Symmetries of the transfer operator for $Γ_0(N)$ and a character deformation of the Selberg zeta function for $Γ_0(4)$

0.28 MB·English

by  Markus Fraczek

Symmetries of the transfer operator for
Γ (N) and a character deformation of the
0
1 Selberg zeta function for Γ (4)
1 0
0
2
M. Fraczek and D. Mayer

n
a
J

9
1 Abstract. The transferoperatorfor Γ0(N) andtrivialcharacter
χ0 possesses a finite group of symmetries generated by permuta-
T] tion matrices P with P2 = id. Every such symmetry leads to a
N factorizationofthe Selbergzeta functionin termsofFredholmde-
terminants of a reduced transfer operator. These symmetries are
.
h related to the group of automorphisms in GL(2,Z) of the Maass
t
a wave forms of Γ0(N) . For the group Γ0(4) and Selberg’s char-
m acter χ there exists just one non-trivial symmetry operator P.
α
[ The eigenfunctions ofthe correspondingreducedtransfer operator
with eigenvalue λ = ±1 are related to Maass forms even respec-
3
tively odd under a corresponding automorphism. It then follows
v
from a result of Sarnak and Phillips that the zeros of the Selberg
1
4 function determined by the eigenvalues λ = −1 of the reduced
4 transferoperatorstayonthe criticalline under the deformationof
4 the character. Fromnumericalresultswe expect thatonthe other
.
1 hand all the zeros corresponding to the eigenvalue λ = +1 leave
1 this line for α turning away from zero.
0
1
:
v Contents
i
X
1. Introduction 2
r
a 2. The transfer operator and Selberg’s zeta function for Hecke
congruence subgroups Γ (N) 3
0
2.1. Symmetries of the transfer operator for Γ (N) 5
0
2.2. A Lewis type functional equation 9
1991 Mathematics Subject Classification. Primary 11M36, 11F72, Sec-
ondary 11F03, 37C30, 37D40, 47B33, 35B25, 35J05 .
Key words and phrases. Hecke congruence subgroups, transfer operator, fac-
torizationofSelberg’szeta function, automorphismsofMaasswaveforms,singular
character deformation, zero’s of Selberg’s zeta function.
ThisworkwassupportedbytheDeutscheForschungsgemeinschaftthroughthe
DFG Research Project “Maass wave forms and the transfer operator approach to
the Phillips-Sarnak conjecture” (Ma 633/18-1).
1
2 M. FRACZEK ANDD. MAYER
3. Automorphism of the Maass forms and their period
functions for Γ (N) 12
0
3.1. The group of automorphisms of Maass forms for Γ (N)
0
and trivial character χ 13
0
3.2. The period functions of Γ (N) and character χ 13
0
3.3. Automorphisms of the period functions 14
4. Selberg’s character χ for Γ (4) 17
α 0
References 22
1. Introduction
In the transfer operator approach to Selberg’s zeta function for a
Fuchsian group Γ this function gets expressed in terms of the Fredholm
determinant of this operator which is constructed from the symbolic
dynamics of the geodesic flow on the corresponding surface of constant
negative curvature. Even if this approach has been carried out up to
now only for certain groups like the modular subgroups of finite index
[2],[3],[4], or the Hecke triangle groups [16], [14],[15] it has lead for
instance to new points of view on this function [22] or the theory of
period functions [12]. Another application of this method is a precise
numerical calculation of the Selberg zeta function [20], which seems
to be impossible by other means at the moment. In this paper we
discuss the transfer operator approach to Selberg’s zeta function for
Hecke congruence subgroups with character, of special interest being
the behaviour of its zeros for Γ (4) under the singular deformation of
0
Selberg’s character [19].
As found numerically by M. Fraczek in [9], certain symmetries of the
transfer operator for these groups play thereby an important role.
These symmetries lead to a factorization of the Selberg zeta function
as known for the full modular group SL(2,Z). There it corresponds to
the involution Ju(z) = u(−z∗) of the Maass forms u for this group [8],
1 0
[12]. Obviously the corresponding element j = ∈ GL(2,Z)
0 −1
!
generates the normalizer group of SL(2,Z) in GL(2,Z). It tuns out
that also the symmetries of the transfer operator for Γ (N) correspond
0
to automorphisms of the Maass forms from its normalizer group in
GL(2,Z).
For the group Γ (4) with a character χ introduced by Selberg in [19]
0 α
and discussed also by Phillips and Sarnak in [18], there is only one
such non-trivial symmetry of the transfer operator. It corresponds to
the generator of Γ (4)’s normalizer group in GL(2,Z) leaving invariant
0
the character χ . Results of Sarnak and Phillips imply that the zeros
α
on the critical line of one factor of Selberg’s function stay on this line
under the deformation of the character, and hence the corresponding
SYMMETRIES OF THE TRANSFER OPERATOR FOR Γ0(N) 3
Maass wave forms for the trivial character remain Maass wave forms.
Numerical results [9] on the other hand imply, that the zeros on the
critical line of the second factor of this function should all leave this
line when the deformation is turned on. A detailed discussion of these
numerical results and their partial proofs is in preparation [1].
The paper is organized as follows: in Section 2 we recall briefly the
0 L+
form of the transfer operator L = β,π for a general finite
β,ρπ L− 0
β,π !
index subgroup Γ of the modular group SL(2,Z) and unitary represen-
0 P
tation π and introduce the symmetries P˜ = of this operator
P 0
!
defined by permutation matrices P. Any such symmetry leads to a
factorization of the Selberg zeta function in terms of the Fredholm de-
terminants of the reduced transfer operator PL+ . The eigenfunctions
β,π
with eigenvalue λ = ±1 of this reduced transfer operator then fulfill
certain functional equations. In Section 3 we discuss the generators
J of the group of automorphisms in GL(2,Z) of the Maass forms u
n,−
for Γ = Γ (N) and π = χ the trivial character. We introduce their
0 0
period functions ψ and derive a formula for the period function J ψ
n,−
of the Maass form J u. In Section 4 we introduce Selberg’s charac-
n,−
ter χ and the non-trivial automorphism J of the Maass forms for
α 2,−
Γ (4). We derive again a formula for the period function J ψ of the
0 2,−
Maass form J u leading to a permutation matrix P which defines
2,− 2,−
a symmetry P˜ of the transfer operator L . From this we con-
2,− β,ρχα
clude that the eigenfunctions with eigenvalue λ = ±1 of the operator
P L+ correspond to Maass forms even respectively odd under the
2,− β,π
involution J . Former results of Phillips and Sarnak then imply that
2,−
the zero’s of the Selberg function on the critical line corresponding to
the eigenfunctions with eigenvalue λ = −1 of this operator stay on this
line under the deformation of the character.
2. The transfer operator and Selberg’s zeta function for
Hecke congruence subgroups Γ (N)
0
The starting point of the transfer operator approach to Selberg’s
zeta function for a subgroup Γ of the modular group SL(2,Z) of index
µ = [SL(2,Z) : Γ] < ∞ is the geodesic flow Φ : SM → SM on the
t Γ Γ
unit tangent bundle SM of the corresponding surface M = Γ\H of
Γ Γ
constant negative curvature. Here H = {z = x+ iy : y > 0} denotes
the hyperebolic plane with hyperbolic metric ds2 = dx2+dy2 on which
y2
a b
the group Γ acts via Möbius transformations gz = az+b if g = .
cz+d c d
!
In the present paper we are mostly working with the Hecke congruence
4 M. FRACZEK ANDD. MAYER
subgroup
a b
Γ (N) = {g ∈ SL(2,Z) : g = }
0 cN d
!
with index µ = N (1+ 1), where p is a prime number. If ρ : Γ →
N p
p|N
end(Cd) is a unitaryQrepresentation of Γ then Selberg’s zeta function
Z is defined as
Γ,ρ
∞
(2.0.1) Z (β) = det(1−ρ(g )exp(−(k +β)l )),
Γ,ρ γ γ
γ k=o
Y Y
where l denotes the period of the prime periodic orbit γ of Φ and
γ t
g ∈ Γ is hyperbolic with g (γ) = γ. In the dynamical approach to this
γ γ
function it gets expressed in terms of the so called transfer operator
well known from D. Ruelle’s thermodynamic formalism approach to
dynamical systems. For general modular groups Γ with finite index µ
and finite dimensional representation π this operator L : B → B
β,π
was determined in [2],[3] as
0 L+
(2.0.2) L = β,ρπ ,
β,π L− 0
β,ρπ !
where B = B(D,Cµ) B(D,Cµ) is the Banach space of holomorphic
functions on the disc D = {z :| z −1 |< 3}, and ρ denotes the repre-
L 2 π
sentation of SL(2,Z) induced from the representation π of Γ whereas
L± is given for ℜβ > 1 by
β,ρπ 2
∞ 1 1
(2.0.3) L± f (z) = ρ (ST±n)f( ),
β,ρπ (z +n)2β π z +n
(cid:16) (cid:17) nX=1
0 −1 1 1
where S = and T = . In the following we restrict
1 0 0 1
! !
ourselves to one dimensional unitary representations π, hence unitary
characters, which we denote as usual by χ. In this case the following
Theorem was proved in [3].
Theorem 2.0.1. The transfer operator L : B → B with L =
β,χ β,χ
0 L+ ∞
β,χ and L± f (z) = 1 ρ (ST±n)f( 1 ) extends to
L− 0 β,χ (z+n)2β χ z+n
β,χ ! n=1
(cid:16) (cid:17)
a meromorphic family of nuclearPoperators of order zero in the entire
complex β plane with possible poles at β = 1−k, k = 0,1,2,.... The
k 2
Selberg zeta function Z for the modular group Γ and character χ
Γ,χ
can be expressed as Z (β) = det(1 − L ) = det(1 − L+ L− ) =
Γ,χ β,χ β,χ β,χ
det(1−L− L+ ).
β,χ β,χ
This shows that the zero’s of Selberg’s function are given by those
β-values for which λ = 1 belongs to the spectrum σ(L ) respectively
β,χ
σ(L− L+ ) = σ(L+ L− ). From Selberg’s trace formula one knows
β,χ β,χ β,χ β,χ
SYMMETRIES OF THE TRANSFER OPERATOR FOR Γ0(N) 5
that there are two kinds of such zeros: the trivial zeros at β = −k, k =
1,2,..., and the so called spectral zeros. They correspond either to
eigenvalues λ = β(1− β) of the automorphic Laplacian with ℜβ = 1
2
or 1 ≤ β ≤ 1 respectively to resonances of the Laplacian, that means
2
poles of the scattering determinant with ℜβ < 1 and ℑβ > 0 [21][10].
2
For arithmetic groups like the congruence subgroups with trivial or
congruent character χ one knows that these resonances are on the line
ℜβ = 1, corresponding to the nontrivial zeros ζ (2β) = 0 of Riemann’s
4 R
zeta function ζ when assuming his hypothesis, respectively on the line
R
ℜβ = 0. For general Fuchsian groups and congruence subgroups with
non-congruent character however these resonances can be anywhere in
the halfplane ℜβ < 1.
2
2.1. Symmetries of the transfer operator for Γ (N). It turns
0
out that there exists for any N a finite number h of µ ×µ permu-
N N N
0 P
tation matrices P with P2 = id such that the matrix P˜ =
µN P 0
!
commutes with the transfer operator L and hence
β,χ
(2.1.1) P L+ = L− P.
β,χ β,χ
Thereby P = (P ) acts in the Banach space B(D,CµN) as
ij 1≤i,j≤µN
µN
(Pf) (z) = P f (z) if f(z) = (f (z)) . We call such a matrix
i ij j i 1≤i≤µN
j=1
P˜ a symmetPry of the transfer operator. As an example consider the
group Γ (4) and Selberg’s character χ ,0 ≤ α ≤ 1, which will be
0 α
described later. Its transfer operator L has the following form
β,χα
∞
L f˜ = f | S˜T1+4q +f | S˜T2+4q +f | S˜T3+4q
β,χα +1 −3 2β −4 2β −5 2β
q=0
X
+ f | S˜T4+4q
−2 2β
∞
L f˜ = e2πi(1+4q)αf | S˜T1+4q +e2πi(2+4q)αf | S˜T2+4q
β,χα +2 −1 2β −1 2β
q=0
X
+ e2πi(3+4q)αf | S˜T3+4q +e2πi(4+4q)αf | S˜T4+4q
−1 2β −1 2β
∞
L f˜ = e−2πiαf | S˜T1+4q +e−2πiαf | S˜T2+4q
β,χα +3 −2 2β −3 2β
q=0
X
+ e−2πiαf | S˜T3+4q +e−2πiαf | S˜T4+4q
−4 2β −5 2β
∞
L f˜ = e−2πiα(1+4q)f | S˜T1+4q +e−2πiα(2+4q)f | S˜T2+4q
β,χα +4 −6 2β −6 2β
q=0
X
+ e−2πiα(3+4q)f | S˜T3+4q +e−2πiα(4+4q)f | S˜T4+4q
−6 2β −6 2β
∞
L f˜ = e2πiαf | S˜T1+4q +e2πiαf | S˜T2+4q
β,χα +5 −4 2β −5 2β
q=0
X
+ e2πiαf | S˜T3+4q +e2πiαf | S˜T4+4q
−2 2β −3 2β
6 M. FRACZEK ANDD. MAYER
∞
L f˜ = f | S˜T1+4q +f | S˜T2+4q +f | S˜T3+4q
β,χα +6 −5 2β −2 2β −3 2β
q=0
X
+ f | S˜T4+4q
−4 2β
∞
L f˜ = f | S˜T1+4q +f | S˜T2+4q +f | S˜T3+4q
β,χα −1 +5 2β +4 2β +3 2β
q=0
X
+ f | S˜T4+4q
+2 2β
∞
L f˜ = e−2πiα(1+4q)f | S˜T1+4q +e−2πiα(2+4q)f | S˜T2+4q
β,χα −2 +1 2β +1 2β
q=0
X
+ e−2πiα(3+4q)f | S˜T3+4q +e−2πiα(4+4q)f | S˜T4+4q
+1 2β +1 2β
∞
L f˜ = e−2πiαf | S˜T1+4q +e−2πiαf | S˜T2+4q
β,χα −3 +4 2β +3 2β
q=0
X
+ e−2πiαf | S˜T3+4q +e−2πiαf | S˜T4+4q
+2 2β +5 2β
∞
L f˜ = e2πiα(1+4q)f | S˜T1+4q +e2πiα(2+4q)f | S˜T2+4q
β,χα −4 +6 2β +6 2β
q=0
X
+ e2πiα(3+4q)f | S˜T3+4q +e2πiα(4+4q)f | S˜T4+4q
+6 2β +6 2β
∞
L f˜ = e2πiαf | S˜T1+4q +e2πiαf | S˜T2+4q
β,χα −5 +2 2β +5 2β
q=0
X
+ e2πiαf | S˜T3+4q +e2πiαf | S˜T4+4q
+4 2β +3 2β
∞
L f˜ = f | S˜T1+4q +f | S˜T2+4q +f | S˜T3+4q
β,χα −6 +3 2β +2 2β +5 2β
q=0
X
+ f | S˜T4+4q
+4 2β
where f˜∈ B(D,Cµ) B(D,Cµ) is given by f˜= (f ,f ) and f =
+ − ±
(f ) and S˜z = 1. The induced representation ρ of the character
±i 1≤i≤6 zL χ
χ on Γ (4) is defined in terms of the coset decomposition of SL(2,Z)
0
6
(2.1.2) SL(2,Z) = Γ (4)R
0 i
i=1
[
as
(2.1.3) ρ (g) = δ (R gR−1)χ(R gR−1), 1 ≤ i,j ≤ 6.
χ ij Γ0(4) i j i j
Thereby we have chosen the following representatives R ∈ SL(2,Z) of
i
the cosets Γ (4)R
0 i
(2.1.4) R = id , R = STi−2,2 ≤ i ≤ 5 and R = ST2S.
1 2 i 6
It turns out that the two permutation matrices P ,i = 1,2 correspond-
i
ing to the permutations σ with
i
1 2 3 4 5 6
(2.1.5) σ =
1 1 2 5 4 3 6
SYMMETRIES OF THE TRANSFER OPERATOR FOR Γ0(N) 7
1 2 3 4 5 6
(2.1.6) σ =
2 6 4 3 2 5 1
fulfill equation (2.1.1) for α = 0 and hence the corresponding matrices
P˜, i = 1,2 commute with the transfer operator L where χ is the
i β,χ0 0
trivial character. The matrix P˜ on the other hand commutes even
2
with the operator L for all α. Indeed the matrix ρ (S) is given by
β,χα χ0
the permutation σ where
S
1 2 3 4 5 6
(2.1.7) σ =
S 2 1 5 6 3 4
and an easy calculation shows that P ρ (S) = ρ (S)P , i = 1, 2. The
i χ0 χ0 i
matrix ρ (T) on the other hand is given by the permutation σ with
χ0 T
1 2 3 4 5 6
(2.1.8) σ = .
T 1 3 4 5 2 6
One then checks that P ρ (T) = ρ (T−1)P , i = 1, 2. Therefore
i χ0 χ0 i
P ρ (STn) = ρ (ST−n)P for all n ∈ N and i = 1, 2. For the
i χ0 χ0 i
character χ analogous relations hold for P .
α 2
For the trivial character χ one can determine for the group Γ (N) the
0 0
number h of matrices P with the above properties and hence defining
N i
symmetries of the transfer operator as follows:
Theorem 2.1.1. For the Hecke congruence subgroup Γ (N) and trivial
0
0 P
character χ ≡ 1 there exist h matrices P˜ = commuting
0 N P 0
!
with the transfer operator L where P is a µ × µ permutation
β,χ0 N N
matrix with P2 = 1 and Pρ (S) = ρ (S)P respectively Pρ (T) =
µN χ0 χ0 χ0
ρ (T−1)P and hence
χ0
P L+ = L− P.
β,χ0 β,χ0
Thereby h = max{k : k | 24 and k2 | N}. The permutation
N
matrices P are determined by the h generators j of the normalizer
N
group N of Γ (N) in GL(2,Z). The Selberg zeta function Z can
N 0 Γ,χ0
be written as
Z = det(1−P L+ )det(1+P L+ ).
Γ,χ0 β,χ0 β,χ0
.
Remark 2.1.2. For Γ (4) obviously h = 2 and there exist according
0 N
to Theorem 2.1.1 two such permutation matrices P and P which in-
1 2
deed are given by the aforementioned permutations σ , i = 1,2. Since
i
P P = P P and P L+ = L− P , i = 1,2 we find
1 2 2 1 i β,χ0 β,χ0 i
P P P L+ = P P L− P = P L+ P P = P L+ P P
1 2 1 β,χ0 1 2 β,χ0 1 1 β,χ0 2 1 1 β,χ0 1 2
8 M. FRACZEK ANDD. MAYER
and the operators P P and P L+ commute, where the operator
1 2 1 β,χ0
P P corresponds to the permutation
1 2
1 2 3 4 5 6
(2.1.9) σ = .
6 4 5 2 3 1
We find also P P L+ = L+ P P . But (P P )2 = id , hence
1 2 β,χ0 β,χ0 1 2 1 2 6
this operator has only the eigenvalues λ = ±1 and the Banach space
B(D,C6) decomposes as B(D,C6) = B(D,C6) ⊕ B(D,C6) with
+ −
P P f = ±f forf ∈ B(D,C6) . Theelementsf ∈ B(D,C6) , ǫ =
1 2 ± ± ± ± ǫ ǫ
± have therefore the form (f ) = f , 1 ≤ i ≤ 3 respectively (f ) =
ǫ i i ǫ σ(i)
ǫf , 1 ≤ i ≤ 3. Denote by
i
L+ : B(D,C6) → B(D,C6)
β,χ0,± ± ±
respectively
P L+ : B(D,C6) → B(D,C6)
1 β,χ0,± ± ±
the restriction of the operators L+ respectively P L+ to the sub-
β,χ0 1 β,χ0
space B(D,C6) , which obiously is isomorphic to the space B(D,C3).
±
Then det(1±P L+ ) = det(1±P L+ )det(1±P L+ ), where
1 β,χ0 1 β,χ0,+ 1 β,χ0,−
the operator P L+ : B(D,C3) → B(D,C3) can be written as
1 β,χ0,ǫ
0 ǫL +L ǫL +L
β,2 β,4 β,1 β,3
(2.1.10) P L+ = L 0 0 .
1 β,χ0,ǫ  β 
0 L +ǫL ǫL +L
β,1 β,3 β,2 β,4
 
 
∞ 4
with L f = f| S˜T1+kq, 1 ≤ k ≤ 4 and L = L . The
β,k 2β β β,k
q=0 k=1
operator L+ Pin the space B(D,C3) on the other handPhas the form
β,χ0,ǫ
0 ǫL +L ǫL +L
β,2 β,4 β,1 β,3
(2.1.11) L+ = L 0 0 .
β,χ0ǫ  β 
0 ǫL +L L +ǫL
β,1 β,3 β,2 β,4
 
To relate the Fredholm determinants of the operators (P L+ )2 and
1 β,χ0,ǫ
(L+ )2 we use the following simple Lemma
β,χ0,ǫ
Lemma 2.1.3. Letbe α,β and γ complexnumbers andǫ = ±1. Then λ
0 α β
is an eigenvalue of the matrix L = γ 0 0 iff ǫλ is an eigenvalue
1  
0 β ǫα
 
0 α β  
of the matrix L = γ 0 0 .
2  
0 ǫβ α
 
 
Proof. Theprooffollowsfromthecharacteristicpolynomialofthe
(cid:3)
two matrices.
3 3
ThisshowsthattraceLn = (Ln) = ǫn traceLn = ǫn (Ln)
1 1 k,k 2 2 k,k
k=1 k=1
foralln ∈ N. ButthenisnottooPdifficulttoseethatalsotraceP(L+ )n =
β,χ0,ǫ
SYMMETRIES OF THE TRANSFER OPERATOR FOR Γ0(N) 9
ǫntrace(P L+ )n for all n ∈ N and hence det(1 − (P L+ )2) =
1 β,χ0,ǫ 1 β,χ0,ǫ
det(1 − (L+ )2) for ǫ = ±. Therefore the Selberg zeta function
β,χ0,ǫ
Z (β) for the group Γ (4) with trivial character χ can be writ-
Γ0(4),χ0 0 0
ten as
Z (β) = det 1−(P L+ )2 = det 1−(L+ )2
Γ0(4),χ0 1 β,χ0 β,χ0
(2.1.12) = det((cid:16)1−L+ )det(1(cid:17)+L+ (cid:16)) (cid:17)
β,χ0 β,χ0
Furthermore this function factorizes in this case also as
Z (β) = det(1−P L+ )det(1−P L+ )
Γ0(4),χ0 1 β,χ0,+ 1 β,χ0,−
(2.1.13) × det(1+P L+ )det(1+P L+ )
1 β,χ0,+ 1 β,χ0−
To prove Theorem 2.1.1 we relate the matrices P to the generating
automorphisms in GL(2,Z) of the Maass wave forms for Γ (N) and
0
can determine this way the explicit form of these matrices P. For
this we derive in a first step a Lewis type functional equation for the
eigenfunctions of the operator P L+ with eigenvalue λ = ±1.
β,χ
2.2. A Lewis type functional equation. Consider any finite
index modular subgroup Γ and any unitary character χ : Γ → C⋆
respectively the induced representation ρ of SL(2,Z). Assume there
χ
0 P
exists a symmetry P˜ = with P a permutation matrix with
P 0
!
the properties analogous to Theorem 2.1.1, and commuting with the
0 L+
transfer operator L = β,ρχ of Γ. If f is an eigenfunction
β,χ L− 0
β,ρχ !
of the operator P L+ with eigenvalue λ = ±1 then one can show
β,χ
Proposition 2.2.1. If P L+ f(ζ) = λf(ζ) with λ = ±1 then the
β,χ
function Ψ(ζ) := Pρ (T−1S)Pf(ζ −1) fulfills the functional equations
χ
1
(2.2.1) Ψ(ζ) = λζ−2βP ρ (S)Ψ( ),
χ
ζ
respectively
ζ
(2.2.2) Ψ(ζ)−ρ (T−1)Ψ(ζ +1)−(ζ +1)−2βρ (T′−1)Ψ( ) = 0,
χ χ
ζ +1
where T′ = ST−1S. On the other hand every solution Ψ of equa-
tions (2.2.1) and (2.2.2) holomorphic in the cut β-plane (−∞,0] with
Ψ (z) = o(z−min{1,2ℜs}) as z ↓ 0, respectively Ψ (z) = o(z−min{0,2ℜs−1})
i i
as z → ∞, determines an eigenfunction f with eigenvalue λ = ±1 of
the operator P L+ .
β,χ
Proof. Let ℜβ > 1. If PL+f(ζ) = λf(ζ), λ = ±1 then obviously
2 β
Pρ (STS)PPL+f(ζ+1) = λPρ (STS)Pf(ζ+1).Subtracting the two
χ β χ
10 M. FRACZEK ANDD. MAYER
equations leads to
1
λf(ζ)−λPρ (STS)Pf(ζ +1)−(ζ +1)−2βPρ (ST)f( ) = 0,
χ χ
ζ +1
and hence the function ψ(ζ) := Pf(ζ −1) fulfills the equation
ζ +1
(2.2.3) ψ(ζ)−ρ (STS)ψ(ζ +1)−λζ−2βρ (ST)Pψ( ) = 0.
χ χ
ζ
Replacing there ζ by 1 and multiplying the resulting equation by
ζ
ζ−2βρ (STS)Pρ (T−1S) gives
χ χ
1 ζ +1
ζ−2βρ (STS)Pρ (T−1S)ψ( )−ζ−2βρ (STS)Pρ (S)ψ( )−
χ χ χ χ
ζ ζ
−λρ (STS)ψ(ζ +1) = 0.
χ
Since ρ (S)P = Pρ (S)onefinds, comparing with equation(2.2.3),
χ χ
1
ψ(ζ) = λζ−2βρ (STS)Pρ (T−1S)ψ( ).
χ χ
ζ
Hence the function ψ˜ := ρ (T−1S)ψ fulfills equation (2.2.1). The same
χ
equation is then fulfilled also by the function
(2.2.4) Ψ(ζ) := Pψ˜(ζ) = P ρ (T−1S)P f(ζ −1),
χ
that is
1
(2.2.5) Ψ(ζ) = λζ−2βPρ (S)Ψ( ).
χ
ζ
Inserting finally ψ(ζ) = ρ (ST)PΨ(ζ) into equation (2.2.3) and using
χ
(2.2.1) leads to the equation
ζ
Ψ(ζ)−Pρ (T)PΨ(ζ +1)−(ζ +1)−2βPρ (T′)PΨ( ) = 0.
χ χ
ζ +1
But by assumption Pρ (T)P = ρ (T−1), hence Pρ (T′)P = ρ (T′−1)
χ χ χ χ
and therefore
ζ
(2.2.6) Ψ(ζ)−ρ (T−1)Ψ(ζ +1)−(ζ +1)−2βρ (T′−1)Ψ( ) = 0.
χ χ
ζ +1
Hence for ℜβ > 1 the first part of the proposition holds. By analytic
2
continuation in β one proves the general case.
To prove the second part we follow the arguments of Deitmar and
Hilgert in [7] (see their Lemma 4.1): if Ψ(ζ) is a solution of the Lewis
equation (2.2.2) with β ∈/ Z then Ψ has the following asymptotic ex-
pansions:
1 ∞
Ψ(ζ) ∼ ζ2βQ ( )+ C∗ζl,
ζ→0 0 ζ l
l=−1
X
∞
Ψ(ζ) ∼ Q (ζ)+ C∗′ζ−l−2β,
ζ→∞ ∞ l
l=−1
X