Table Of ContentSkew-orthogonal polynomials, differential systems and random matrix theory
Saugata Ghosh∗
The Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste, Italy.
(Dated: February 7, 2008)
We study skew-orthogonal polynomials with respect to the weight function exp[−2V(x)], with
V(x)=P2Kd=1(uK/K)xK,u2d >0,d>0. Afinitesubsequenceofsuchskew-orthogonalpolynomials
arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system
of differential-difference-deformation equation. The vectors formed by such subsequence has the
rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain
compatibility condition and hence admit a simultaneous fundamental system of solutions.
PACSnumbers: 02.30.Gp,05.45.Mt
7
0
0
2
1. INTRODUCTION
n
a A. Random matrices
J
9
The concept of ‘Universality’ in random matrix theory and its various applications in real physical systems have
2 attracted both mathematicians and physicists in the last few decades [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
v From mathematical point of view, the study of ‘Universality’ in the energy level correlations of random matrices
7 require a good understanding of the asymptotic behavior of certain families of polynomials. For example Unitary
0 ensembles ofrandommatricesrequireanunderstandingofthe largen behavioroforthogonal(the one-matrixmodel)
0
and bi-orthogonal (the two matrix model) polynomials, while Orthogonal and Symplectic ensembles require that of
7
the skew-orthogonal polynomials. The rich literature available on orthogonal polynomials [6, 7, 8, 9, 10, 11, 12, 13,
0
14,15,16,17,18]andbi-orthogonalpolynomials[19,20,21,22,23]havecontributedalotinourunderstandingofthe
6
0 Unitary ensembles. Our aim is to develop the theory of skew-orthogonal polynomials [3, 4, 5, 24, 25, 26, 27] so that
/ we canhave further insight into the one-matrix model for Orthogonaland Symplectic ensembles of randommatrices.
h
Inthisdirection,previousexperiencewithorthogonalandbi-orthogonalpolynomialsarisinginthetwo-matrixmodel
p
makes us believe that perhaps the most logical and rigorous way to study the asymptotic properties of these skew-
-
h orthogonal polynomials is to do the following: (1). First, we must look for a finite subsequence of skew-orthonormal
t
a vectors which will satisfy a mutually compatible system of differential-difference-deformationequation. This requires
m one to fix the ‘size’ or rank of this finite sub-sequence. The Generalized Christoffel-Darboux formula helps in this
regardas it gives an indication of the number of terms near the so-called‘Fermi-level’ which are actually required to
:
v studythissystem. (2). Thenextstepistolookforamutuallycompatiblesystemofdifferential-difference-deformation
i
X equation satisfied by the finite subsequence of these vectors and hence find the so-called ‘fundamental solution’. (3).
Finally, to formulate a quaternion matrix Riemann-Hilbert problem and by applying the steepest descent method,
r
a obtain the Plancherel-Rotach-type [15] formula for skew-orthogonalpolynomials.
Reference [5] achieved the first goal. In this paper, we try to address the second issue of the existence of the
‘fundamental solution’ of a system of differential-difference-deformation equation. One of the by-products is that
we observe certain duality property between the skew-orthonormalvectors of Orthogonaland Symplectic ensembles.
(In fact, this justifies further the use of skew-orthogonalpolynomials to ordinary orthogonalpolynomials in studying
these ensembles [6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36].) The third part, whichis also the mostcrucialone, is notyet
fully understood.
We shall consider an ensemble of 2N dimensional matrices H with probability distribution
1
P (H)dH = exp[−[2TrV(H)]]dH, (1.1)
β,N
Z
βN
wheretheparameterβ =1and4correspondstothejointprobabilitydensityforOrthogonalandSymplecticensembles
∗Electronicaddress: [email protected]
2
of random matrices. The “potential” V(x) is a polynomial of degree 2d with positive leading coefficient:
2d
u
V(x)= KxK, u >0, d>0, (1.2)
2d
K
K=1
X
where u is called the deformation parameter. Z is the so called ‘partition function’:
K βN
N−1
Z := exp[−[2TrV(H)]]dH =N! g(β), (1.3)
βN j
ZH∈M2(βN) jY=0
where M(β) is a set of all 2N ×2N real symmetric (β = 1) and quaternion real self dual (β = 4) matrices. dH
2N
is the standard Haar measure. g(β) is the skew-normalization constant for polynomials related to Orthogonal and
j
Symplectic ensembles [24].
Remark on notation. Before entering the details of calculation, we must mention that throughout this paper, we
havefollowedto agreatextentnotations usedin[17, 19,20, 21, 25]. Apartfromhavingthe advantageofusing awell
established and compact notation, this strategy will also highlight the striking similarity (as well as the difference)
between skew-orthogonalpolynomials and their bi-orthogonal counterpart.
B. Skew-Orthogonal Polynomials: relevance in Orthogonal and Symplectic ensembles
Definition:
For Orthogonaland Symplectic ensembles of random matrices, we define semi-infinite vectors:
Φ(β)(x)=(Φ(β)t(x)...Φ(β)t(x)...)t, Φ(β)(x)=−Φ(β)t(x)Z, (1.4)
0 n
Ψ(β)(x)=(Ψ(β)t(x)...Ψ(β)t(x)...)t, Ψ(β)(x)=−Ψ(β)t(x)Z, (1.5)
0 n b
where
b
Ψ(4)(x)=Φ′(4)(x), Ψ(1)(x)= Φ(1)(y)ǫ(x−y)dy. (1.6)
n n n n
R
Z
Each entry in these semi-infinite vectors is a (2×1) matrix:
φ(β)(x) ψ(β)(x)
Φ(β)(x)= 2n , Ψ(β)(x)= 2n , (1.7)
n φ(2βn)+1(x) ! n ψ2(βn)+1(x) !
where
1
φ(β)(x) = π(β)(x)exp[−V(x)], (1.8)
n n
(β)
g
n
q
is the weighted skew-normalized polynomial (often called the quasi-polynomial) and
n
π(β)(x) = c(n,β)xk, c(n,β) =1, n∈N, (1.9)
n k n
k=0
X
is monic skew-orthogonalpolynomial of order n.
0 1
Z = ∔...∔ (1.10)
−1 0
(cid:18) (cid:19)
is semi-infinite anti-symmetric block-diagonalmatrix with Z2 =−1 and
|r|
ǫ(r) = (1.11)
r
3
is the step function. These vectors satisfy skew-orthonormalityrelation:
1 0
(Φ(β),Ψ(β))≡ Φ(β)(x)Ψ(β)(x)dx=δ , n,m∈N. (1.12)
n m n m nm 0 1
R
Z (cid:18) (cid:19)
Remark: This definition differsbfrom that of [3, 4b, 26] for β = 4 by a factor of 2, which is incorporated in the
normalization constant.
These skew-orthonormalvectors form the necessary constituents of the kernel functions required to study different
statistical properties of Orthogonal ensembles and Symplectic ensembles of random matrices. For example, the two
point correlationfunction can be written in terms of the 2×2 kernel function [26, 27]:
σ(β)(x,y) = S2(βN)(x,y) D2(βN)(x,y) , (1.13)
2 I2(Nβ)(x,y)+ δ12,βǫ(x−y) S2(βN)(y,x) !
where δ is the kronecker delta and
S(β)(x,y)=Ψ(β)(x) Φ(β)(y), D(β)(x,y)=Φ(β)(x) Φ(β)(y), (1.14)
2N 2N
N N
Y Y
b b
I(β)(x,y)=Ψ(β)(x) Ψ(β)(y), S(β)(y,x)=Φ(β)(x) Ψ(β)(y). (1.15)
2N 2N
N N
Y Y
b b
The level density is given by
ρ(β)(x,x)=S(β)(x,x). (1.16)
2N
The matrix
=diag(1,...,1,0,...,0) (1.17)
N
Y N
is formed by N (2×2) unit matrices (i.e. a unit matr|ix {ozf si}ze 2N in real space). In general, this will be used to
truncate semi-infinite matrices and in the above case, the semi-infinite vectors of the finite sum in Eqs.(1.14) and
(1.15).
C. The Generalised Christoffel Darboux sum
(β)
Here, we present a summary of the main results of Ref.[5], where we studied the kernel function S (x,y). We
2N
′ ′
expand (xΦ(β)(x)), (Φ(β)(x)) and (xΦ(β)(x)) in terms of Φ(β)(x) (and hence introduce the semi-infinite matrices
Q(β), P(β) and R(β) respectively):
xΦ(β)(x) = Q(β)Φ(β)(x), (1.18)
Ψ(4)(x) = P(4)Φ(4)(x), xΨ(4)(x)=R(4)Φ(4)(x), (1.19)
2Φ(1)(x) = P(1)Ψ(1)(x), 2xΦ(1)(x)=R(1)Ψ(1)(x), (1.20)
where Eq.(1.20) is obtained by multiplying the above expansion by ǫ(y−x) and integrating by parts. Here
R(4) =P(4)Q(4), R(1) =Q(1)P(1). (1.21)
The matrices P(β), Q(β) and R(β) are quaternion matrices. For a nice introduction to the subject, the reader is
referred to the book by Prof. M. L. Mehta [27].
Defining the quaternion-matrix
R(4)(x):=R(4)−xP(4), R(1)(x):= 1 R(1)−xP(1) (1.22)
2
h i
4
the Generalized Christoffel Darboux formula for the Symplectic ensembles (β =4) can be written as:
(x−y)S(4)(x,y)=−Φ(4)(x) R(4)(x), Φ(4)(y). (1.23)
2N
" #
N
Y
b
For Orthogonalensembles (β =1) the Generalized Christoffel Darboux is given by:
(y−x)S(1)(x,y)=−Ψ(1)(x) R(1)(y), Ψ(1)(y). (1.24)
2N
" #
N
Y
b
(β)
The Generalized Christoffel-Darboux matrix R (x), in terms of the elements of the quaternion matrix takes
N
the form h i
Q
(β)
0 0 0 R (x) 0 0
2N−2d,2N
0 0 0 ... ... 0
(β) (β)
[R(β)(x), ]= 0 0 0 R2N−1,2N(x) ... R2N−1,2N+2d−1(x) ,N ≥d(.1.25)
−R(β) (x) ... −R(β) (x) 0 0 0
N 2N,2N−2d 2N,2N−1
Y 0 ... ... ... 0 0
(β)
0 0 −R (x) 0 ... 0
2N+2d−1,2N−1
Here, we must point out that there is a small difference with the notation used in the second part of Ref.[5], where
we used a matrix of size 2N+2 to provethe ‘Universality’in the Gaussiancase. Also the definition of R(1) and P(1)
differs by a factor of 2.
D. Difference-differential-deformation equations
A quick glance at Eq.(1.25) reveals that the relevant vectors contributing to the correlation function are
φ(β) (x)...φ(β) (x). This prompts us to define a finite subsequence(or window: W ⇛{N−d,...,N+d−
(2N−2d) (2N+2d−1)
1},N ≥d) of skew-orthogonalvectors of size 2d (in the quaternionic sense), Φ(β)(x) and Ψ(β)(x):
W W
φ(β) (x) Φ(β) (x)
2N−2d N−d
Φ(Wβ)(x):= ... ≡ ... , N ≥d, (1.26)
(β) (β)
φ2N+2d−1(x) ΦN+d−1(x)
ψ(β) (x) Ψ(β) (x)
2N−2d N−d
Ψ(Wβ)(x):= ... ≡ ... , N ≥d, (1.27)
ψ2(βN)+2d−1(x) ΨN(β+)d−1(x)
such that
Φ(β) (x):= Φ(β)t (x)...Φ(β)t (x) t, ∀j ∈N, (1.28)
W±j N−d±j N+d−1±j
Ψ(β) (x):=(cid:16)Ψ(β)t (x)...Ψ(β)t (x)(cid:17)t, ∀j ∈N. (1.29)
W±j N−d±j N+d−1±j
(cid:16) (cid:17)
The rank of the window is equal to the degree of the potential V(x) in the quaternionic space and twice that of the
degree of V(x) in the real space.
The recursion relations connecting the finite subsequence (or window) with the upper or lower one is through the
ladder operator
Φ(4) (x)=A(4)(x)Φ(4)(x), Φ(4) (x)=(A(4) (x))−1Φ(4)(x), (1.30)
W+1 N W W−1 N−1 W
5
Ψ(1) (x)=A(1)(x)Ψ(1)(x), Ψ(1) (x)=(A(1) (x))−1Ψ(1)(x), (1.31)
W+1 N W W−1 N−1 W
and
Ψ(4) (x)=B(4)(x)Ψ(4)(x), Ψ(4) (x)=(B(4) (x))−1Ψ(4)(x), (1.32)
W+1 N W W−1 N−1 W
Φ(1) (x)=B(1)(x)Φ(1)(x), Φ(1) (x)=(B(1) (x))−1Φ(1)(x), (1.33)
W+1 N W W−1 N−1 W
The vectors Φ(β)(x) and Ψ(β)(x) also satisfy a system of ODEs:
W W
d d
Ψ(4)(x)=D(4)(x)Ψ(4)(x), Φ(1)(x)=D(1)(x)Φ(1)(x),
dx W N W dx W N W
d d
Φ(4)(x)=D(4)(x)Φ(4)(x), Ψ(1)(x)=D(1)(x)Ψ(1)(x). (1.34)
dx W N W dx W N W
Under aninfinitesimalchangeofthe deformationparameteru (the coefficientsofthepolynomialpotentialV(x)),
K
these vectors satisfy a system of PDE’s given by
∂ Ψ(4)(x)=UN(4)(x)Ψ(4)(x), ∂ Φ(1)(x)=UN(1)(x)Φ(1)(x),
∂u W K W ∂u W K W
K K
∂ Φ(4)(x)=UN(4)(x)Φ(4)(x), ∂ Ψ(1)(x)=UN(1)(x)Ψ(1)(x). (1.35)
∂u W K W ∂u W K W
K K
Thusthefinitesubsequenceofskew-orthogonalvectorssatisfyasystemofdifference-differential-deformationequation.
Remark. The two pairs of matrices D(4)(x) and D(1)(x) (similarly (D(1)(x) and D(4)(x)), UN(4)(x) and UN(1)(x)
N N N N K K
(similarly (UN(1)(x) and UN(4)(x)) and A(4)(x) and A(1)(x) (similarly B(4)(x) and B(1)(x)) are dual in the sense
K K N N N N
thatthey remaininvariantunderaninterchangeofΨ(4)(x)7→Φ(1)(x) andQ(4) 7→Q(1) (andsimilarlyforΨ(1)(x)and
Φ(4)(x)).
E. Compatibility
The existence of recursion relations, differential equations and deformation equations for vectors arising in the
Orthogonal ensembles and Symplectic ensembles of random matrices can be viewed as just a projection of the semi-
infinitefunctionsΦ(β)(x)andΨ(β)(x)ontothefinite“window”Φ(β)(x)andΨ(β)(x)respectively. However,wemayalso
W W
consider these equations as defining an overdetermined system of finite difference-differential-deformation equations
ofthevectorfunctionsandseethatthesesystemsarecompatible. Thisleadstotheexistenceofafundamentalmatrix
solution, denoted by Φ(β)(x) and Ψ(β)(x) where all the column vectors satisfy the above difference-differential-
[W] [W]
deformation equations simultaneously.
The compatibility of the deformationanddifference equationwith the differential equationimply that the general-
ized monodromy of the operator d −D(β)(x) and d −D(β)(x) is invariant under u deformations and shifts
dx N dx N K
in N. (cid:16) (cid:17) (cid:16) (cid:17)
Outline of the article
-Section 2 deals with the properties of different finite band matrices related to skew-orthogonalpolynomials.
-Insection3,westudy the systemofPDE’sarisingfromthe infinitesimalchangeofthedeformationparameteru .
K
-We derive the difference relations satisfied by the finite subsequence of skew-orthogonalvectors in section 4.
-In section 5, we derive the folding function which is used to project any given vector Φ(β) (or Ψ(β)) onto its finite
n n
subsequence or window.
-In section 6, we obtain the folded deformation matrix, using the results of section 5.
-Thedifferentialequationforskew-orthonormalvectors,usingtheresultsobtainedinSection6,isderivedinsection
7.
-In section 8, we discuss the existence of the Cauchy-like transforms of the skew-orthogonal vectors of order n as
the other solutions to the differential-deformation-difference equations, for fairly large n.
-We prove compatibility conditions for these difference-deformation-differentialsystem in section 9.
6
2. RECURSION RELATIONS AND FINITE BAND MATRICES.
In this section, we will study in detail the different recursion relations satisfied by the skew-orthonormal vectors.
Here, we must note that unlike the orthogonal polynomials, skew-orthonormal vectors do not satisfy a three-term
recursion relation and hence do not give birth to the tri-diagonalJacobi matrix.
Differentiating Eqs.(1.19)-(1.20), we get
d d
Φ(β)(x)=P(β)Φ(β)(x), Ψ(β)(x)=P(β)Ψ(β)(x), β =1,4, (2.1)
dx dx
d d
x Φ(4)(x)=R(4)Φ(4)(x), x Ψ(4)(x)= R(4)−1 Ψ(4)(x), (2.2)
dx dx
d d h i
x Ψ(1)(x)=R(1)Ψ(1)(x), x Φ(1)(x)= R(1)−1 Φ(1)(x), (2.3)
dx dx
h i
where Φ(β)(x) and Ψ(β)(x) are semi-infinite column vectors defined in Eqs.(1.4,1.5). This is equivalent to saying
[R(4)−xP(4)]Φ(4)(x) = 0, [R(4)−xP(4)−1]Ψ(4)(x)=0, (2.4)
[R(1)−xP(1)]Ψ(1)(x) = 0, [R(1)−xP(1)−1]Φ(1)(x)=0. (2.5)
These finite band matrices satisfy the following commutation relations:
[Q(β),P(β)]=1, [R(β),P(β)]=P(β). (2.6)
Here, each entry is a 2×2 quaternion of the form:
A˜ A˜
A := 2n,2m 2n,2m+1 . (2.7)
n,m "A˜2n+1,2m A˜2n+1,2m+1 #
Using (ψ(4),ψ(4)) and (xψ(4),ψ(4)) for β =4, and replacing ψ(4)(x) by φ(1)(x) for β =1, we get
n m n m
P(β) =−P(β)D, R(β) =−R(β)D, (2.8)
where dual of a matrix A is defined as [27]:
AD =−ZAtZ. (2.9)
Remark. The condition of anti-self duality imposes a much lighter restriction on a diagonal quaternion than
antisymmetry condition on a diagonal matrix element. For example, it leaves the off-diagonal entry of the diagonal
quaternionarbitrary. This is the reasonwhy the odd skew-orthogonalpolynomials are arbitrary upto the addition of
a lower even polynomial.
However starting with (xφ(β),ψ(β)) and using (2.6), we get
n m
(Q(β)−Q(β)D)P(β) =P(β)(Q(β)−Q(β)D)=1. (2.10)
As pointed out in the beginning of the section, this essentially means that like the orthogonal polynomials, we do
not havea tri-diagonalJacobi-matrixfor skew-orthogonalpolynomials. It is this relationthatcauses the Generalized
Christoffel-Darboux sum to have a local behavior (i.e. the rank has dependence on the measure or weight function).
Inasense,thisisamajorsetbackinourhopeofdefininga2×2matrixRiemann-Hilbertproblemforskew-orthogonal
vectors, similar to that of the orthogonalpolynomials. Also from the dependence of P(β) and R(β) on 2d (the degree
of the polynomial V(x)), we can conclude that the ‘size’ of our q-matrix Riemann-Hilbert problem will depend on d.
From [5], we also have
P(β)+[V′(Q(β))]=lower, R(β)+[Q(β)V′(Q(β))]=lower , (2.11)
+
where ‘lower’and ‘lower ’ denotes a strictly lowertriangularmatrix anda lower triangularmatrix with the principal
+
diagonal respectively.
7
Having obtained the recursion relations for the skew-orthonormal vectors, we introduce a convenient notation to
express band-matrices P(β) and R(β):
n+d d n+d d
P(β)Φ(β)(x)= ζ(β)(n)Φ(β) (x), R(β)Φ(β)(x)= η(β)(n)Φ(β) (x), (2.12)
nm m k n−k nm m k n−k
m=n−d k=−d m=n−d k=−d
X X X X
and
n+d n+d d
R(β)Φ(β)(x)−x P(β)Φ(β)(x)= α(β)(n)Φ(β) (x), (2.13)
nm m nm m k n−k
m=n−d m=n−d k=−d
X X X
where
η(β) :=diag(η(β)(0),...,η(β)(n),...), ζ(β) :=diag(ζ(β)(0),...,ζ(β)(n),...), α(β) :=diag(α(β)(0),...,α(β)(n),...).
k k k k k k k k k
(2.14)
Here, we have suppressed the x-dependence of α(β)(n). This notation will be useful in deriving the folding function.
k
More explicitly, with this notation, we can write
ζ(β)(0) ζ(β)(0) ... ... ζ(β)(0) 0 0 ...
0 −1 −d
ζ(β)(1) ζ(β)(1) ζ(β)(1) ... ζ(β) (1) ζ(β)(1) 0 ...
1 0 −1 −d+1 −d
ζ(β)(2) ζ(β)(2) ζ(β)(2) ... ζ(β) (2) ζ(β) (2) ζ(β)(2) 0
2 1 0 −d+2 −d+1 −d
P(β) = ... ... ... ... ... ... ... ... , (2.15)
ζ(β)(d) ζ(β) (d) ... ... ζ(β)(d) ... ... ...
d d−1 0
0 ζ(β)(d+1) ... ... ... ... ... ...
d
0 0 ζ(β)(d+2) ... ... ...
d
and
η(β)(0) η(β)(0) ... ... η(β)(0) 0 0 ...
0 −1 −d
η(β)(1) η(β)(1) η(β)(1) ... η(β)(1) η(β)(1) 0 ...
1 0 −1 −d −d
η(β)(2) η(β)(2) η(β)(2) ... η(β) (2) η(β)(2) η(β)(2) 0
2 1 0 −d+1 −d −d
R(β) = ... ... ... ... ... ... ... ... , (2.16)
η(β)(d) η(β) (d) ... ... η(β)(d) ... ... ...
d d−1 0
0 η(β)(d+1) ... ... ... ... ... ...
d
0 0 η(β)(d+2) ... ... ...
d
We note that for even V(x), the quaternions of the outermost band are such that
(β)
0 0 (η (n)) 0
∀n, ζ(β)(n)≡ , η(β)(n)≡ ±d 00 . (2.17)
±d " (ζ±(βd)(n))10 0# ±d "(η±(βd)(n))10 (η±(βd)(n))11 #
From (2.6), we get the following quadratic relation between the coefficients (ζ(β),η(β))
j j
d−1 d−1
(β) (β) (β) (β) (β)
η (n)ζ (n−l+j)− ζ (n)η (n−l+j)=ζ (n). (2.18)
l−j j l−j j l
j=l j=l
X X
This is the compatibility relation for the difference-differential system and will be used in Section IX.
Remark. Here, one can notice the basic difference between the properties of bi-orthogonal polynomials and their
skew-orthogonal counterparts. In the skew-orthogonal vectors, the semi-infinite matrices are symmetric around the
principal diagonal. This is not true for the bi-orthogonalpolynomials where the matrices have an asymmetry.
8
3. THE DEFORMATION MATRIX
In this section, we will consider infinitesimal deformation corresponding to changes in u , the coefficient of the
K
potential V(x). Using the definition of the semi-infinite matrices Φ(β)(x) and Φ(β)(x) (and Ψ(β)(x), Ψ(β)(x)), the
deformation matrix can be defined as:
b b
∂ Φ(β)(x) = U(β)Φ(β)(x), ∂ Φ(β)(x)=Φ(x)[U(β)]D, K =1,...2d, (3.1)
∂u K ∂u K
K K
∂ Ψ(β)(x) = U(β)Ψ(β)(x), ∂ Ψb(β)(x)=bΨ(x)[U(β)]D, K =1,...2d. (3.2)
∂u K ∂u K
K K
The matrix U(β) is anti-self dual, i.e.: b b
K
(β) (β) D
U =−[U ] . (3.3)
K K
Moreover,they satisfy the following relationswith P andR (we dropthe superscriptβ fromthe matrix Q(β), P(β)
and R(β) for simplicity):
∂u P =[U(β),P], ∂u R=[U(β),R]. (3.4)
K K K K
Explicitly, U(β) can be written as
K
1 1
U(β) =− (QK) −(QK)D − (QK) −(QK)D , (3.5)
K K + − 2K 0 0
(cid:0) (cid:1) (cid:0) (cid:1)
where Q+, Q− denote the upper and lower triangular quaternion matrix while Q0 is the diagonal quaternion.
Proof:
DifferentiatingEq.(1.12)w.r.t. u andusingEqs. (3.1)and(3.2),wegetEq.(3.3). Eq.(3.4)followsbyinterchanging
K
the operators ∂/∂x and x∂/∂x with ∂/∂u . Finally to prove Eq.(3.5), we start with the (2×1) normalized quasi-
K
polynomial
1
Φ(β)(x):= e−V(x)Π (x), n∈N, (3.6)
n n
g(β)
n
q
where V(x) is defined in Eq.(1.2) and
π (x)
Π (x)= 2n (3.7)
n
π2n+1(x) !
isthemonicskew-orthogonalpolynomial,definedinEq.(1.9). Tosavecluttering,wehavesuppressedtheβ dependence
of π (x). We have used g(β) =g(β) . Differentiating with respect to the deformation parameter, we get
n 2n 2n+1
1 e−V(x) xK
∂u Φ(β)(x)=− ∂u (g(β))Φ(β)(x)+ ∂u Π (x)− Φ(β)(x). (3.8)
K n 2gn(β) K n n g(β) K n K n
n
q
Also differentiating the skew-normalizationcondition, we get
∂
− g(β) = − ∂u (Φ(β)(x))Ψ(β)(x)+Φ(β)(x)∂u Ψ(β)(x) dx,
∂u n K n n n K n
K
Z h i
= 1 QK +(QK)D bg(β). b (3.9)
K n,n n
h i
The diagonal elements have the form:
(U(β)) = − QK + 1 (QK) +(QK)D ,
K n,n K 2K n,n n,n
(cid:18) (cid:19)n,n (cid:16) (cid:17)
= − 1 QK −(QK)D . (3.10)
2K n,n n,n
h i
9
Thus, we get
U(β) =− 1 QK −(QK)D − 1 (QK)−(QK)D . (3.11)
K K + − 2K 0 0
(cid:0) (cid:1) (cid:16) (cid:17)
Remark. Here the operation (QK)D means that the dual is taken first and then the lower triangular part collected.
−
Also, due to the arbitrariness in the definition of an anti-self dual matrix, we may choose the lower off-diagonal
element of the diagonal quaternions in U(β) zero. With this choice, we can remove the arbitrariness in the definition
K
of the odd skew-orthonormalpolynomials.
4. THE DIFFERENCE EQUATION
In this section, we introduce a sequence of companion-likematrices A(β)(x) and B(β)(x) ofsizes (2d)×(2d). Using
N N
the relations in Eqs. (2.4) and (2.5), we can write:
d d
α(4)(n)Φ(4) (x)=0 α(1)(n)Ψ(1) (x)=0 ∀k ∈N, (4.1)
k n−k k n−k
k=−d k=−d
X X
and
d d
α(1)(n)−δ Φ(1) (x)=0 α(4)(n)−δ Ψ(4) (x)=0 ∀k ∈N, (4.2)
k 0,k n−k k 0,k n−k
kX=−d(cid:16) (cid:17) kX=−d(cid:16) (cid:17)
we get for N ≥d:
Φ(4) (x) Φ(4) (x)
N−d+1 N−d
. .
.. ..
Φ(4) (x) := =A(4)(x) =A(4)(x)Φ(4)(x), (4.3)
W+1 . N . N W
. .
. .
Φ(N4)+d(x) Φ(N4)+d−1(x)
and
Ψ(1) (x) Ψ(1) (x)
N−d+1 N−d
. .
Ψ(1) (x) := .. =A(1)(x) .. =A(1)(x)Ψ(1)(x), (4.4)
W+1 . N . N W
.. ..
Ψ(N1)+d(x) Ψ(N1)+d−1(x)
where (suppressing the β dependence of α (n))
j
0 1 0 ... 0
(β) 0 0 1 ... 0
A (x) = . (4.5)
N . . . .
.. .. .. .. 1
−(α−d(N))−1αd(N) ... −(α−d(N))−1(α0(N)) ... −(α−d(N))−1α−d+1(N)
Remark. For potentialV(x) with evendegreeand ∀n, α±d(n)=Rn(β,n)±d−xPn(β,n)±d andits inversehas the quaternion
structure:
α±d(n) := ab 0c !; α±d(n)−1 = −a1b 01 !, a,c6=0. (4.6)
ac c
Hence they are invertible. This is the criteria for the existence of the skew-orthogonalpolynomials corresponding to
even potential. We leave it to the reader to verify that α±d(n) is non-invertible for potentials with odd degree.
10
Using this, one can easily see that
Φ(4) (x) Φ(4) (x)
N−d−1 N−d
. .
Φ(4) (x) := .. =(A(4) (x))−1 .. =(A(4) (x))−1Φ(4)(x), (4.7)
W−1 . N−1 . N−1 W
. .
. .
Φ(N4)+d−2(x) Φ(N4)+d−1(x)
and
Ψ(1) (x) Ψ(1) (x)
N−d−1 N−d
. .
Ψ(1) (x) := .. =(A(1) (x))−1 .. =(A(1) (x))−1Ψ(1)(x), (4.8)
W−1 . N−1 . N−1 W
.. ..
Ψ(N1)+d−2(x) Ψ(N1)+d−1(x)
where
−(αd(N −1))−1αd−1(N −1) ... −(αd(N −1))−1(α0(N −1)) ... −(αd(N −1))−1α−d(N −1)
1 0 ... ... 0
(A(Nβ−)1(x))−1 = 0 ... ... ... 0 (4.9).
. . .
.. .. .. 1 0
Similarly, for Ψ(4)(x) and Φ(1)(x), A(β)(x) is replaced by B(β)(x), where
0 1 0 ... 0
B(β)(x) = 0 0 1 ... 0 . (4.10)
N . . . .
.. .. .. .. 1
−(α−d(N))−1αd(N) ... −(α−d(N))−1(α0(N)−1) ... −(α−d(N))−1α−d+1(N)
More generally, this ladder operator can be used successively to obtain:
Φ(β) (x)=(A(β) (x))−1...(A(β) (x))−1Φ(β)(x), ∀j ∈N, (4.11)
W−j N−j N−1 W
and
Φ(β) (x)=A(β) (x)...A(β)(x)Φ(β)(x), ∀j ∈N. (4.12)
W+j N+j−1 N W
This is the underlying idea behind folding which will be discussed in details in the next section.
5. FOLDING
Remark. Fromthispointonwefocusonthe skew-orthogonalvectorsΦ(4)(x)(anditsdualΨ(1)(x)), buteverything
n n
being said can be immediately extended to the vectors Φ(1)(x), (and hence Ψ(4)(x)) by interchanging the rˆoles of
n n
matrices R(β) by R(β)−1 .
Thenotionof“folding”isthefollowing: weexpressanyquasipolynomialΦ(β)(x)(orΨ(β)(x))asalinearcombination
n n
of 2d fixed consecutive vectors Φ(β)(x)≡Φ(4) (x) , j =d,···−d+1 with polynomial coefficients. We now provide a
W N−j
way of computing the folding function on the same line as done in [20] for bi-orthogonalpolynomials.
For a fixed subsequence (or window) W, such that W ⇛ {N −d,...,N +d−1},N ≥ d, we seek to describe the
folding of the infinite wave-vector Φ(β)(x) onto the window W by means of a single quaternion matrix F(x) of size
∞×(2d) with polynomial entries such that:
N+d−1
∀n, Φ(β)(x)= F (x)Φ(β)(x) . (5.1)
n n,k k
k=N−d
X
