On the eigenvalues of the Chandrasekhar-Page angular equation

35 Pages·2005·0.47 MB·English

by  Batic D.

JOURNALOFMATHEMATICALPHYSICS46,012504(2005)
On the eigenvalues of the Chandrasekhar–Page angular
equation
Davide Batica) and Harald Schmidb)
NWFI–Mathematik,UniversitätRegensburg,D-93040Regensburg,Germany
Monika Winklmeierc)
FB3–Mathematik,UniversitätBremen,D-28359Bremen,Germany
(Received17February2004;accepted10September2004;publishedonline3January2005)
Inthispaperwestudyforagivenazimuthalquantumnumberktheeigenvaluesof
the Chandrasekhar–Page angular equation with respect to the parameters m am
“
and n av, where a is the angular momentum per unit mass of a black hole, m is
“
the rest mass of the Dirac particle and vis the energy of the particle (as measured
at infinity). For this purpose, a self-adjoint holomorphic operator family Ask;m,nd
associatedtothiseigenvalueproblemisconsidered.Atfirstweprovethatforfixed
kPR\s−1,1dthespectrumofAsk;m,ndisdiscreteandthatitseigenvaluesdepend
2 2
analytically on sm,ndPC2. Moreover, it will be shown that the eigenvalues satisfy
a first order partial differential equation with respect to mand n, whose character-
istic equations can be reduced to a Painlevé III equation. In addition, we derive a
powerseriesexpansionfortheeigenvaluesintermsofn−mandn+m,andwegive
a recurrence relation for their coefficients. Further, it will be proved that for fixed
sm,ndPC2theeigenvaluesofAsk;m,ndarethezerosofaholomorphicfunctionQ
whichisdefinedbyarelativelysimplelimitformula.Finally,wediscusstheprob-
lem if there exists a closed expression for the eigenvalues of the Chandrasekhar–
Page angular equation. © 2005 American Institute of Physics.
[DOI:10.1063/1.1818720]
I. INTRODUCTION
The angular eigenvalue problem of a spin-1 particle in the Kerr–Newman geometry is given
2
by the Chandrasekhar–Page angular equation
L+ S =sam cos u−ldS , s1d
1/2 +1/2 −1/2
L− S =sam cos u+ldS , s2d
1/2 −1/2 +1/2
see Chandrasekhar (1998, Chap. 10, Sec. 104), where the Kerr parameter a is the angular mo-
mentum per unit mass of a black hole and m is the rest mass of the Dirac particle. Moreover, the
differential operators L±1/2 are defined by
cot u k
L1±/2=]u±Qsud+ 2 , Qsud“avsin u+ sin u, uPs0,pd,
where v is the energy of the particle (as measured at infinity) and kis a half-integer, i.e.,
k=k−1 withsomekPZ.AparameterlPRiscalledaneigenvalueofthisspectralproblemifthe
2
a)Electronicmail:[email protected]
b)Electronicmail:[email protected]
c)Electronicmail:[email protected]
0022-2488/2005/46(1)/012504/35/$22.50 46,012504-1 ©2005AmericanInstituteofPhysics
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012504-2 Batic,Schmid,andWinklmeier J.Math.Phys.46,012504(2005)
systemgivenby(1)-(2)hasanontrivialsolutionwhichissquare-integrableons0,pd withrespect
to the weight function sin u. In this paper we study for fixed k the eigenvalues of the
Chandrasekhar–Page angular equation as a function of the parameters m am and n av.As a
“ “
main result, we will prove that the eigenvalues satisfy a first order quasilinear partial differential
equation, and we will derive a power series expansion for the eigenvalues in terms of n−mand
n+m.
Forthispurposeitisnecessarytoconsiderthesystem(1)-(2)inamoregeneralcontextwhere
kis real, ukuø1, and m, nare complex numbers. At first we rewrite this system for fixed k
PR\s−1,1d as 2an eigenvalue problem for some self-adjoint holomorphic operator family
2 2
A=Ask;m,nd depending on the parameters sm,ndPC2. In the special case where sm,ndPR2 the
differentialoperatorAsk;m,ndisself-adjointandhaspurelydiscretespectrum.InSec.IIweprove
that for a given kthe eigenvalues lsk;m,nd of A are holomorphic functions in sm,nd, and we
j
derive some basic estimates for them. Furthermore, we transform the system (1)-(2) to a matrix
differential equation
F G
1 1
y8sxd= B + B +C ysxd s3d
x 0 x−1 1
on the interval s0,1d with coefficient matrices
1 2 1 2
k 1 k 1 S D
− − m−l + 0
2 4 2 4 −2n −2m
B = , B = , C= ,
0 k 1 1 k 1 2m 2n
0 + m−l − −
2 4 2 4
which can be extended to the complex domain C\h0,1j. In this way we obtain a further charac-
terizationoftheeigenvaluesofAandsomeusefulestimatesforthecorrespondingeigenfunctions.
Applying analytic perturbation theory, we show in Sec. III that the eigenvalues lsk;m,nd satisfy
j
the partial differential equation
]l ]l
sm−2nld +sn−2mld +2km+2mn=0. s4d
]m ]n
In particular, this result can be used to obtain a recurrence relation for the coefficients c of a
m,n
power series expansion
‘
o
lsk;m,nd= c sn−mdmsn+mdn.
j m,n
m,n=0
In Sec. IV we solve the PDE (4) by the method of characteristics. First, we derive an explicit
formula for the eigenvalues in the case umu=unu. Moreover, in the regions where umu(cid:222)unu we
reduce the characteristic equations of (4) to a Painlevé III equation
vv8+tvv9−tsv8d2−2ksv2±1dv−tsv4−1d=0
with parameters a=±b=2kand g=−d=1 according to the notation in Milne et al. (1997) and
Mansfield and Webster (1998).As this differential equation is in general not solvable in terms of
elementary functions, we cannot expect a closed expression for the eigenvalues of the
Chandrasekhar–Page angular equation for all sm,ndPR2. However, if kis a half-integer, i.e.,
k=k−1 with some positive integer k, then a±b=2s2k−1d, and there are integrals of polynomial
2
type for the third Painlevé equation in this special case, cf. Milne et al. (1997). Hence, if
k=±1,±3,..., there exist algebraic solutions of the partial differential equation (4), and the
2 2
question arises if these explicit solutions are in fact eigenvalues of the Chandrasekhar–Page
angular equation. It turns out that there is another type of “special values” associated to the
operator A, called monodromy eigenvalues, which belong to the algebraic solutions of the PDE
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012504-3 TheChandrasekhar–Pageangularequation J.Math.Phys.46,012504(2005)
(4). For a half-integer k, the monodromy eigenvalues are introduced in Sec. V by requiring that
the system (3) has a fundamental matrix of the form
fxs1−xdg−sk/2d−1/4Hsxd
with an entire matrix function H:C!M2sCd. This property turns out to be equivalent to the
existence of special solutions of the form
fxs1−xdg−sk/2d−1/4p±sxde±2tx,
˛
where p±:C!C2arepolynomialsandt=± n2−m2.Forcomparisonpurposes,aneigenvalueofA
can be characterized by the property that (3) possesses a nontrivial solution of the form
fxs1−xdgsk/2d+1/4hsxd
with some entire vector function h:C!C2. We prove that the monodromy eigenvalues are zeros
of a polynomial with degree 2k−1 whose coefficients are polynomials in mand n. Moreover, it
can be shown that monodromy eigenvalues and “classical” eigenvalues are distinct at least in a
neighborhood of sm,nd=s0,0d. Nevertheless, they are both characterized by the fact that certain
monodromy data of the system (3) are preserved for all parameters sm,nd. In fact, l is a mono-
dromyeigenvalueofA ifandonlyifthemonodromymatricesof(3)attheregular-singularpoints
0 and1 arediagonal,whereasl isaclassicaleigenvalueofA ifandonlyifacertainnondiagonal
entry of the connection matrix for the fundamental matrices at 0 and 1 vanishes. Hence, for the
Chandrasekhar–Pageangularequationthemonodromyaswellastheclassicaleigenvalueproblem
is closely related to the isomonodromy problem for the differential equation (3). Monodromy
preserving deformations for such a system were studied by Jimbo et al. (1981), but only if the
eigenvalues of B0 and B1 do not differ by an integer, i.e., k+21„Z. In Sec. VI we consider the
isomonodromyproblemfor(3)inthecasethatkisahalf-integer.Asaconsequence,weshowthat
the monodromy eigenvalues of A satisfy the partial differential equation (4), and we obtain an
alternative derivation of (4) for the classical eigenvalues of A. Unlike the proof in Sec. III, which
relies on the particular structure of the Chandrasekhar–Page angular equation, the method pre-
sented in Sec. V is more general and based on finding suitable deformation equations for
parameter-dependent differential equations. Thus, we expect that this technique is applicable to
other eigenvalue problems as well.
II.ASELF-ADJOINT HOLOMORPHIC OPERATOR FAMILYASSOCIATED TO THE
CHANDRASEKHAR–PAGEANGULAR EQUATION
By introducing the notations
S D
˛ S sud
m am, n av, Ssud sin u +1/2 , uPs0,pd,
“ “ “ S sud
−1/2
the Chandrasekhar–Page angular equation (1)-(2) takes the form
1 2
k
S D
−mcos u − −nsin u
0 1 sin u
sASdsud S8sud+ Ssud=lSsud s5d
“ −1 0 k
− −nsin u mcos u
sin u
with fixed kPR\s−1,1d and parameters sm,ndPC2. We can associate the so-called minimal
2 2
operator A to the formal differential expression A, which acts in the Hilbert space
0
H“L2ss0,pd,C2d of square integrable vector functions with respect to the scalar product
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012504-4 Batic,Schmid,andWinklmeier J.Math.Phys.46,012504(2005)
E
p
sS ,S d S sud*S suddu, S ,S PH. s6d
1 2 “ 2 1 1 2
0
The operator A0 given by DsA0d=C0‘ss0,pd,C2d and A0S“AS for SPDsA0d is densely defined
andclosable.Forukuø1 andsm,ndPR2theformaldifferentialoperatorin(5)isinthelimitpoint
2
case at 0 and p, hence A is even essentially self-adjoint. In the following we denote the closure
0
of A by A=Ask;m,nd.According to Weidmann (1987, Theorem 5.8) the domain of Ask;0,0d is
0
given by
DsAd=hSPH : S is absolutely continuous and Ask;0,0dSPHj.
Since Ask;m,nd=Ask;0,0d+Tsm,nd with the bounded multiplication operator
S D
−mcos u −nsin u
Tsm,nd= ,
−nsin u mcos u
its domain of definition DsAd is independent of sm,ndPC2, see Kato (1966, Chap. IV, §1, Theo-
rem 1.1). Moreover, if sm,ndPR2, then Tsm,nd is a symmetric perturbation of Ask;0,0d, and
Theorem4.10inKato(1966,Chap.V,§4)yieldsthatAsk;m,ndisself-adjoint.Thus,accordingto
the classification in Kato (1966, Chap. VII, §3), Ask;m,nd forms a self-adjoint holomorphic
operator family of type (A) in the variables sm,ndPC2. Further, the spectrum of Ask;0,0d is
discrete and consists of simple eigenvalues given by
ljsk;0,0d=sgnsjdsuku− 21 +ujud, jPZ\h0j s7d
(for the details we refer to Appendix A). This means, in particular, that Ask;0,0d has compact
resolvent, and from Theorem 2.4 in Kato (1966, Chap. V, §2) it follows that Ask;m,nd has
compactresolventforallsm,ndPC2.Asaconsequence,thespectrumofAsk;m,nd,sm,ndPC2,is
discrete, and since Ask;m,nd is in the limit point case at u=0 and u=p, its spectrum consists of
simpleeigenvaluesforsm,ndPR2.Now,Theorem3.9inKato(1966,Chap.V,§3)impliesthatthe
eigenvalues lj=ljsk;m,nd, jPZ\h0j, of Ask;m,nd are simple and depend holomorphically on
sm,nd in a complex neighborhood of R2. Moreover, the partial derivatives of A with respect to m
and nare given by
S D S D
]A −cos u 0 ]A 0 −sin u
= , = ,
]m 0 cos u ]n −sin u 0
which yields the following estimates for the growth rate of the eigenvalues, compare Kato(1966,
Chap. VII, §3, Sec. 4):
I I I I
U U U U
]l ]A ]l ]A
j ł ł1, j ł ł1.
]m ]m ]n ]n
Here i·i denotes the operator norm of a s232d matrix. In addition, by Theorem 4.10 in Kato
(1966, Chap. V, §3), we have
min ul−lsk;0,0dułiTsm,ndiłmaxhumu,unuj s8d
j
jPZ\h0j
for each eigenvalue l of Ask;m,nd. Finally, by interchanging the components of Ssud, we obtain
that a point l is an eigenvalue of Ask;m,nd if and only if −l is an eigenvalue of As−k;m,−nd.
Since the eigenvalues depend holomorphically on mand n, the identity
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012504-5 TheChandrasekhar–Pageangularequation J.Math.Phys.46,012504(2005)
lsk;m,nd=−l s−k;m,−nd
j −j
holds for all sm,nd in a neighborhood of R2. Therefore, we restrict our attention to the case
kPf1,‘d. Note that lPC is an eigenvalue of Ask;m,nd if and only if the system (5) has a
2
nontrivial solution Ssud satisfying
E
p
uSsudu2 du,‘. s9d
0
By means of the transformation
1˛ 2
u
S D
tan 0
2 u
Ssud= ˛ y sin2 , uPs0,pd, s10d
u 2
0 cot
2
the differential equation (5) is equivalent to the system
F G
1 1
y8sxd= B + B +C ysxd s11d
x 0 x−1 1
on the interval s0,1d with coefficient matrices
1 2 1 2
k 1 k 1 S D
− − m−l + 0
2 4 2 4 −2n −2m
B , B , C , s12d
0“ k 1 1“ k 1 “ 2m 2n
0 + m−l − −
2 4 2 4
and the normalization condition (9) becomes
1 2
1
E 0
1 1−x
ysxd* ysxddx,‘. s13d
1
0 0
x
Ifweconsiderthedifferentialequation(11)forafixedkPs0,‘dinthecomplexplane,thenithas
two regular singular points, one at x=0 and one at x=1 with characteristic values ±fsk/2d+1g.
4
From the theory of asymptotic expansions [see Wasow (1965), for example], it follows that for
each lPC there exists a nontrivial solution
y sx,ld=xsk/2d+1/4hsx,ld, xPB s14d
0 0
of (11) in the open unit disk B0,C with center 0, where hs·,ld:B0!C2 is a holomorphic
function,
S D
o‘ m−l
hsx,ld= xnh sld, h sld . s15d
n 0 “ k+ 1
n=0 2
Here h sld is an eigenvector of B for the eigenvalue sk/2d+1, and the coefficients h sld, n.1,
0 0 4 n
are uniquely determined by the recurrence relation
sB −a−ndh sld=sB +B −C+1−a−ndh sld+Ch sld s16d
0 n 0 1 n−1 n−2
witha sk/2d+1 andh sld 0.SincethematricesB andB dependholomorphicallyonl,the
“ 4 −1 “ 0 1
coefficientshn:C!C2areholomorphicfunctions.ByslightlymodifyingtheproofofTheorem5.3
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012504-6 Batic,Schmid,andWinklmeier J.Math.Phys.46,012504(2005)
in Wasow (1965), it can be shown that the series (15) converges uniformly in every compact
subset of B03C. Thus, by a theorem of Weierstrass, h:B03C!C2 is a holomorphic vector
function in the variables sx,ld. Now, let
S D S D
1 fsld
h ,l ,
2 ‹ gsld
and we define the holomorphic function D:C!C by
Dsld“fsld2−gsld2, lPC. s17d
The following lemma provides a connection between the eigenvalues of A and the zeros of D.
Lemma 1: For fixed kPf1,‘d and sm,ndPC2, a point lPC is an eigenvalue of Ask;m,nd if
2
andonlyifl isazeroofthefunctionD givenby(17).Thisisequivalenttothestatementthatthe
differential equation (11) has a nontrivial solution of the form
ysxd=fxs1−xdgsk/2d+1/4hsxd, xPC\h0,1j, s18d
where h:C!C2 is an entire vector function. As a consequence, if S is an eigenfunction of
Ask;m,nd for some eigenvalue l, then
uSsudułC sinku, uPs0,pd, s19d
with some constant C.0.
Proof: Defining
S D
0 1
K , s20d
“ 1 0
we have K−1=K and KB K=B , KCK=−C. Hence, y is a solution of the system (11) if and only
0 1
if the function Kys1−xd satisfies (11). In particular, y sxd Ky s1−xd is a solution of (11) in the
1 “ 0
unit disk B1,C with center 1, and y1 has the form
y sx,ld=s1−xdsk/2d+1/4Khs1−x,ld, xPB .
1 1
Moreover, by the Levinson theorem, see Eastham (1989, Theorem 1.3.1), any solution of (11)
which is linearly independent of y in s0,1d behaves asymptotically like x−sk/2d−1/4fv +os1dg as
0 0
x!0, where v is an eigenvector of B for the eigenvalue −sk/2d−1. Similarly, any solution of
0 0 4
(11) which is linearly independent of y in s0,1d has the asymptotic behavior sx−1d−sk/2d−1/4
1
3fv +os1dg as x!1 with an eigenvector v of B for the eigenvalue −sk/2d−1. Now, if l is an
1 1 1 4
eigenvalue of Ask;m,nd, then the system (11) has a nontrivial solution y satisfying (13), and it
follows that ysxd=yasx,ldca holds in s0,1d with some constants caPC\h0j, aPh0,1j. Thus, y0
and y are linearly dependent, and the Wronskian Wsx,ld detsy sx,ld,y sx,ldd vanishes iden-
tically1 for all xPs0,1d. In particular, 0=Ws1,ld=2−k−1“/2Dsld.0Convers1ely, if Dsld=0, then
Ws1,ld=0, which implies that y and y are lin2early dependent. Hence, y sxd=y sxdc with some
2 0 1 0 1
constant cPC\h0j, and therefore y0 is a solution of (11) satisfying the condition (13) on the
interval s0,1d. Moreover, we immediately obtain that y has the form (18) with a holomorphic
0
vectorfunctionh:B0łB1!C2,andsince(11)isregularinC\h0,1j,wecanextendh:C!C2to
an entire function by the existence and uniqueness theorem. Finally, by means of the transforma-
tion (10), an eigenfunction S of Ask;m,nd has to be a constant multiple of
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012504-7 TheChandrasekhar–Pageangularequation J.Math.Phys.46,012504(2005)
1 2
u
S D
sin 0
2 u
sinku hsin2 , uPs0,pd,
u 2
0 cos
2
and this yields the estimate (19). h
Lemma 2: For fixed kPf21,‘d and jPZ\h0j, the jth eigenvalue ljsk;m,nd of Ask;m,nd has
a power series expansion of the form
‘
o
lsk;m,nd= l mmnn, l =lsk;0,0d, s21d
j m,n 0,0 j
m,n=0
which is uniformly convergent in the polydisc C“hsm,ndPC2:umu,unuł21j. Moreover, for all
integers m and n, the following estimate holds:
ul ułsuku+ujud2n+m. s22d
m,n
Proof:Sincethecoefficientmatricesin(11)dependholomorphicallyonsl,m,ndPC3,wecan
modifyTheorem5.3inWasow(1965)appropriatelyinordertoobtainthathin(14)andtherefore
D=Dsl,m,nd as given by (17) are holomorphic functions on C3. By a similar reasoning as in the
proof of Lemma 1, we can show that for fixed sm,ndPC2 the eigenvalues of Ask;m,nd coincide
withthezerosofthefunctionl(cid:176)Dsl,m,nd.Inparticularforthecasesm,ndPR2 thesezerosare
simple because Ask;m,nd has only simple eigenvalues. Hence, by solving the equation
Dsl,m,nd=0 and using the implicit function theorem, an eigenvalue lsk;m,nd of the operator
j
Ask;m,nd dependsholomorphicallyonsm,nd inacomplexneighborhoodofR2.Furthermore,the
estimate (8) implies that the set hlPC:minj(cid:222)0ul−ljsk;0,0duø21j contains no eigenvalues of
Ask;m,nd for all sm,ndPC. Thus there exists a holomorphic solution l:C!C of the equation
Dsl,m,nd=0, which is uniquely determined by ls0,0d=lsk;0,0d. Consequently, lsk;m,nd is
j j
holomorphicinC,andthereforeithasapowerseriesexpansioninCoftheform(21).Inaddition,
by Cauchy’s formula,
R
1 lsk;m,nd
l =− j dmdn,
m,n 4p2 mm+1nn+1
]C
and applying (8) and (7), it follows that
ulsk;m,ndułulsk;0,0du+maxhumu,unujłuku+uju,
j j
which gives the estimate (22). h
According to Lemma 1, for fixed parameters sm,ndPC2 the eigenvalues of Ask;m,nd are
exactly the zeros of the function Dsld given by (17). In principle, this result can be used for
numerical computation of the eigenvalues. However, in order to calculate Dsld at some point
lPC,wefirsthavetodeterminethecoefficientshnsldwiththehelpoftherecurrencerelation(16)
andsubsequentlyweneedtoevaluatehsx,ldatx=1 bymeansofthepowerseriesexpansion(15).
2
Unfortunately,thismethodrequiresthecalculationoftwoconsecutivelimits,makingthingsrather
complicated. In the remaining part of this section we show that there is yet another function Q
which encodes the eigenvalues of Ask;m,nd.The main advantage of Q is, that it can be obtained
by only one limit process.
By setting ysxd xas1−xd1−ayˆsxd with a sk/2d+1, the system (11) becomes
“ “ 4
F G
1 1
yˆ8sxd= Bˆ + Bˆ +C yˆsxd s23d
x 0 x−1 1
with the coefficient matrices
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012504-8 Batic,Schmid,andWinklmeier J.Math.Phys.46,012504(2005)
S D S D S D
−k− 1 m−l k− 1 0 −2n −2m
Bˆ 2 , Bˆ 2 , C= .
0“ 0 0 1“ m−l −1 2m 2n
Now, there exists a holomorphic solution of (23) in B given by
1
S D
o‘ m−l
yˆsx,ld= xnd sld, d sld , s24d
n 0 “ k+ 1
n=0 2
where d sld is an eigenvector of Bˆ for the eigenvalue 0. In addition, the coefficients d sld, n
0 0 n
.1, are uniquely determined by the recurrence relation
d sld=sBˆ −nd−1fsE−ndd sld+Cd sldg
n 0 n−1 n−2
with
S D
2n 3m−l
E , d sld 0.
“ −m−l −2n −1 “
Finally, we denote by Q sld the second component of d sld.
n n
Lemma 3: Let kPf1,‘d and sm,ndPC2 be fixed. Then, for each lPC, the limit
2
Qsld lim Q sld s25d
“ n
n!‘
exists, and Q:C!C is a holomorphic function. Moreover, a point lPC is an eigenvalue of
Ask;m,nd if and only if Qsld=0.
Proof: For fixed lPC, the differential equation (23) has a regular singular point at x=1 with
characteristic values −1 and k−1. First, let us assume that their difference k+1 is not an integer.
2 2
In this case the system (23) has a fundamental system of solutions in a complex neighborhood of
x=1, which can be written as
‘ ‘
o o
yˆ sx,ld=s1−xd−1 s1−xdnd1sld, yˆ sx,ld=s1−xdk−1/2 s1−xdnd2sld, s26d
1 n 2 n
n=0 n=0
where
S D S D
0 k+ 1
d1sld= e , d2sld= 2
0 1 ‹ 2 0 m−l
are eigenvectors of Bˆ for the eigenvalues −1 and k−1, respectively. Now, yˆ can be written as a
1 2
linear combination
yˆsx,ld=gsldyˆ sx,ld+gsldyˆ sx,ld
1 1 2 2
with connection coefficients g1sld,g2sldPC. Applying Corollary 1.6 in Schäfke and Schmidt
(1980) to the system (23) gives
lim d sld=gslde , s27d
n 1 2
n!‘
and therefore the limit (25) exists. Furthermore, l is an eigenvalue of Ask;m,nd if and only if
gsld=0, i.e., if and only if Qsld becomes zero. Finally, it can be shown that the functions d
1 n
converge uniformly in every compact subset of C, and Weierstrass’theorem implies that Q is an
entire function.
Now, suppose that k k+1 is a positive integer. In this case, a fundamental system of the
“ 2
form(26)maynotexist.Nevertheless,itcanbeproved(seeLemma6inSec.VI)thatthesystem
(23) has a fundamental matrix,
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012504-9 TheChandrasekhar–Pageangularequation J.Math.Phys.46,012504(2005)
‘
o
Yˆsx,ld=Gsld H slds1−xdns1−xdDs1−xdJsld,
n
n=0
in a complex neighborhood of x=1, where D diags−1,k−1d, H sld=I, and
“ 0
S D S D
0 k+ 1 0 0
Gsld= 2 , Jsld=
1 m−l qsld 0
with some qsldPC. In particular, we can write Yˆ in the form
‘
Yˆsx,ld=Hˆsx,lds1−xd˜Jsld, Hˆsx,ld=os1−xdnD sld,
n
n=0
where
S D S D
0 0 −1 0
D sld= , ˜Jsld= .
0 1 0 qsld −1
Since yˆ solves the system (23), there exists a vector csldPC2 such that yˆsx,ld=Yˆsx,ldcsld, and
Theorem 1.1 in Schäfke (1980) implies
1 1
d sld=D sld s−˜JslddGsn+1d sn−˜Jslddcsld+Osnd−1d s28d
n 0 G G
forarbitraryd.0.Forthedefinitionanddiscussionofthereciprocalgammafunctionformatrices
we refer to theAppendix in Schäfke (1980). Particularly, for the Jordan-type matrices −˜Jsld and
n−˜Jsld we obtain
1 2
S D 1
0
1 1 0 1 Gsn+1d
s−˜Jsldd= , sn−˜Jsldd= .
G * 1 G 1
*
Gsn+1d
Now,ifgslddenotesthefirstcomponentofcsld,then(28)implies(27).Sincelisaneigenvalue
1
of Ask;m,nd if and only if gsld=0, the proof of Lemma 3 is complete. h
1
III.APARTIALDIFFERENTIALEQUATION FOR THE EIGENVALUES
Theorem 1: For fixed kPf21,‘d and jPZ\h0j, the jth eigenvalue l=ljsk;m,nd of A is an
analyticalfunctioninsm,ndPR2satisfyingthefirstorderquasilinearpartialdifferentialequation
]l ]l
sm−2nld +sn−2mld +2km+2mn=0, s29d
]m ]n
where lsk;0,0d is given by (7).
j
Proof: Let
S D
S sud
Ssud 1 , uPs0,pd,
‹ S sud
2
be that eigenfunction of Ask;m,nd for the eigenvalue l=lsk;m,nd which is normalized by the
j
condition sS,Sd=1. Introducing the functions
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012504-10 Batic,Schmid,andWinklmeier J.Math.Phys.46,012504(2005)
Usud S sud2+S sud2, Vsud S sud2−S sud2, Wsud 2S sudS sud,
“ 1 2 “ 2 1 “ 1 2
a straightforward calculation shows that U, V, and W are solutions of the system of differential
equations
S D
k
U8sud=2 nsin u+ Vsud+2mcos uWsud, s30d
sin u
S D
k
V8sud=2 nsin u+ Usud+2lWsud, s31d
sin u
W8sud=2mcos uUsud−2lVsud. s32d
Now, from analytic perturbation theory, compare Kato (1966, Chap. VII, §3, Sec. 4), it follows
that
S D E S D E
]l ]A p −cos u 0 p
= S,S = Ssud* Ssuddu= cos uVsuddu, s33d
]m ]m 0 cos u
0 0
S D E S D E
]l ]A p 0 −sin u p
= S,S = Ssud* Ssuddu=− sin uWsuddu. s34d
]n ]n −sin u 0
0 0
In addition, from (19) we obtain the estimates
uUsudu,uVsudu,uWsudułC sin2ku
with some constant C.0. Since kis positive, U, V, and W vanish at u=0 and u=p. If we
integrate (33) by parts and replace V8sud with the right-hand side (rhs) of (31), then we get
E E
]l p p
=− sin uV8suddu=− s2nsin2 u+2kdUsud+2l sin uWsuddu
]m
0 0
E E E
p p p
=−s2n+2kd Usuddu−2l sin uWsuddu+2n cos2 uUsuddu.
0 0 0
Taking into account that
E E
p p ]l
Usuddu=sS,Sd=1, sin uWsuddu=− ,
]n
0 0
we have
E
]l ]l p
m =−ms2n+2kd+2ml +2mn cos2 uUsuddu. s35d
]m ]n
0
Moreover, Eq. (32) implies
2mcos2 uUsud=cos uW8sud+2l cos uVsud,
and integration by parts gives
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