aa r X i v : . [ m a t h - ph ] J un Skew-orthogonal polynomials: the quartic case.
Saugata Ghosh ∗ F 253, Sushant Lok, Part II, Sector 57, Gurgaon, India. (Dated: November 2, 2018)We present an iterative technique to obtain skew-orthogonal polynomials with quartic weight,arising in the study of symplectic ensembles of random matrices.
PACS numbers: 02.30.Gp, 05.45.Mt
1. INTRODUCTION
After the publication of [1] and [2], late Prof. M. L. Mehta was not particularly happy with the formal nature ofthe presentation. Following his suggestion, this paper attempts to give an explicit workout of the skew-orthogonalpolynomials (SOP) corresponding to the quartic potential.We study SOP arising in the study of symplectic ensembles of random matrices. The corresponding weight functionis w ( x ) = exp[ − V ( x )], where V ( x ) = x αx , α ∈ R , (1.1)is the quartic potential. In this paper, we outline an iterative technique to develop these polynomials which can beused to obtain the level-density and 2-point function for the symplectic ensembles of random matrices. Here, weemphasize that this method can be easily extended to all potentials of the form V ( x ) = d X k =1 u k x k k , u d = 1 . (1.2)Without loss of generality, we will be dealing with monic SOP of the form φ n ( x ) = w ( x ) n X k =0 c ( n ) k x k , c ( n ) n = 1 , (1.3)and define ψ n ( x ) := ddx φ n ( x ) . (1.4)They satisfy the skew-orthonormalization relation Z R φ n ( x ) ψ m ( x ) dx = g n Z nm , (1.5)where Z = (cid:18) − (cid:19) ∔ . . . ∔ , (1.6)is an anti-symmetric block-diagonal matrix with Z = − g n is the normalization constant with the property g n = g n +1 . (1.7)Here ψ n ( x ) is a polynomial of order n + 2 d − −
1. We have dropped the superscript β (as in[1]) and [2]) since we will only be interested in SOP arising in the study of symplectic ensembles of random matrices. ∗ Electronic address: [email protected]
As explained in Ref.[1] and [2], we expand ψ ( x ) in terms of φ ( x ) and write xψ n ( x ) = k X m = j R nm φ m ( x ) , (1.8)where from Eqs.(1.2, 1.3, 1.4), one can say that k = n + 2 d . Furthermore, using the integral R R xψ n ( x ) ψ m ( x ) dx , wecan show that the matrix R satisfy anti-self dual relation R = ZR t Z ≡ − R D . (1.9)Summing up these results, we can write for any polynomial weight xψ n ( x ) = R n, n +2 d φ n +2 d ( x ) + . . . + R n, n − d φ n − d ( x ) , (1.10) xψ n +1 ( x ) = R n +1 , n +2 d +1 φ n +2 d +1 ( x ) + . . . + R n +1 , n − d φ n − d ( x ) , (1.11)where φ − m = 0, m being a positive integer. Our choice of the potential ( u d = 1) ensures that R m,m +2 d = − , m ≥ . (1.12)Also from Eq.(1.9), we can show that R n, n − d = − R n − d +1 , n +1 = 1 , n ≥ d, = 0 , n < d,R n +1 , n − d +1 = − R n − d, n = 1 , n ≥ d, = 0 , n < d. (1.13)Following this brief recapitulation of [1] and [2], we will outline our plan of action.1. We will calculate the SOP’s φ n ( x ) for 0 ≤ n < d (for the particular case d = 2) using generalized Gram-Schmidtorthogonalization.2. Using these polynomials, we will use Eqs.(1.10) and (1.11) to calculate the higher order polynomials recursively.The coefficients will be expressed in terms of integrals of the form R R x n exp[ − V ( x )] dx .3. We show that these coefficients themselves satisfy a set of difference equations.4. Using these difference equations, we also obtain the normalization constants.5. Finally, we talk briefly about the zeros of these polynomials.
2. THE GRAM-SCHMIDT TECHNIQUE
Putting n = 0 in Eqs.(1.10) and (1.11) (keeping in mind Eq.(1.13)), we can see that it is useless, unless we haveinformation about the SOP’s φ m for 2 d > m ≥
0. To overcome this problem, we will obtain these 2 d polynomials( φ ( x ) , . . . , φ d − ( x )) using the Gram-Schmidt technique for SOP. From here onward, we will focus our attention onthe specific weight function defined in Eq.(1.1), although the method outlined can be extended to any d . For d = 2,the first 4 monic polynomials can be written as φ ( x ) = w ( x ) , φ ( x ) = xw ( x ) , (2.1) φ ( x ) = ( x + c (2)0 ) w ( x ) , φ ( x ) = ( x + c (3)1 x ) w ( x ) . (2.2)Correspondingly ψ ( x ) = − V ′ ( x ) w ( x ) , ψ ( x ) = (1 − xV ′ ( x )) w ( x ) , (2.3) ψ ( x ) = [2 x − ( x + c (2)0 ) V ′ ( x )] w ( x ) , ψ ( x ) = [3 x + c (3)1 − V ′ ( x )( x + c (3)1 x )] w ( x ) . (2.4)Using Z ∞−∞ φ ( x ) ψ ( x ) dx = 0 , (2.5)we get C (2)0 = − R ∞−∞ x (1 − xV ′ ( x )) w ( x ) dx R ∞−∞ (1 − xV ′ ( x )) w ( x ) dx . (2.6)Similarly, using Z ∞−∞ φ ( x ) ψ ( x ) dx = 0 , (2.7)we get C (3)1 = − R ∞−∞ x V ′ ( x ) w ( x ) dx R ∞−∞ xV ′ ( x ) w ( x ) dx . (2.8)We also get g = g = Z ∞−∞ φ ( x ) ψ ( x ) dx, g = g = Z ∞−∞ φ ( x ) ψ ( x ) dx. (2.9)
3. RECURSION RELATION
Now, let us look at Eqs.(1.10) and (1.11). Since V ( x ) = V ( − x ), we have ψ n ( x ) = − ψ n ( − x ) , ψ n +1 ( x ) = ψ n +1 ( − x ) . (3.1)Thus the odd (even) terms will be absent in Eqs.(1.10) ((1.11)) since R n, n +2 k +1 = − g n +2 k Z ∞−∞ xψ n ( x ) ψ n +2 k ( x ) dx = 0 , and R n +1 , n +2 k = 1 g n +2 k Z ∞−∞ xψ n +1 ( x ) ψ n +2 k +1 ( x ) dx = 0 . (3.2)Using these results and Eq.(1.12), we can rewrite Eqs. (1.10) and (1.11) for n = 0 and d = 2 as xψ ( x ) = − φ ( x ) + R , φ ( x ) + R , φ ( x ) , (3.3) xψ ( x ) = − φ ( x ) + R , φ ( x ) + R , φ ( x ) . (3.4)Also, using the skew-orthogonality property, we get R , = 1 g Z ∞−∞ xψ ( x ) ψ ( x ) dx, (3.5) R , = 1 g Z ∞−∞ xψ ( x ) ψ ( x ) dx. (3.6)Similarly, R , = − g Z ∞−∞ xψ ( x ) ψ ( x ) dx = − R , , (3.7) R , = − g Z ∞−∞ xψ ( x ) ψ ( x ) dx. (3.8)Here, we note that R , = − R , can be obtained directly from Eq.(1.9). Once the coefficients are known, one cancalculate φ ( x ) and φ ( x ) using Eqs. (3.3) and (3.4). Using (1.4), we can calculate ψ ( x ) and ψ ( x ). We can alsocalculate g = g = Z ∞−∞ φ ( x ) ψ ( x ) dx. (3.9)We will now calculate φ ( x ) and φ ( x ) using xψ ( x ) = − φ ( x ) + R , φ ( x ) + R , φ ( x ) + R , φ ( x ) , (3.10) xψ ( x ) = − φ ( x ) + R , φ ( x ) + R , φ ( x ) + R , φ ( x ) . (3.11)Again, using Eq.(1.9), we have R , = − R , , R , = − R , , R , = − R , . (3.12) R , and R , has already been calculated in (3.6) and (3.8) respectively. Also R , = 1 g Z ∞−∞ xψ ( x ) ψ ( x ) dx, (3.13) R , = − g Z ∞−∞ xψ ( x ) ψ ( x ) dx, (3.14)and R , = − R , = 1 g Z ∞−∞ xψ ( x ) ψ ( x ) dx. (3.15)With these, we can calculate φ ( x ), φ ( x ) and correspondingly ψ ( x ) and ψ ( x ) and have g = g = Z ∞−∞ φ ( x ) ψ ( x ) dx. (3.16)Following the same technique, we can obtain the polynomials for all n . For n ≥ d = 2, we may rewrite Eqs.(1.10)and (1.11) as xψ n ( x ) = X m = − R n, n +2 m φ n +2 m ( x ) , xψ n +1 ( x ) = X m = − R n +1 , n +2 m +1 φ n +2 m +1 ( x ) . (3.17)From Eq.(1.12), we have R n, n +4 = R n +1 , n +5 = − . (3.18)Also from Eq.(1.13), we have R n, n − = − R n − , n +1 = 1 , R n +1 , n − = − R n − , n = 1 . (3.19) R n, n − (for the even case) and R n +1 , n − (for the odd case) is already known since R n, n − = − R n − , n +1 , R n +1 , n − = − R n − , n . (3.20)Finally, we are left with terms of the form R k,k +2 and R k,k (for both k odd and even). They are given by R n, n +2 k = 1 g k +2 n Z ∞−∞ xψ n ( x ) ψ n +2 k +1 ( x ) dx, k = 0 , R n +1 , n +2 k +1 = − g k +2 n Z ∞−∞ xψ n +1 ( x ) ψ n +2 k ( x ) dx, k = 0 , . (3.22)Here, we note that by expanding xψ n (or xψ n +1 ), we can evaluate R n, n +2 k (or R n +1 , n +2 k +1 ) analytically. Butthat will involve terms like c (2 n +4)2 n +2 (or c (2 n +5)2 n +3 ) and thereby cannot be used to obtain the polynomials recursively. Wemight recall that for monic orthogonal polynomials, this problem does not exist [5]. However it is possible to evaluateEqs.(3.21) and (3.22) numerically, although it may be a tiresome process.
4. THE NORMALIZATION CONSTANT
So far, we have outlined a formalism to obtain the polynomials recursively, where the recursion coefficients areexpressed in terms of certain integrals which needs to be evaluated at every iteration. It is not practical to use thisprocess for the study of large n behavior of these polynomials. Nor is it convenient to study potentials with larger d ,since the number of terms in the recursion relation increases with d .In this section, we present an alternative technique to obtain both the recursion coefficients and the normalizationconstant. We begin with the identity (obtained by integration by parts) Z [ x ( xψ j ( x )) ′ ] ′ φ k ( x ) dx = − Z ( xψ k ( x ))( xψ j ( x )) ′ dx, j, k = 0 , , . . . . (4.1)The only criteria to use this formalism is that as initial condition, we need to know these polynomials for n =0 , . . . , d −
1. Taking j = 2 n and k = 2 n + 1 in Eq.(4.1), we get the recursion relation for the normalization constant.We get g + γ g − (1 + γ ) g = 0 , n = 0 , (4.2) g + γ g − (1 + γ + γ ) g + γ g = 0 , n = 1 , (4.3) g n +4 + γ n g n +2 − (2 + γ n + γ n − ) g n + γ n − g n − + g n − = 0 , n ≥ , (4.4)where γ n := R n, n +2 R n +1 , n +3 . (4.5)Here, we have expanded xψ n ( x ) and xψ n +1 ( x ) (using Eq.(3.17)), and using Eqs.(3.18), (3.19) and (3.20)), get theresult. We have assumed φ − n and consequently g − n and γ − n is zero, n being a positive integer. Here, we must alsoremember that for these SOP, g n = g n +1 .To evaluate γ , we need to know R k,k +2 for both k odd and even. This can be calculated from Eq.(4.1) by putting j = 2 n , k = 2 n + 3 and j = 2 n + 1, k = 2 n + 2. We get R n +3 , n +5 ( g n +4 − g n +2 ) − R n +1 , n +1 R n, n +2 ( g n − g n +2 ) + R n − , n +1 ( g n − − g n +2 ) = 0 , (4.6) R n +2 , n +4 ( g n +4 − g n +2 ) − R n, n R n +1 , n +3 ( g n − g n +2 ) + R n − , n ( g n − − g n +2 ) = 0 . (4.7)These three equations can be used together to obtain g k for all k . For example, a knowledge of R , and R , (whichwe get from Eqs.(3.6) and (3.8)) will give γ . This will give g from (4.2) (again we need g and g which we willextract from Eq.(2.9)). A knowledge of g in turn yields R , and R , (where we use R , and R , , calculated fromEq.(3.7)) from Eqs.(4.6) and (4.7), which gives γ . This in turn can be used to calculate g (4.3) and so on.However, we still need to know R j,j for j ≥
2. This can be obtained by writing j = 2 n + 3 and k = 2 n in Eq.(4.1).We get R n +3 , n +3 R n, n +2 ( g n +2 − g n ) + R n +3 , n +5 ( g n − g n +4 ) + R n − , n +1 ( g n − g n − ) = 0 . (4.8) R n +2 , n +2 can be calculated from the relation R n +2 , n +2 = − R n +3 , n +3 . Thus a knowledge of g , R , and R , gives R , (4.6). Then using g , g , R , and R , , we can obtain R , (and hence R , ) from (4.8).To summarize, we have outlined a recursive technique to obtain the normalization constant. This necessitates aknowledge of coefficients of the form R k,k +2 . Again to evaluate this (also needed in Eq.(3.17)), we need R k,k . Weobtain a series of self-consistent recursion relations to obtain these coefficients. Comment:
In the context of random matrix theory, we may point out that now we can obtain various statisticalproperties like the level-density and 2-point functions for even finite dimensional symplectic ensembles of randommatrices ([1] [3] [4]).
5. THE ZEROS OF THESE POLYNOMIALS
Having obtained the polynomials for the specific case of d = 2, we will discuss briefly about the zeros of these SOPin this section. We know that in a given interval [ a, b ] ∀ n = 0 , Z ba ψ n ( x ) w ( x ) dx = 0 , n ≥ . (5.1)So ψ n ( x ) should have atleast one point in the interior of [ a, b ] where it changes sign. Let there be k such points x , x , . . . , x k . Then the function ψ n ( x )( x − x ) . . . ( x − x k ) is positive or negetive definite for k ≤ n + 2 d −
1, since ψ n ( x ) is a polynomial of order n + 2 d −
1. This implies Z ba ψ n ( x )( x − x ) . . . ( x − x k ) dx = 0 , ∀ k ≤ n + 2 d − . (5.2)However, from skew-normalization relation, this condition is satisfied if and only if1. For n = 2 m , k = 2 m + 1.2. For n = 2 m + 1, k = 2 m .3. ( x − x ) . . . ( x − x k ) = φ k ( x ).This implies thata. φ k ( x ) is a polynomial of order k with k real zeros.b. ψ k ( x ) is a polynomial of order k + 2 d −
1, with ψ m ( x ) having 2 m + 1 and ψ m +1 ( x ) having 2 m real zeros.
6. CONCLUSION
This paper gives a formalism to derive the SOP arising in the study of symplectic ensembles of random matrices[6]. Here, we would like to point out that this paper contains almost no new results which were not present in [1] and[2]. However, a lot of the properties were overlooked. For example, the explicit form of the recursion coefficients, thenecessity to use Gram-Schmidt method to obtain polynomials of order n < d , the zeros of φ n ( x ) and ψ n ( x ) etc. Atthis point, one can easily obtain these polynomials, although a deeper insight into the recursion coefficients (speciallyits large n behavior) is an absolute necessity. However, with all its limitations, we hope that this paper will take usfurther towards our goal, which is “to develop the theory of skew orthogonal polynomials until it becomes a workingtool as handy as the existing theory of orthogonal polynomials” [7]. Acknowledgment : The author is grateful to Mr. T. K. Bhaumik for his support during the completion of thiswork. [1] Ghosh S., 2006, Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory, J.Phys. A: Math. Gen. .[3] Ghosh S., 2004, Long-range interactions in the quantum many-body problem in one dimension: Ground state, Phys. Rev.E 69, 036118 (2004)[4] Ghosh S., Pandey A, Puri S., and Saha R, 2003, Non-Gaussian random-matrix ensembles with banded spectra, Phys. Rev.E 67, 025201 (2003)[5] Orthogonal Polynomials , Szego G., 1939 (American Mathematical Society, Providence)[6]
Random Matrices , Mehta M. L., 2004 (The Netherlands, Elsevier, 3rd ed.)[7] Dyson F. J., 1972, A Class of Matrix Ensembles, J. Math. Phys.13