation of this result based on the explicit WKB analysis of (6).
We have therefore obtained Marchenko-Pastur distribution as an exact, semiclassical limit of the quantum mechanical hydrogen atom problem. It is intriguing to speculate why such link has not been exploited (to the authors knowledge) in the literature. Perhaps the reason is that Bohr-Sommerfeld quantization condition does not reproduce correctly the ground state of the hydrogen atom, and not even the Bohr quantization condition [34]. It is amusing to notice, that if one replaced l(l + 1) by (l + 1/2) 2 in the numerator of the centrifugal potential, this would be the case and B-S approximation would lead to the exact result for the hydrogen spectra [35]. Of course, in the large l limit it does not matter which of the equations (5) or (6) we use, however, at the microscopic level, additional square root in Laguerre function for the Wishart will play the crucial role in getting the proper scaling of the hard edge.
We complete this part with the observation, that in the case of harmonic oscillator, similar construction is ambiguities free, since Bohr-Sommerfeld quantization condition yields exact spectrum. The Wigner semicircle, or rather semi-ellipse, is just the similar projection of the ellipse p 2 + x 2 /4 = 1 onto the x axis in the phase space. The Bohr-Sommerfeld quantization condition just reads ρ(x)dx = 1, where ρ = 1 2π √ 4 − x 2 (in units where 2µ = 1) [19]. Again, the rigid argument comes from the fact, that the Wigner function for harmonic oscillator is explicitly known [33], and yields a direct relation between the momenta and positions at the semi-classical level.
1 For this analogy it is even better to take the 2D radial momentum pr = i T d dx + 1 x , but this eventually leads to the same result in large T limit. Correlations of eigenvalues probed on the scale of the typical separation between them are independent on the probability density function of matrix elements. They fall into several classes, depending on the point x 0 of the spectrum at which their behavior is probed. The shape of the spectral density, in turn, determines the microscopic scale s by demanding that in the interval [x 0 , x 0 + s] one expects one eigenvalue to occur. Looking at the form of the Marchenko-Pastur distribution (see Fig. 1), we immediately identify three distinct regions corresponding to microscopic scalings.
A Hard edge. In the limit when N, T → ∞ but α = T − N remains fixed (c → 1), the turning point r − approaches zero, and the eigenvalue density near this point behaves like 1/ √ x. Asking how many out of original N eigenvalues will appear in a narrow bin of size s around zero, we get Demanding that n hard ∼ 1, we set the proper microscopic scale to s ∼ N −2 .
B Bulk. Between the endpoints, at some x 0 , when counting the number of eigenvalues in a narrow interval of length s, one can approximate the density as locally constant ρ(x 0 ). This leads to which implies that the bulk microscopic scale is s ∼ 1 N ρ(x0) .
C Soft edge. When c = 1, the macroscopic spectral density around both turning points vanishes like |r ± − x|.
Counting the eigenvalues close to the edge, leads to thus the edge microscopic scale is set to s ∼ N −2/3 .
Following the generic arguments by Borodin and Olshanski [24] and inspired by Tao [19] presentation for the Gaussian Unitary Ensemble, we will now obtain the microscopic, universal kernels for the complex Wishart ensemble. We remark that this case belongs to the generic class of Sturm-Liouville operators, considered recently by Bornemann [20]. However, in this note, we attempt to use the insights from quantum mechanics rather than abstract mathematics.
The complete set of eigenfunctions provides a resolution of identity 1 = ∞ k=0 |ψ k ψ k |. The random matrix kernel is obtained by truncating this sum to first N eigenstates and is therefore a projection. The range of this projection can be formally written asĤ ≤ E N −1 or, using the explicit form of (6), as The microscopic scalings provide further deformations of the projection range, which in the large N, T limit gives rise to the universal microscopic kernels, which we work out in details beneath.
A Bessel kernel. Using the hard edge scaling x/T → sN −2 , and performing the large N limit (note that k ∼ N ), we obtain the equation Changing variables z = 2 √ s converts the above bound onto more familiar form where on the l.h.s. we recognize Bessel operator, appearing in quantum mechanical problems with polar angle symmetry. To see the deformation caused by microscopic scaling at the hard edge, we invoke the Hankel transform and its inverse Since the Hankel transform of the Bessel operator reads [36], the spectral deformation in dual variable t (note that t cannot be negative) reads simply Hankel transform and its inverse give a representation of the identity operator The deformation condition (19) restricts the range of the parameter t and therefore turns the above identity operator into the projection Changing variables once more as z = √ s and introducing t = √ y and t = √ x, we rewrite the above as so the kernel, understood as a projection, reads where on the r.h.s. we presented the more familiar form of the kernel based on the Lommel integral and primes denote differentiation with respect to the argument. Hard edge scaling deforms the upper half plane in s variable onto the strip between the parallel lines s = 0 and s = 1.
B Sine kernel. Combining the rescaling needed for the finite support and the microsopic scaling we define the new variable s as x/T = x 0 + s N ρ(x0) . Upon taking the large N, T limit, the bound (14) in this new variable reads Using the explicit form of the Marcenko-Pastur density (10), the above bound is simplified to On the l.h.s. we recognize the Schrödinger operator for a free particle, therefore the natural procedure for resolving this bound is to use plane waves, i.e. move to the momentum space via the Fourier transformation: The spectral deformation in the momentum space reads therefore Combination of Fourier transforms provides a representation of an identity operator The deformation (27) projects the above identity operator onto Microscopic scaling in the bulk restricts the range of momenta to − 1 2 ≤ q ≤ 1 2 . Calculation of the integral in square brackets yields the projection in the position basis, which is the sine kernel C Airy kernel. At the soft edge we introduce the scaling variable s as x/T = r ± ± s √ c(r±N ) 2/3 . In the large N and T limit generic bound (14) is transformed into On the l.h.s. we recognize the Schrödinger operator with the linear potential. This condition in the positionmomentum space (s, q) restricts the range of integration to the parabola 4π 2 q 2 + s ≤ 0, which is not well suited for reading out the limiting kernel. To circumvent this problem, Tao introduced a similarity transformation in the momentum space [19]. Alternatively, since we identify the differential Airy operator in (31), we can directly resort to the Airy transform [37] and its inverse Using the Airy transform for the operator bound (31), and the fact that Airy function fulfills Ai (x) = xAi(x) we express the spectral deformation in dual variable t simply as Combining both Airy transforms we obtain the identity operator The deformation condition (34) turns the above identity operator into a projection so the kernel reads where on the r.h.s. we presented the more familiar form of the Airy kernel based on relation d dz We summarize this section in Fig. 2, by plotting the domain of the projection operator before and after the pertinent microscopic scalings.

A. Bochner theorem
In a short paper written in 1929 [21], Salomon Bochner has noticed, that if an infinite sequence of polynomials P n (x) satisfies an eigenequation to the second order self-adjoint differential operator p(x)P n (x) + q(x)P n (x) + r(x)P n (x) = λ n P n (x), then p(x), q(x), r(x) must be polynomials of degree 2, 1, and 0, respectively. If additionally polynomials are orthogonal and their support is real, the only solutions are polynomials of Jacobi, Laguerre or Hermite type.
These orthogonal polynomials are associated with classical random matrix ensembles: Gaussian Unitary Ensemble (Hermite), Laguerre Unitary Ensemble (also known as complex Wishart) and Jacobi Unitary Ensemble (complex MANOVA). Recently, Bornemann [20], using the spectral projection method, classified the scaling limits of determinantal processes arising from Sturm-Liouville operators 2 . They do not lead to any new universality class than what is known for Wishart ensemble.
On the other hand, it is known that there are other scaling limits of the kernel in unitary matrix models. These are related to different vanishing of the spectral density at the edge or at the closing gap in the bulk, see [38] for a review. This raises a question whether such limits can be related to spectral projections. To avoid limitations of the Bochner theorem one may look at the class of Hamiltonians with higher powers of momentum operator. Self-adjointness constrains these Hamiltonians to have only even powers of momentum and Krall [39] provided complete classification of orthogonal polynomials to the problem with quartic momenta. However, classification of higher order Bohner-Krall polynomial systems remains still an open problem. While there are some particular examples of sixth [40] and eighth order systems [41], the corresponding weights are only modifications of classical Gaussian, Laguerre and Jacobi weights by Heaviside theta and Dirac delta functions (see [42] for review), which makes them uninteresting from the random matrix theory perspective.

B. Non-Hermitian 'Hamiltonians'
Relaxing the self-adjointness condition admits a broader class of operators. Then one deals with non-Hermitian 'Hamiltonian' and two eigenequations to each eigenvalue: Here |P k and Q k | are called left and right eigenfunctions, in the analogy to non-Hermitian matrices. They are no longer orthogonal, but bi-orthogonal The adjoint Hamiltonian H † is defined in a standard way Now, because of biorthogonality the two sets of eigenfunctions cannot be both polynomials, enlarging the space of possible solutions. Preiser [43] considered a higher order generalization of Bochner-Krall theorem with restriction that P k (x) are polynomials in x, while Q k (x) are polynomials in x m multiplied by some weight. He found that for the Hamiltonian with third derivative there exists only one such set, which was discovered earlier by Spencer and Fano [44].
Biorthogonal structures appear in multi matrix models, where the correlation kernel is built from biorthogonal functions P k and Q k Biorthogonality ensures that the kernel is a projection. It is therefore tempting to ask whether such kernels ale built of eigenfunctions of a certain 'Hamiltonian' and if so, is it possible to obtain the microscopic scaling using spectral projections?

C. Singular values of products of complex Gaussian matrices
Let us consider X k a rectangular matrices of size (N + ν k−1 ) × (N + ν k ) with complex Gaussian iid entries of zero mean and unit variance. Without loss of generality we assume ν 0 = 0 and ν k > 0 for k > 0. The squared singular values of the product Y M = X 1 X 2 . . . X M form a biorthogonal ensemble with the correlation kernel (43). The biorthogonal functions are explicitly given by [22] P k (x) = G 1,0 Here G stands for the Meijer-G function (see Appendix C). From the differential equation (85) we deduce that polynomials P k satisfy the eigenproblem (H M P k = λ k P k with λ k = k) of the following differential operator (Hamiltonian) With the help of the identity d dx The explicit form (45) and the differential equation (85) prove that Q k satisfy the eigenequation H † M Q k = kQ k . Therefore P k and Q k are left and right eigenfunctions of a non-Hermitian Hamiltonian.
To probe the microscopic scaling at the edge, we rescale x = z N , which turns the eigenequation for H M into As k is always smaller than N , in the large N limit we obtain the condition In order to continue the analogy to the deformation of the phase-space of Hermitian operators, we have to find the suitable transformation, which will convert the operator-valued inequality into an algebraic constraint.

D. The Narain transform
In a series of papers [25][26][27] Narain introduced a broad class of asymmetric transforms, which include many known classical transforms. The Narain transform and its inverse are defined as where the integral kernels read k(s, y) = 2γx γ−1/2 G m,p p+q,m+n a 1 , . . . , a p , b 1 , . . . , b q c 1 , . . . , c m , d 1 , . . . , d n (sy) 2γ , h(y, s) = 2γx γ−1/2 G n,q p+q,m+n If f has a discontinuity at x, then In the space of the dual variable s, the operator ∆ (M +1) ν acts by multiplying by s, as can be easily proven, using identities from Appendix C. The hard edge scaling of the kernel reduces therefore the range of parameter s to s ≤ 1. Alike in the Hermitian case, the identity operator is deformed to h(x, s)k(s, y)ds g(y)dy.
We obtain this way the limiting form of the microscopic kernel at the hard edge Note that G 1,0 This form slightly differs from (23). To understand this discrepancy, let us note that biorthogonal functions can be rescaled as without altering their biorthogonality. Under such a rescaling kernel is transformed K(x, y) → 1 f (x) K(x, y)f (y). In our case it is sufficient to take f (x) = x ν/2 and further rescale (x, y) → 1 4 (x, y). The Narain transform can therefore be viewed as a generalization of the Hankel transform at the hard edge. As another example we consider the joint pdf of eigenvalues introduced by Muttalib [23] and elaborated later by Borodin [13] P (λ 1 , . . . , with α > −1, and θ ≥ 0. Eigenvalues form a determinantal point process with a correlation kernel given by the bi-orthogonal functions (43). Here P k is a polynomial of order k, while Q k is a polynomial in x θ multiplied by the Laguerre weight. For integer values of θ Konhauser provides the explicit form of Q [45, eq. (5)] while Carlitz gives the explicit form of polynomials [46, eq. (9)] function in terms of Laguerre functions 3 . Two years later (1928) Wishart introduced his ensemble in multivariate statistics, as a generalization of the χ 2 ensemble [29]. The original paper deals with the real random variables, but his ideas were later generalized to complex variables [48]. A year later (1929), Bochner has proven his theorem [21] for Sturm-Liouville operators, without any direct references to Schrödinger equation. At that time spectral properties of random matrices were not considered at all. Laguerre polynomials appeared explicitly in random matrix theory only after the Mehta and Gaudin used the orthogonal polynomial trick to disentangle the Van der Monde determinant [49]. This technique has also paved the way for classical universal kernels. However, the link to the uniqueness of the determinantal triality of soft, edge and bulk microscopic universalities of Sturm-Liouville operators have been cleared out only recently [20].
In 1967 Marchenko and Pastur derived the spectral density for the Wishart ensemble [50]. Interestingly, they used the ideas borrowed from hydrodynamics [51]. The fact that the Marchenko-Pastur distribution can be interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom was not, to the best of or knowledge, explicitly stated in the literature. Such a link is intuitively expected, because the Dyson electrostatic analogy in the limit of large matrices allows one to solve the random matrix model using the saddle point approximation -the same mathematical method which gives the WKB approximation in Quantum Mechanics, with the correspondence 1 ↔ N . The relation between momentum and the spectral density requires, however, additional knowledge of the properties of Wigner functions, as we point out in this paper.
With introducing non-trivial initial conditions for Dyson Brownian motion, new universality classes emerged in random matrix theory. In the 90's of the previous century, collision of soft edges in GUE led Brezin and Hikami [52] to the Pearcey kernel. In a similar collision of chiral fronts at the hard edge of the chiral random matrix model one of the authors found the Bessoid kernel universality [53]. While still determinantal [54], such models break rotational invariance, and require non-standard tools. Later it was discovered that such ensembles can be solved by polynomials that are orthogonal to more than one weight [55].
The bi-orthogonality method of Muttalib and Borodin opened a new way for treating a broader class of random matrix models, to which the orthogonal polynomials method does not apply. Historically, it is again puzzling that biorthogonality was not linked to random matrices earlier. Already in 1951, Fano and Spencer [44] studying propagation of the X-rays through the matter, have introduced bi-orthogonal Laguerre polynomials. These ideas were further developed in mathematics by Preiser [43] and Konhauser [45]. In particular, Preiser's construction corresponds exactly to the case of Muttalib-Borodin ensemble. This is precisely that intertwining of ideas and the lack of explicit ideas, which prompted us to reexamine Bochner theorem. Rapid progress in random matrix theory in last three decades has brought plethora of new microscopic universality classes. Despite so many examples of microscopic universalities, there is lack of their systematic classification. The spectral projection method adopted to non-Hermitian Hamiltonians and possible generalizations of Bochner theorem for higher order differential operators 4 offer a new perspective on this problem. Certainly, this program is a challenging mathematical problem, which we do not attempt to solve.
This work raises a series of fundamental questions related to possible generalization of Bochner theorem in the context of random matrix theory. Is it possible to reframe all universality classes in this language? Will this classification be predictive for constructing new types of random matrix models? Can one infer the microscopic kernels of non-Hermitian ensembles from a 'complex version' of Bochner theorem? We leave these questions open but we think that the presented method has also pedagogical value. It offers an easy and intuitive way to recover not only the the classical universality classes, but also more involving Meijer-G functions. Combining physical intuition with mathematics may provide in such a way new insights even in standard problems.
The turning points are x ± (t) = 1 − c + 2ct ± 2 ct(1 + ct − c). (79) The spectral density is therefore given by where χ A is equal to 1 when A is true and 0 for A false. Setting C = 1 π normalizes the density.