Determinantal Correlations for Classical Projection Processes

0.31 MB·English

by  Peter J. Forrester

Determinantal Correlations for Classical Projection Processes
Peter J. Forrester† and Taro Nagao∗
8 †Department of Mathematics and Statistics, University of Melbourne,
0 Victoria 3010, Australia
0
2
∗ Graduate School of Mathematics, Nagoya University,

n
Chikusa-ku, Nagoya 464-8602, Japan
a
J

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] Abstract
h
p
-
h
t Recentapplications inqueuingtheoryandstatistical mechanics have isolated theprocessformed
a
by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in
m
general from the eigenvalues of nested sequences of matrices resulting from random corank 1
[
projections of classical random matrix ensembles, are identified for the LUE and JUE. The

1
correlations for all these processes can be computed in a unified way. The resulting expressions
v
0 can then be analyzed in various scaling limits. At the soft edge, with the rank of the minors
0 differing by an amount proportional to N2/3, the scaled correlations coincide with those known
1
from the soft edge scaling of the Dyson Brownian motion model.
0
.
1
0
8
0
:
v
i
X
r
a
1
1 Introduction
SincethepioneeringdaysofWigner,Gaudin,MehtaandDyson(seePorter[31]foracollection of
papersfromthisperiod),randommatrixtheory hasshownitself tobeperhapstherichestsource
of exact solutions for correlation and distribution functions of all statistical mechanical models.
Many of the discoveries of this type have their motivation in new applications of random matrix
theory, unknown in the pioneering days. A case in point is the recent work of Johansson and
Nordenstam [22, 27], who compute the exact form of the correlation functions for the coupled
eigenvalue sequences obtained from the principal minors of Gaussian unitary ensemble (GUE)
matrices.
The motivation for studying this GUE minor process begins with a work of Baryshnikov [1].
Some years earlier Glynn and Whitt [15] had studied the problem of computing the distribution
of exit times from a queueing system, in the limit the number of queues tends to infinity but
the number of jobs remains finite. It was proved that for general i.i.d. service times the scaled
distributions D of the exit time of the kth customer from the final queue could be written as
k
k−1
D = sup B (t ) B (t ) .
k i i+1 i i
−
X
Xi=0(cid:16) (cid:17)
Here each B denotes an independent standard Brownian motion, while the condition X is that
i
0 = t < t < < t = 1.
0 1 k
···
By studying the particular case of exponential waiting times Baryshnikov was able to show
that D could alternatively be specified as the joint distribution of µ , where µ is
k k=1,2,... k k
{ } { }
the largest eigenvalue of the kth principal minor of an infinite GUE matrix X with probability
densityfunction(PDF)proportionaltoexp( X2/2). JohanssonandNordenstam[22]give other
−
occurrences in statistical mechanics of essentially the same process relating to the eigenvalues of
minors of GUE matrices (referred to as the GUE minor process). These are in the specification
of certain point processes relating to the boundary region (neighbourhood of the frozen zone) of
random domino tilings of the Aztec diamond [20], and to random lozenge-tilings of a hexagon
[21]. Intheequivalentlanguageofsteppedsurfaces,OkounkovandReshetikhin[29]makesimilar
observations. The recent work of Borodin, Ferrari and Sasamoto [5] encounters this process in
the context of studying the dynamics of the asymmetric exclusion process.
The broader setting of the measures encountered in the studies of the above statistical
mechanical modelsrelates totheRobinson-Schensted-Knuth(RSK)correspondence(seeSection
2.1) below. Consideration of the structures inherent therein [2, 14] identifies natural extensions
of the GUE minor process. One such class of extensions replaces the Gaussian weight in the
latter by the Laguerre or Jacobi weights, which can all be realized by a sequence of projections
onto random complex hyperplanes, giving rise to so called classical projection processes (the
Gaussian, Laguerre and Jacobi weights are all classical from the viewpoint of the theory of
orthogonal polynomials). Our main point in the present paper is that the correlations for the
classical projection processes can becomputed exactly in a unified way. The essential ingredient
hereistheRodriguesformulaforclassical orthogonalpolynomials. Moreover, knownasymptotic
formulas for the latter allow for the evaluation of scaling limits of the correlations.
WebegininSection2byrecallinghowtheRSKcorrespondencerelatestoacertainstatistical
mechanical model of last passage times. In Section 3 the occurrence of special cases of the joint
PDF in some random matrix setting is noted. The correlations for the classical projection
process are calculated in Section 4, and their scaling limits analyzed in Section 5.
2
2 A joint probability density associated with RSK
2.1 The case of general parameters
The Robinson-Schensted-Knuth (RSK) correspondence gives a bijection between n n non-
1 2
×
negative integer matrices [x ] (rows countered from the bottom) with entry (ij) weighted
i,j
(aibj)xi,j, and pairs of weighted semi-standard tableaux (weights ai , bj ) of shape µ =
{ } { }
(µ ,...,µ )). Results from [18, 2] associate a probabilistic model to the RSK correspondence.
1 n
Identify with each lattice site (ij) a random non-negative integer variable x chosen from the
i,j
geometric distribution with parameter a b , so that
i j
Pr(x = k) = (1 a b )(a b )k. (2.1)
i,j i j i j
−
For given non-negative parameters a , b , and given n ,n Z+, a probabilistic quantity of
i j 1 2
{ } { } ∈
interest is the sequence of last passage times
L(l)(n ,n ) = max x , l = 1,...,max(n ,n ). (2.2)
1 2 i,j 1 2
(Xrd∗)l
Here (rd∗)l denotes the set of l disjoint (no common lattice points) rd∗ lattice paths, which
in turn are defined as either a single point, or points connected by segments formed out of
arbitrarypositive integer multiples of stepsto theright andsteps upin therectangle 1 i n ,
1
≤ ≤
1 j n .
2
≤ ≤
A crucial feature of the RSK correspondence is that the length of the first row of the semi-
standard tableaux pair is equal to (2.2) in the case l = 1, and thus µ = L(1)(n ,n ). More
1 1 2
generally, all row lengths are determined by (2.2) according to [17]
µ = L(l)(n ,n ) L(l−1)(n ,n ) (2.3)
l 1 2 1 2
−
with L(0)(n ,n ) := 0. Thus the distribution of the last passage times is fully determined by
1 2
the distribution of µ .
l
{ }
Awell knownfact relating totheRSKcorrespondenceis thattheprobability ann n non-
1 2
×
negative matrix with elements chosen according to (2.1) corresponds to a pair of semi-standard
tableaux with shape µ, one of content n , the other of content n , is given by [24]
1 2
n1 n2
P(µ) = (1 a b )s (a ,...,a )s (b ,...,b ), (2.4)
− i j µ 1 n1 µ 1 n2
i=1j=1
YY
where s denotes the Schur polynomial. A lesser known fact is that for n > n the joint
µ 1 2
probability that an n (n +1) non-negative integer matrix with elements chosen according
1 2
×
to (2.1) corresponds to a pair of semi-standard tableaux with shape µ, content n and n +1,
1 2
and the bottom left sub-matrix corresponds to a pair of semi-standard tableaux with shape κ,
content n and n is [14]
1 2
n1 n2+1(1 a b )s (a ,...,a )s (b ,...,b )bPnj=21(µj−κj)+µn2+1χ(µ > κ) (2.5)
− i j µ 1 n1 κ 1 n2 n2+1
i=1 j=1
Y Y
where, with χ(A) = 1 if A is true, χ(A) = 0 otherwise,
χ(µ > κ) := χ(µ κ µ κ µ κ 0). (2.6)
1 ≥ 1 ≥ 2 ≥ 2 ≥ ··· ≥ n1 ≥ n1 ≥
3
Itfollows from(2.4)and(2.5)thatgiven thepairofsemi-standardtableauxcorrespondingto
an n n matrix (n > n ) has shape κ, the probability of the pair of semi-standard tableaux
1 2 1 2
×
corresponding to the n (n + 1) matrix, obtained by adding an extra row to the existing
1 2
×
matrix, having shape µ is
P(µ,κ) := χ(µ > κ) n1(1 a b )sµ(a1,...,an1)bPnj=21(µj−κj)+µn2+1. (2.7)
− i n2+1 s (a ,...,a ) n2+1
i=1 κ 1 n1
Y
Let us now seek the joint probability that with n n + p, an n (n + p) non-negative
1 2 1 2
≥ ×
integer matrix with elements chosen according to (2.1) is such that the principal n (n +s)
1 2
×
sub-blocks (s = 0,1,...,p) correspond to pairs of semi-standard tableaux with shape µ(s). This
is computed from (2.4) and (2.7) according to
p−1 n1 n2+p
P(µ(0)) P(µ(s+1),µ(s)) = (1 a b )s (a ,...,a )s (b ,...,b )
− i j µ(p) 1 n1 µ(0) 1 n2
s=0 i=1 j=1
Y Y Y
p
Pn2+s−1(µ(s)−µ(s−1))+µ(s)
b j=1 j j n2+sχ(µ(s) >µ(s−1)). (2.8)
× n2+s
s=1
Y
2.2 Specializing the parameters
In [14] the joint probability (2.5) is specialized to the case of a geometrical progression of
parameters
(a ,...,a ) = (z,zt,zt2,...,ztn1−1)
1 n1
(b ,...,b ) = (z,zt,zt2,...,ztn2−1). (2.9)
1 n2
This is a preliminary step for taking the so called Jacobi limit, in which a joint probability of a
type known from random matrix theory is obtained. In this subsection the parameters will be
specialized according to (2.9) for the more general joint probability (2.8).
Now for(2.8)to benon-zerowerequireℓ(µ(p)) n +p. Underthis circumstance, wededuce
2
≤
from [14, Eq. (2.26)] with κ µ(p), n n , n n +p, r h(p) := µ(p)+n +p j that
7→ 2 7→ 1 1 7→ 2 j 7→ j j 2 −
s (1,t,...,tn1−1) = t−Pnj=1−1n2−pj(j−1)t−(n2+p)Pnj=1−1n2−pjt−Pnj=11(j−1)(n2+p−j)
µ(p) n1−1(t;t)
l=1 l
n1−n2−p−1(t;t)in2+p (t;t)h(ip)+n1−n2−pQn2+p(th(jp) th(ip)).(2.10)
× (t;t) −
i=1 i=1 h(p) i<j
Y Y i Y
This makes explicit the first Schur polynomial factor in (2.8). In regards to the second Schur
polynomial factor, making use of [14, Eq. (2.18)] with n n , n∗ n , λ µ(0), h h(0) :=
7→ 2 7→ 2 7→ j 7→ j
(0)
µ +n j gives
j 2−
sµ(0)(1,t,...,tn2−1)= t−Pnjn=221−(j1−(1t);(tn)2−j) n2(th(j0) −th(i0)). (2.11)
l=1 l i<j
Y
Q
Use of these results in (2.8) allows the following results to be deduced.
4
Proposition 1. Let n n + p. On each site of the n (n + p) square lattice specify a
1 2 1 2
≥ ×
random non-negative integer x according to the specification
i,j
Pr(x = k) = (1 z2ti+j−2)(z2ti+j−2)k, j n
i,j 2
− ≤
Pr(x = k) = (1 α zti−1)(α zti−1)k (s = 1,...,p).
i,n2+s − s s
(p) (p)
Introduce the notations h := µ +n +p i and
i i 2 −
χ˜(h(p),h(p−1)) := χ(h(p) h(p−1) > h(p) h(p−1) > > h(p) h(p−1) > h(p) ).
1 ≥ 1 2 ≥ 2 ··· n2+p−1 ≥ n2+p−1 n2+p
In this setting, the joint probability that the matrix [x ] is such that the sub-matrices
i,j n1×(n2+p)
[x ] (s = 0,...,p) correspond, under RSK, to pairs of tableaux of shape µ(s) has the
i,j n1×(n2+s)
explicit form
p
Kn1,n2,p({αs},z,t)zPnj=2+1ph(jp)+Pnj=21h(j0) αPs nj=2+1sh(js)−Pnj=2+1s−1h(js−1)χ˜(h(s),h(s−1))
s=1
Y
n2+p(t;t)h(ip)+n1−n2−p n2+p(th(jp) th(ip)) n2(th(j0) th(i0)) (2.12)
× (t;t) − −
i=1 h(p) i<j i<j
Y i Y Y
with
p
Kn1,n2,p({αj},z,t) = z−2Pnj=21(n2+p−j)z−Pps=1(p−s) αss−p t−Pnj=1−1n2−pj(j−1)
(cid:16)sY=1 (cid:17)
t−(n2+p)Pnj=1−1n2−pjt−Pnj=21(j−1)(n2−j)t−Pnj=11(j−1)(n2+p−j)
× n2−1(t;t) n1−1(t;t)
l=1 l l=1 l
n1−n2−p−1 n1 n2Q p Qn1
(t;t) (1 z2ti+j−2) (1 α zti−1).
l s
× − −
l=1 i=1j=1 s=1i=1
Y YY YY
2.3 The Jacobi limit
TheJacobilimitofthesettingofProposition1correspondstoeachsiteofthen (n +p)square
1 2
×
lattice specifying a non-negative continuous exponential random variable with site dependent
variance j n given by
2
≤
Pr(x [y,y+dy]) = (i+j 2+2a)e−y(i+j−2+2a)dy, j n
i,j 2
∈ − ≤
Pr(x [y,y+dy]) = (i 1+a+a )e−y(i−1+a+as)dy, (s = 1,...,p). (2.13)
i,n2+s ∈ − s
This can be obtained from (2.12) by setting
t = e−1/L, z = e−a/L, α = e−as/L, h(s)/L = x(s) (2.14)
s j j
andtakingthelimitL . Thejointprobability(2.12),scaledbymultiplyingbyL(1+p)(n2+p/2),
→ ∞
has the following limiting form.
Corollary 1. The PDF obtained by the limiting procedure (2.14) applied to (2.12) is equal to
p
K˜n1,n2,p({aj},a)e−a(Pjn=2+1px(jp)+Pnj=21x(j0)) e−as(Pnj=2+1sx(js)−Pnj=2+1s−1x(js−1))χ(x(s) > x(s−1))
s=1
Y
n2+p
(1 e−x(ip))n1−n2−p (e−x(jp) e−x(ip)) (e−x(j0) e−x(i0)) (2.15)
× − − −
Yi=1 1≤i<Yj≤n2+p 1≤iY<j≤n2
5
where
n1−n2−p−1l! p Γ(a +a+n ) n1 Γ(2a+i+n 1)
K˜ ( a ,a) = l=1 s 1 2−
n1,n2,p { j} ( nlQ=11−1l!)( ln=21−1l!) s=1 Γ(as +a) i=1 Γ(2a+i−1)
Y Y
and Q Q
χ(x(s) > x(s−1)) := χ(x(s) > x(s−1) > > x(s) > x(s−1) > x(s) > 0). (2.16)
1 1 ··· n2+s−1 n2+s−1 n2+s
Changing variables e−xj(s) = y(s) the PDF (2.15) reads
j
n2+p n2 p
K˜ ( a ,a) (y(p))a−1(1 y(p))n1−n2−p (y(0))a χ(y(s) < y(s−1))
n1,n2,p { j} i − i j
Yi=1 jY=1 sY=1(cid:16)
n2+s(y(s))as
j=1 j (p) (p) (0) (0)
(y y ) (y y ) (2.17)
× n2Q+s−1(y(s−1))as+1 j − i j − i
j=1 j (cid:17)1≤i<Yj≤n2+p 1≤iY<j≤n2
where Q
χ(y(s) < y(s−1)):= χ(0 < y(s) < y(s−1) < < y(s−1) <y(s) < 1). (2.18)
1 1 ··· n2+s−1 n2+s
In the special case a = a j (s =1,...,p) this simplifies to the functional form
s
−
1 n2+p p
w(y(p)) (y(p) y(p)) (y(0) y(0)) χ(y(s) < y(s−1)), (2.19)
C l j − i j − i
Yl=1 1≤i<Yj≤n2+p 1≤iY<j≤n2 sY=1
with C the normalization (C will be used generally below for this purpose and so its explicit
valuemayvaryfromequationtoequation)andw(y) = yα(1 y)β forcertainαandβ. Thelatter
−
is the Jacobi weight, which in a functional form involving Vandermonde products is typical of
a PDF arising in random matrix theory. Indeed, this functional form for each of the Gaussian,
Laguerre and Jacobi weights can be obtained as eigenvalue PDFs.
Before turning to such random matrix interpretations, it should be pointed out that in the
setting of Proposition 1, choosing each site of the n (n + p) square lattice according to
1 2
×
continuous exponential random variables
Pr(x [y,y+dy]) = e−ydy (2.20)
i,j
∈
leads to (2.19) with the particular Laguerre weight w(y) = yn1−(n2+p)e−y. Further, rescaling
(p) (p)
the variables y n (1+y 2/n ) therein, and taking n gives (2.19) back with the
j 7→ 1 j 1 1 → ∞
Gaussianweightw(y) = e−y2. Inthecasep = 1bothofthesefactsareexplicitlydemonstratedin
p
[14, Props. 4&5]; the extension to general p involves straightforward limiting procedures applied
to (2.17).
3 Joint eigenvalue PDF for some nested sequences of random
matrices
3.1 Gaussian unitary ensemble
By definition, matrices M from the N N GUE satisfy the recurrence
N
×
M w~
M = N (3.1)
N+1 w~† a
(cid:20) (cid:21)
6
where a N[0,1/√2] and each component w of w~ has distribution w N[0,1/2]+iN[0,1/2].
j j
∼ ∼
†
Further, with U the unitary matrix which diagonalizes M , and thus M = U D U , where
N N N N N N
D is the diagonal matrix of the eigenvalues of M , one has
N N
U ~0 M w~ U ~0 † D w~
N N N N . (3.2)
~0T 1 w~† a ~0T 1 ∼ w~† a
(cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
Thisborderedformisthekeytostudyingthejointdistributionoftheeigenvalues ofthesequence
ofGUE matrices M , orequivalently that of thesequenceof principalminorsofasingle
j j=1,2,...
{ }
infinite GUE matrix [1, 14, 11].
In particular, it follows from (3.2) that the characteristic polynomials p (λ),p (λ) for
N N+1
M ,M are related by
N N+1
p (λ) N w 2
N+1 i
= λ a | |
pN(λ) − − λ λ(N)
Xi=1 − i
(N)
where λ denotes the eigenvalues of M assumed ordered
{ i } N
(N) (N) (N)
λ <λ < < λ .
1 2 ··· N
(N)
With λ regarded as given the PDF for the zeros of this random rational function, and thus
{ i }
(N+1)
the PDF for the distribution of the eigenvalues λ of M , can be computed to beequal
{ i } N+1
to
N+1(λ(N+1) λ(N+1))
e−PNj=+11(λ(jN+1))2 j<k j − k χ(λ(N+1) < λ(N))
Q N (λ(N) λ(N))
j<k j − k
where χ(λ(N+1) < λ(N)) is specified aQs in (2.18) except that the first and last inequalities are
replaced by < and < respectively. From this result, together with the fact that the
−∞ ∞
eigenvalue PDF for N N GUE matrices is proportional to
×
N
e−(λ(lN))2 (λ(N) λ(N))2,
k − j
l=1 1≤j<k≤N
Y Y
it follows that the joint eigenvalue PDF for the sequence of GUE matrices M is
n n=N,...,N+p
{ }
given by (2.19) with n = N and w(y) = e−y2, and the definition (2.16) of χ(y(s) < y(s−1))
2
modified appropriately.
3.2 Laguerre unitary ensemble
Matrices A from the N N LUE with parameter a = n N are constructed from n N
(n)
× − ×
rectangular complex Gaussian matrices X with entries N[0,1/√2]+iN[0,1/√2] according to
(n)
†
A = X X . Such matrices satisfy the recurrence
(n) (n) (n)
A = A +~x~x†, A = [0] (3.3)
(n+1) (n) (0) N×N
where ~x is an N 1 column vector of complex Gaussians. Our interest is in the case n N of
× ≤
A , for which there are N n zero eigenvalues. Then, after making use too of the invariance of
(n)
−
~x~x† by a unitary similarity transformation, the recursion (3.3) can be written in the equivalent
form
A = diag(a ,...,a ,0,...,0)+~x~x† (3.4)
(n+1) 1 n
N−n
| {z }
7
where a are the non-zero eigenvalues of A , assumed ordered so that 0 < a < <
i i=1,...,n (n) 1
{ } ···
a . From (3.4) it follows that the corresponding characteristic polynomials are such that
n
det(λ1N −A(n+1)) = 1 n |xj|2 Nj=n+1|xj|2. (3.5)
det(λ1 A ) − λ a − λ
N − (n) j=1 − j P
X
Theconditional PDF for the zeros of this random rational function, andthus the conditional
PDF for the eigenvalues of A , can be computed exactly as [14, 11]
(n+1)
n+1λN−(n+1) n 1 e−Pnj=+11λj+Pnj=1aj ni<+j1(λi−λj)χ(λ < a)
i=1 i j=1ajN−n Qni<j(ai−aj)
Y Y
Q
Recalling that the non-zero eigenvalue PDF for the matrices A is proportional to
(n)
n n
aN−ne−aj (a a )2
j i− k
j=1 i<k
Y Y
it follows that the joint eigenvalue PDF for the sequence of LUE matrices A with
(r) r=n,...,n+p
{ }
n+p N is given by (2.19) with n = n, w(y) = yN−(n+p)e−y.
2
≤
3.3 Corank 1 random projections
Let A be an n n matrix with eigenvalues a < a < < a , and let ~x be an n 1 random
n 1 2 n
× ··· ×
complex Gaussian normalized column vector. The matrix
M := Π A Π , Π := 1 ~x~x† (3.6)
n n n n n n
−
then represents a corank 1 random projection of A . We know from [14] that in general M has
n n
a single zero eigenvalue, while the non-zero eigenvalues λ have the conditional PDF
j j=1,...,n−1
{ }
n−1(λ λ )
(n 1)! i<j j − k χ(a < λ). (3.7)
− n (a a )
Qi<j j − k
Q
Introduce a nested sequence of matrices A by setting A equal to the diagonal
i i=1,...,n n−1
{ }
matrix formed from the non-zeroeigenvalues of M , computingM according to (3.6), setting
n n−1
A equal to the diagonal matrix formed from the the non-zero eigenvalues of M , and
n−2 n−1
repeating. With the eigenvalues of A having PDF proportional to
n
n
w(a ) (a a )2 (3.8)
l k j
−
l=1 1≤j<k≤n
Y Y
it follows immediately from (3.7) that the joint PDF for A is given by (2.19) with
i i=n−p,...,n
{ }
n +p = n.
2
We remark (see e.g. [8]) that the Laguerre and Jacobi cases can be realized by the matrix
†
structure A = X X for certain matrices (recall the discussion of the previous subsection for
n n n
the Laguerre case). It follows that the above construction is then equivalent to applying a
sequence of corank 1 projections directly to the matrix X .
n
8
4 Correlation for the classical projection process
4.1 Approach via a general formula
Instudyingcorrelations associated with (2.19), the mostgeneral case occurs whenn = 1. After
2
(p) (p+1)
then changing notation by setting p = N 1, y x , (2.19) reads
− l 7→ p+2−l
N N−1
1
w(x(N)) (x(N) x(N)) χ(x(s+1) > x(s)). (4.1)
C l j − k
l=1 1≤j<k≤N s=1
Y Y Y
Here χ(x(s+1) > x(s)) is defined as
χ(x(s+1) > x(s)) = χ(x(s+1) > x(s) > > x(s+1) > x(s) > x(s+1))
1 1 ··· s s s+1
(cf. (2.16)) and w(x) involves a factor χ . Hereafter we consider more general cases in
0<x<1
which w(x) does not always involve such a factor (see (4.7) below). Of course the product of
differences can be written in terms of the Vandermonde determinant, giving
N−1
(N) (N) (N) (N) (N)
w(x ) (x x ) det[w(x )p (x )] ,
l j − k ∝ k N−j k j,k=1,...,N
l=1 1≤j<k≤N
Y Y
with p (x) a set of arbitrary polynomials, p (x) of degree j. Furthermore, we know
j j=0,...,N−1 j
{ }
from [13, Lemma 1] that
χ(x(s+1) >x(s)) = det[χ(x(s+1) > x(s))] (4.2)
j k j,k=1,...,s+1
(s)
where x := . Consequently (4.1) can be written in the form
s+1 −∞
N−1
1
det[φ(x(s),x(s+1))] det[ΨN (x(N))] (4.3)
C j k j,k=1,...,s+1 N−j k j,k=1,...,N
s=1
Y
with
φ(x,y) := χ , ΨN(x) := w(x)p (x). (4.4)
y>x j j
The general structure (4.3) is precisely that for which the correlations have been determined
∞
intherecentwork[4,Lemma3.4]. Toapplythisresult,with(a b)(x,y) := a(x,z)b(z,y)dz,
∗ −∞
it is necessary to compute the quantities
R
φ(n1,n2)(x,y) := (φ φ)(x,y), n < n
1 2
∗···∗
n2−n1times
(for n n , φ(n1,n2)(x,y) := 0), and | {z }
1 2
≥
Ψn (x) := (φ(n,N) ΨN )(x) (1 n < N, j = 1,...,N).
n−j ∗ N−j ≤
Use of (4.4) shows
1
φ(n1,n2)(x,y) = χ (y x)n2−n1−1 (4.5)
y>x
(n n 1)! −
2 1
− −
9
(with the convention that 1/( p)! = 0 for p Z+, this vanishes for n n ) and
1 2
− ∈ ≥
1 ∞
Ψn (x) = w(y)p (y)(y x)N−n−1dy. (4.6)
n−j (N n 1)! N−j −
− − Zx
To proceed further, we choose w(y) to be one of the classical weight functions
e−y2, Gaussian
w(y) = yae−yχ , Laguerre (4.7)
 y>0
 ya(1 y)bχ0<y<1, Jacobi.
−
We further choose p (y) to be proportional to the corresponding orthogonal polynomials, as
j
specified by their Rodrigues formulas
H (y), Gaussian
j
1 dj
p (y)= w(y)(Q(y))j = L(a)(y), Laguerre (4.8)
j e w(y)dyj  j
j (cid:16) (cid:17)  P(a,b)(1 2y), Jacobi
j −
with the quantities e and Q(y) defined in the various cases by the pairs
j
(( 1)j,1), Gaussian
−
(e ,Q(y)) = (j!,y), Laguerre (4.9)
j

(2jj!,y(1 y)), Jacobi.

−
Substituting (4.8) in (4.6) and integrating by parts shows that for j 0 (n = N)
≥ 6
w(x)H (x), Gaussian
j
Ψnj(x) = (−1)N−neN−ejn+j  ww((xx))||aa7→7→aa++NN−−nnPLj((jaa++NN−−nn,)b+(xN)−, n)(1−2x), JLaacgoubeir,re (4.10)
b7→b+N−n

while for j < 0 
( 1)N−n+j 1 ∞
Ψn(x) = − (y x)−j−1w(y)(Q(y))N−n+j dy. (4.11)
j e ( j 1)! −
N−n+j − − Zx
As further required by [4, Lemma 3.4], one introduces the polynomials Φn(x) , n =
{ j }j=0,...,n−1
1,...,N 1 by the orthogonality requirement
−
∞
Φn(x)Ψn(x)dx = δ .
j k j,k
Z−∞
From (4.10) we see that
1 H (x), Gaussian
Nj j
Φnj(x) = (−1)N−neN−ejn+j  Nj|a7→1a1+N−nLP(ja(a++NN−−nn),(bx+)N,−n)(1 2x), LJaacgoubeir,re (4.12)
 Nj|a7→a+N−n j −
b7→b+N−n


where 
2jj!√π, Gaussian
Γ(j+a+1)
= , Laguerre (4.13)
j  Γ(j+1)
N
 Γ(j+a+1)Γ(j+b+1) , Jacobi
j!(2j+a+b+1)Γ(j+a+b+1)


10