Volume 2011, Article ID 743135,27pages doi:10.1155/2011/743135
Research Article
Hierarchies of Difference Boundary Value Problems
Sonja Currie and Anne D. Love
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Correspondence should be addressed to Sonja Currie,sonja.currie@wits.ac.za Received 25 November 2010; Accepted 11 January 2011
Academic Editor: Olimpio Miyagaki
Copyrightq2011 S. Currie and A. D. Love. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper generalises the work done in Currie and Love2010, where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with various combinations of Dirichlet, non-Dirichlet, and affineλ-dependent boundary conditions at the end points, whereλis the eigenparameter. We now consider generalλ-dependent boundary conditions. In particular we show, using one of the Crum-type transformations, that it is possible to go up and down a hierarchy of boundary value problems keeping the form of the second- order difference equation constant but possibly increasing or decreasing the dependence onλof the boundary conditions at each step. In addition, we show that the transformed boundary value problem either gains or loses an eigenvalue, or the number of eigenvalues remains the same as we step up or down the hierarchy.
1. Introduction
Our interest in this topic arose from the work done on transformations and factorisations of continuous Sturm-Liouville boundary value problems by Binding et al. 1and Browne and Nillsen 2, notably. We make use of analogous ideas to those discussed in 3–5 to study difference equations in order to contribute to the development of the theory of discrete spectral problems.
Numerous efforts to develop hierarchies exist in the literature, however, they are not specifically aimed at difference equations per se and generally not for three-term recurrence relations. Ding et al.,6, derived a hierarchy of nonlinear differential-difference equations by starting with a two-parameter discrete spectral problem, as did Luo and Fan7, whose hierarchy possessed bi-Hamiltonian structures. Clarkson et al.’s,8, interest in hierarchies lay in the derivation of infinite sequences of systems of difference equations by using the B¨acklund transformation for the equations in the second Painleve’ equation hierarchy.
Wu and Geng,9, showed early on that the hierarchy of differential-difference equations possesses Hamiltonian structures while a Darboux transformation for the discrete spectral problem is shown to exist.
In this paper, we consider a weighted second-order difference equation of the form
cnyn1−bnyn cn−1yn−1 −cnλyn, 1.1
wherecn>0 represents a weight function andbna potential function.
Our aim is to extend the results obtained in 10, 11 by establishing a hierarchy of difference boundary value problems. A key tool in our analysis will be the Crum-type transformation 2.1. In 10, it was shown that 2.1 leaves the form of the difference equation 1.1 unchanged. For us, the effect of 2.1 on the boundary conditions will be crucial. We considerλeigenparameter-dependent boundary conditions at the end points. In particular, the eigenparameter dependence at the initial end point will be given by a positive Nevanlinna function, Nλ say, and at the terminal end point by a negative Nevanlinna function,Mλsay. The case ofNλ Mλ 0 was covered in10and the the case of Nλ Mλ constant was studied in11. Applying transformation2.1to the boundary conditions results in a so-called transformed boundary value problem, where either the new boundary conditions have moreλ-dependence, lessλ-dependence, or the same amount of λ-dependence as the original boundary conditions. Consequently the transformed boundary value problem has either one more eigenvalue, one less eigenvalue, or the same number of eigenvalues as the original boundary value problem. Thus, it is possible to construct a chain, or hierarchy, of difference boundary value problems where the successive links in the chain are obtained by applying the variations of2.1given in this paper. For instance, it is possible to go from a boundary value problem withλ-dependent boundary conditions to a boundary value problem withλ-independent boundary conditions or vice versa simply by applying the correct variation of2.1an appropriate number of times. Moreover, at each step, we can precisely track the eigenvalues that have been lost or gained. Hence, this paper provides a significant development in the theory of three-term difference boundary value problems in regard to singularities and asymptotics in the hierarchy structure. For similar results in the continuous case, see12.
There is an obvious connection between the three-term difference equation and orthogonal polynomials. In fact, the three-term recurrence relation satisfied by orthogonal polynomials is perhaps the most important information for the constructive and computa- tional use of orthogonal polynomials13.
Difference equations and operators and results concerning their existence and construction of their solutions have been discussed in14, 15. Difference equations arise in numerous settings and have applications in diverse areas such as quantum field theory, combinatorics, mathematical physics and biology, dynamical systems, economics, statistics, electrical circuit analysis, computer visualization, and many other fields. They are especially useful where recursive computations are required. In particular see 16 9, Introduction for three physical applications of the difference equation1.1, namely, the vibrating string, electrical network theory and Markov processes, in birth and death processes and random walks.
It should be noted that G. Teschl’s work,17, Chapter 11, on spectral and inverse spectral theory of Jacobi operators, provides an alternative factorisation, to that of10, of a second-order difference equation, where the factors are adjoints of one another.
This paper is structured as follows.
InSection 2, all the necsessary results from10are recalled, in particular how1.1 transforms under2.1. In addition, we also recap some important properties of Nevanlinna functions.
The focus of Section 3 is to show exactly the effect that 2.1 has on boundary conditions of the form
y−1 Nλy0, ym−1 Mλym. 1.2
We give explicitly the new boundary conditions which are obeyed, from which it can be seen whether theλ-dependence has increased, decreased, or remained the same.
Lastly, inSection 4, we compare the spectrum of the original boundary value problem with that of the transformed boundary value problem and show under which conditions the transformed boundary value problem has one more eigenvalue, one less eigenvalue, or the same number of eigenvalues as the original boundary value problem.
2. Preliminaries
In10, we considered1.1forn 0, . . . , m−1, where the values ofy−1andymare given by boundary conditions, that is,ynis defined forn −1, . . . , m.
Let the mappingy→wbe defined by wn: yn−yn−1 zn
zn−1, n 0, . . . , m, 2.1
where, throughout this paper,znis a solution to1.1for λ λ0 such thatzn > 0 for alln −1, . . . , m. Whether or notznobeys the various given boundary conditionsto be specified lateris of vital importance in obtaining the results that follow.
From10, we have the following theorem.
Theorem 2.1. Under the mapping2.1,1.1transforms to
cwnwn1−bwnwn cwn−1wn−1 −λcwnwn, 2.2 where forn 0, . . . , m
cwn cn−1zn−1
zn ,
bwn
cn−1zn−1
cnzn zn
zn−1
cn−1zn−1
zn .
2.3
We now recall some properties of Nevanlinna functions.
IThe inverse of a positive Nevanlinna function is a negative Nevanlinna function, that is
1
Nλ −Bλ, 2.4
whereNλ, Bλare positive Nevanlinna functions. This follows directly from the fact that Iz≥0 if and only ifI−1/z≥0.
IIIf
Nλ b−s
j 1
cj
λ−dj
, cj>0, b /0, 2.5
then
1
Nλ β−s
j 1
σj
λ−δj, σj>0, β /0. 2.6
This follows byItogether with the fact that sinceNλhasszeros 1/Nλhasspoles. Also Nλ → basλ → ±∞so 1/Nλ → 1/b : βas λ → ±∞. Thus, ifNλis a positive Nevanlinna function of the form 2.5, then for b /0, 1/Nλ is a negative Nevanlinna function of the same form.
IIIIf
Nλ aλb−s
j 1
cj
λ−dj, aj, cj>0, 2.7
then
1
Nλ −s1
j 1
σj
λ−δj, σj>0, 2.8
sinceNλhass1 zeros so 1/Nλhass1 poles andNλ → aλb → ±∞asλ → ±∞
so 1/Nλ → 1/aλb → 0 asλ → ±∞.
For the remainder of the paper,Ns,j λwill denote a Nevanlinna function where sis the number of terms in the sum;
jindicates the value ofnat which the boundary condition is imposed and
⎧⎨
⎩
± if the coefficient ofλis positive or negative respectively,
0 if the coefficient ofλis zero. 2.9
3. General λ-Dependent Boundary Conditions
In this section, we show how y obeying general λ-dependent boundary conditions transforms, under2.1, to w obeying various types ofλ-dependent boundary conditions.
The exact form of these boundary conditions is obtained by considering the number of zeros and poles singularities of the various Nevanlinna functions under discussion and these correlations are illustrated in the different graphs depicted in this section.
Lemma 3.1. Ifyobeys the boundary condition
y−1
b−s
k 1
ck
λ−dk y0: R0s,−1λy0, 3.1
then the domain ofwnmay be extended fromn 0, . . . , mton −1, . . . , mby forcing the condition w−1
w0 U, 3.2
where
U bw0−λcw0
cw−1 − cw0 cw−1
b0/c0−λ−z1/z0−c−1/c0R0s,−1λ
1−R0s,−1λz0/z−1 3.3
withcw−1 c−1.
Proof. The transformed equation2.2, forn 0, together with3.2gives
cw0w1 cw−1Uw0 bw0−λcw0w0. 3.4
Also the mapping2.1, together with3.1, yields
w0 y0
1−R0s,−1λ z0 z−1
. 3.5
Substituting3.5into3.4, we obtain
cw0w1 cw−1U
1−R0s,−1λ z0 z−1
y0 bw0−λcw0
1−R0s,−1λ z0 z−1
y0.
3.6
Now2.1, withn 1, gives
w1 y1−y0z1
z0 3.7
which when substituted into3.6and dividing through bycw0results in
y1−y0z1
z0cw−1 cw0 U
1−R0s,−1λ z0 z−1
y0
bw0 cw0 −λ
1−R0s,−1λ z0 z−1
y0.
3.8
This may be rewritten as
y1−y0 z1
z0−
cw−1
cw0 U−bw0 cw0
1−R0s,−1λ z0 z−1
−λ
1−R0s,−1λ z0 z−1
y0.
3.9
Using1.1, withn 0, together with3.1, gives
y1− b0
c0−c−1
c0 R0s,−1λ
y0 −λy0. 3.10
Subtracting3.10from3.9results in
y0 b0
c0 −c−1
c0 R0s,−1λ− z1
z0
cw−1
cw0 U−bw0 cw0
1−R0s,−1λ z0 z−1
y0
−λ
1−R0s,−1λ z0 z−1
λ
.
3.11
Rearranging the above equation and dividing through by1−R0s,−1λz0/z−1cw−1/
cw0yields
cw0/cw−1
b0/c0−c−1/c0R0s,−1λ−z1/z0−λ 1−R0s,−1λz0/z−1
U− bw0
cw−1 −λ cw0 cw−1
3.12
and hence
U bw0−λcw0
cw−1 − cw0 cw−1
b0/c0−λ−z1/z0−c−1/c0R0s,−1λ
1−R0s,−1λz0/z−1 . 3.13
Thuswobeys the equation on the extended domain.
The remainder of this section illustrates why it is so important to distinguish between the two cases ofzobeying or not obeying the boundary conditions.
Theorem 3.2. Consider yn obeying the boundary condition 3.1 where R0s,−1λ is a positive Nevanlinna function, that is, ck > 0 fork 1, . . . , s. Under the mapping2.1,y obeying3.1 transforms towobeying3.2as follows.
AIfzdoes not obey3.1thenwobeys
i
w−1 Uw0
β−s
t 1
γt
λ−qt w0: Ts,−10 λw0, b 0, 3.14
ii
w−1 Uw0
αλβ−s
t 1
γt
λ−qt w0: Ts,−1 λw0, z−1
z0 > b >0.
3.15
BIfzdoes obey3.1forλ λ0thenwobeys
i
w−1 Uw0
β−s−1
t 1
γt
λ−vt w0: Ts−1,−10 λw0, b 0, 3.16
ii
w−1 Uw0
αλβ−s−1
t 1
γt
λ−vt w0: Ts−1,−1 λw0, z−1
z0 > b >0,
3.17
where γt,γt, α,α > 0, that is,Ts,−10 λ, Ts,−1 λ, Ts−1,−10 λ, Ts−1,−1 λare positive Nevanlinna functions.
In (A) and (B),b <0 is not possible.
Proof. The fact thatw−1 Uw0is by construction, seeLemma 3.1. We now examine the form of U in Lemma 3.1. Let Γ1 : bw0/cw−1,Γ2 : cw0/cw−1, Γ3 : b0/c0− z1/z0andΓ4: c−1/c0then
w−1
w0 U Γ1−λΓ2−Γ2
Γ3−λ−Γ4R0s,−1λ 1−z0/z−1R0s,−1λ
Γ1−λΓ2−Γ2
Γ4z−1
z0 Γ3−λ−Γ4z−1/z0 1−z0/z−1R0s,−1λ
Γ1−λΓ2−Γ2Γ4z−1
z0 Γ2z−1/z0λ−Γ3 Γ4z−1/z0 z−1/z0−R0s,−1λ .
3.18
But
Γ3−Γ4z−1 z0
b0 c0 −z1
z0 −c−1
c0 λ0 3.19
thus
w−1
w0 U Γ1−λΓ2−Γ2Γ4z−1
z0 Γ2 z−1/z0λ−λ0
z−1/z0−R0s,−1λ. 3.20
Nowλ−λ0/z−1/z0−R0s,−1λhas the expansion
fλ− p
t 1
rt
λ−qt, 3.21
wherert>0 and theqt’s correspond to wherez−1/z0 R0s,−1λ, that is, the singularities of3.20.
SinceR0s,−1λis a positive Nevanlinna function it has a graph of the form shown in Figure 1.
Clearly, the gradient ofR0s,−1λatqtis positive for allt, that is,
∂
∂λR0s,−1λ
qt
>0, t 1, . . . , p. 3.22
R0s,−1λ
z−1 z0
b
q1 q2 q3
d1 d2 d3
λ
Figure 1:R0s,−1λ.
Ifzdoes not obey3.1, then the zeros of
λ−λ0
z−1/z0−R0s,−1λ 3.23 are the poles ofR0s,−1λ, that is, thedk’s andλ λ0wheredk/λ0fork 1, . . . , s. It is evident, fromFigure 1, that the number ofqt’s is equal to the number ofdk’s, thus in3.21,p s.
We now examine the form offλin3.21. Asλ → ±∞it follows thatR0s,−1λ → b.
Thus
λ−λ0
z−1/z0−R0s,−1λ −→ λ−λ0
z−1/z0−b. 3.24
Therefore
fλ λ−λ0
z−1/z0−b. 3.25
Hence, substituting into3.20gives w−1
w0 U Γ1−λΓ2−Γ2Γ4z−1
z0 Γ2z−1 z0
fλ−s
t 1
rt λ−qt
Γ1−λΓ2−Γ2Γ4z−1
z0 Γ2z−1 z0
λ−λ0
z−1/z0−b−s
t 1
rt
λ−qt Γ1−Γ2Γ4z−1
z0 − λ0Γ2
1−bz0/z−1λ
−Γ2 Γ2
1−bz0/z−1
−Γ2z−1 z0
s t 1
rt
λ−qt.
3.26
Let
β: Γ1−Γ2Γ4z−1
z0 − λ0Γ2
1−bz0/z−1, α: −Γ2 Γ2
1−bz0/z−1
Γ2b z−1/z0−b, γt: Γ2z−1
z0 rt.
3.27
Then sinceΓ2 >0,z−1/z0>0 andrt>0 we have thatγt>0 and clearly ifb 0 thenα 0 giving3.14, that is,
w−1 Uw0
β−s
t 1
γt
λ−qt w0: Ts,−10 λw0. 3.28
Ifb /0 then we wantα >0 so that we have a positive Nevanlinna function, that is Γ2b
z−1/z0−b >0 3.29 which means that either,
Γ2b >0, z−1
z0 −b >0, 3.30
giving that, sinceΓ2>0,
b >0, z−1
z0 > b, 3.31
which is as shown inFigure 1, or,
Γ2b <0, z−1
z0 −b <0, 3.32
giving that
b <0, z−1
z0 < b, 3.33
but this means thatz−1/z0<0 which is not possible.
Thus,α >0 forz−1/z0> b >0, that is, givenb, the ratioz−1/z0must be chosen suitably to ensure thatTs,−1 λis a positive Nevanlinna function as required. Hence we obtain 3.15, that is
w−1 Uw0
αλβ−s
t 1
γt
λ−qt w0: Ts,−1 λw0. 3.34
If zobeys 3.1, for λ λ0, then z−1/z0 R0s,−1λ0. Thus in Figure 1, one of theqt’s t 1, . . . , sis equal toλ0 and sinceλ0is less than the least eigenvalue of the boundary value problem1.1,3.1together with a boundary condition atm−1specified laterit follows thatq1 λ0, asλ0< dkfor allk 1, . . . , s.
Now λ−λ0
z−1/z0−R0s,−1λ
λ−λ0 R0s,−1λ0−R0s,−1λ
1
R0s,−1λ0−R0s,−1λ
/λ−λ0 3.35
and asλ → λ0
1
R0s,−1λ0−R0s,−1λ
/λ−λ0 −→ − ∂
∂λR0s,−1λ
λ0
<0. 3.36
Thusλ λ0 q1is a removable singularity. Alternatively, λ−λ0
R0s,−1λ0−R0s,−1λ
λ−λ0 b−s
k 1ck/λ0−dk−bs
k 1ck/λ−dk s −1
k 1ck/λ0−dkλ−dk,
3.37
which illustrates that the singularity atλ λ0 q1is removable.
We now have that the number of nonremovable singularities,qt, in3.20is one less than the number ofdk’sk 1, . . . , s, seeFigure 1. Thus3.21becomes
fλ−s
t 2
rt
λ−qt, rt>0 3.38
which may be rewritten as
fλ−s−1
t 1
rt λ−vt
, rt>0, 3.39
wherevn qn1,rn rn1forn 1, . . . , s−1.
We now examine the form offλin 3.39. As λ → ±∞, we have that, as before, R0s,−1λ → b. Thus
fλ λ−λ0
R0s,−1λ0−b. 3.40
Hence, from3.20, w−1
w0 U Γ1−λΓ2−Γ2Γ4R0s,−1λ0 Γ2R0s,−1λ0
fλ−s−1
t 1
rt λ−vt
Γ1−λΓ2−Γ2Γ4R0s,−1λ0 Γ2R0s,−1λ0
λ−λ0
R0s,−1λ0−b−s−1
t 1
rt
λ−vt
Γ1−Γ2Γ4R0s,−1λ0− λ0Γ2
1−b
1/R0s,−1λ0λ
⎡
⎢⎣−Γ2 Γ2
1−b
1/R0s,−1λ0
⎤
⎥⎦
−Γ2R0s,−1λ0s−1
t 1
rt
λ−vt.
3.41
Let
β: Γ1−Γ2Γ4R0s,−1λ0− λ0Γ2
1−b
1/R0s,−1λ0,
α: −Γ2 Γ2
1−b
1/R0s,−1λ0 Γ2b R0s,−1λ0−b,
γt: Γ2R0s,−1λ0rt.
3.42
Then sinceΓ2 >0,R0s,−1λ0>0 andrt>0 we have thatγt>0 and clearly ifb 0 thenα 0 giving3.16, that is,
w−1 Uw0
β−s−1
t 1
γt
λ−vt w0: Ts−1,−10 λw0. 3.43
Ifb /0 then we needα > 0 so that we have a positive Nevanlinna function, that is Γ2b
R0s,−1λ0−b>0 3.44
which means that either
Γ2b >0, R0s,−1λ0−b >0, 3.45
giving that, sinceΓ2>0,
b >0, R0s,−1λ0 z−1
z0 > b, 3.46
which is as shown inFigure 1, or,
Γ2b <0, R0s,−1λ0−b <0, 3.47
giving that
b <0, R0s,−1λ0< b, 3.48
but this means thatR0s,−1λ0 z−1/z0<0 which is not possible.
Thus,α > 0 forR0s,−1λ0> b >0, that is, givenb, the ratioz−1/z0 R0s,−1λ0must be chosen suitably to ensure that Ts−1,−1 λis a positive Nevanlinna function as required.
Hence, we obtain3.17, that is,
w−1 Uw0
αλβ−s−1
t 1
γt
λ−vt w0: Ts−1,−1 λw0. 3.49
In the theorem below, we increase theλdependence by introducing a nonzeroλterm in the original boundary condition. As inTheorem 3.2, theλdependence of the transformed boundary condition depends on whether or notzobeys the given boundary condition. In addition, to ensure that theλ dependence of the transformed boundary condition is given by a positive Nevanlinna function it is necessary that the transformed boundary condition is imposed at 0 and 1 as opposed to−1 and 0. Thus the interval under consideration shrinks by one unit at the initial end point. By routine calculation it can be shown that the form of theλdependence of the transformed boundary condition, if imposed at−1 and 0, is neither a positive Nevalinna function nor a negative Nevanlinna function.
Theorem 3.3. Considerynobeying the boundary condition
y−1
aλb−s
k 1
ck
λ−dk y0: Rs,−1λy0, 3.50
whereRs,−1λis a positive Nevanlinna function, that is,a >0 andck>0 fork 1, . . . , s. Under the mapping2.1,yobeying3.50transforms towobeying the following.
1 Ifzdoes not obey3.50thenwobeys
w0
β−s1
t 1
γt
λ−δt
w1: Ts1,00 λw1. 3.51
2 Ifzdoes obey3.50, forλ λ0, thenwobeys
w0
β−s
t 1
γt λ−δt
w1: T0s,0λw1, 3.52
whereγt, γt>0.
Proof. Since w0 and w1 are defined we do not need to extend the domain in order to impose the boundary conditions3.51or3.52.
The mapping2.1, atn 0, together with3.50gives
w0 y0
1−Rs,−1λ z0 z−1
. 3.53
Also2.1, atn 1, is
w1 y1−y0z1
z0. 3.54
Substituting in fory1from1.1, withn 1, and using3.50, we obtain that
w1 y0
b0 c0 −z1
z0−λ− c−1
c0 Rs,−1λ
. 3.55
From3.53and3.55, it now follows that w0
w1
1−Rs,−1λz0/z−1
b0/c0−z1/z0−λ−c−1/c0Rs,−1λ. 3.56
As inTheorem 3.2, letΓ3 b0/c0−z1/z0andΓ4 c−1/c0. Then3.56becomes w0
w1
1−Rs,−1λz0/z−1 Γ3−λ−Γ4Rs,−1λ
z0/z−1 Γ3−λ−Γ4Rs,−1λ
/
z−1/z0−Rs,−1λ
z0/z−1 Γ4−−Γ3λ Γ4z−1/z0/
z−1/z0−Rs,−1λ.
3.57
Rs,−1λ
z−1 z0
q1 q2 q3
q4
d1 d2 d3
aλb
λ
Figure 2:Rs,−1λ.
FromTheorem 3.2, we have thatΓ3−Γ4z−1/z0 λ0so
w0 w1
z0 z−1
⎡
⎢⎣ 1 Γ4−λ−λ0/
z−1/z0−Rs,−1λ
⎤
⎥⎦. 3.58
Also, as inTheorem 3.2,
λ−λ0
z−1/z0−Rs,−1λ, 3.59
has the expansion
fλ − p
t 1
rt
λ−qt, rt>0, 3.60
whereqtcorresponds toz−1/z0 Rs,−1λ, that is, the singularities of3.59. NowRs,−1λ is a positive Nevanlinna function with graph given inFigure 2.
Clearly, the gradient ofRs,−1λatqt is positive for allt 1, . . . , p, that is,
∂
∂λRs,−1λ
qt
>0, t 1, . . . , p. 3.61
Ifzdoes not obey3.50then the zeros of
λ−λ0
z−1/z0−Rs,−1λ 3.62
are the poles ofRs,−1λ, that is, thedk’s andλ λ0wheredk/λ0fork 1, . . . , s. It is evident, fromFigure 2, that the number of qt’s is one more than the number of dk’s, thus in 3.60, p s1.
We now examine the form offλ in3.60. Asλ → ±∞it follows thatRs,−1λ → aλb, thus
λ−λ0
z−1/z0−Rs,−1λ −→ λ−λ0
z−1/z0−aλb −→ −1
a. 3.63
Hence,fλ −1/a.
Using3.58we now obtain
w0 w1
z0 z−1
⎡
⎢⎣ 1 Γ4−
fλ −s1
t 1
rt/
λ−qt
⎤
⎥⎦
z0 z−1
1 Γ41/as1
t 1
rt/ λ−qt
1
z−1/z0Γ4 z−1/z01/a s1
t 1
rtz−1/z0/
λ−qt
.
3.64
Note thatrtz−1/z0>0. Let
Δ: z−1
z0 Γ4z−1 z0
1
a, 3.65
then
w0 w1
1 Δ−s1
t 1
−rtz−1/z0/
λ−qt
. 3.66
NowΔ/0 since ifΔ 0 thenΓ −1/a, that is,c0/c−1 −abuta >0 andc0/c−1>0 so this is not possible. Therefore bySection 2, Nevanlinna resultII, we have that
w0
β−s1
t 1
γt
λ−δt w1: Ts1,00 λw1, 3.67 that is,3.51holds.
Ifzdoes obey3.50forλ λ0 thenz−1/z0 Rs,−1λ0. Thus, inFigure 2, one of theqt’s,t 1, . . . , pis equal toλ0and sinceλ0is less than the least eigenvalue of the boundary value problem1.1,3.50together with a boundary condition atm−1specified laterit follows thatq1 λ0, asλ0< dkfor allk 1, . . . , s.
Now3.59can be written as λ−λ0 Rs,−1λ0−Rs,−1λ
1
Rs,−1λ0−Rs,−1λ
/λ−λ0 3.68
and asλ → λ0 1
Rs,−1λ0−Rs,−1λ
/λ−λ0 −→ − ∂
∂λRs,−1λ
λ0
<0. 3.69
Thusλ λ0 q1is a removable singularity. Alternatively, we could substitute in forRs,−1λ andRs,−1λ0to illustrate that the singularity atλ λ0 q1is removable, seeTheorem 3.2.
Hence the number of nonremovableqt’s is the same as the number ofdk’s, seeFigure 2. So 3.60becomes
fλ −s1
t 2
rt λ−qt
, rt>0, 3.70
which may be rewritten as
fλ −s
t 1
rt
λ−qt, rt>0, 3.71
wherern rn1andqn qn1forn 1, . . . , s.
We now examine the form offλin3.70. Asλ → ±∞, we have that Rs,−1λ → aλb, thus
λ−λ0
Rs,−1λ0−Rs,−1λ −→ λ−λ0
Rs,−1λ0−aλb −→ −1
a. 3.72
Hence,fλ −1/a. So, from3.58withz−1/z0 Rs,−1λ0, we obtain w0
w1
1 Rs,−1λ0
Γ41/as
t 1 rt/
λ−qt 1
Δ−s
t 1
−rtRs,−1λ0/
λ−qt,
3.73
where, as before,
Δ: z−1
z0 Γ4z−1 z0
1
a Rs,−1λ0Γ4Rs,−1λ01
a/0. 3.74
Thus, bySection 2, Nevanlinna resultII, we have that
w0
β−s
t 1
γt
λ−δt w1: T0s,0λw1, γt>0, 3.75 that is,3.52holds.
InTheorem 3.4, we impose a boundary condition at the terminal end point and show how it is transformed according to whether or notzobeys the given boundary condition.
Theorem 3.4. Consideryobeying the boundary condition atn mgiven by
ym−1 ym
gλh−l
k 1
sk
λ−pk : ymR−l,mλ, 3.76
whereR−l,mλis a negative Nevanlinna function, that is,g <0 andsk<0 fork 1, . . . , l. Under the mapping2.1,yobeying3.76transforms towobeying the following.
I Ifzdoes not obey3.76thenwobeys
wm−1 wm
φλϕ−l1
t 1
t
λ−σt : wmTl1,m− λ. 3.77 IIIfzdoes obey3.76thenwobeys
wm−1 wm
φλ ϕ−l
t 1
t
λ−σt : wmTl,m− λ, 3.78
whereφ,φ, k,k<0.
Proof. Sincewm−1andwmare defined we do not need to extend the domain ofw in order to impose the boundary conditions3.77or3.78.
The mapping2.1, atn m−1, gives
wm−1 ym−1−ym−2zm−1
zm−2. 3.79
From1.1, withn m−1, we can substitute in forym−2in the above equation to get
wm−1 ym−1
1λzm−1cm−1
zm−2cm−2−zm−1bm−1 zm−2cm−2
zm−1cm−1 zm−2cm−2ym.
3.80
Using3.76, we obtain
wm−1 ym
R−l,mλ
1λzm−1cm−1
zm−2cm−2−zm−1bm−1 zm−2cm−2
zm−1cm−1 zm−2cm−2
. 3.81
Butzobeys1.1atn m−1, forλ λ0, so that3.81becomes
wm−1 ymzm−1cm−1 zm−2cm−2
R−l,mλλ−λ0−R−l,mλ zm zm−11
. 3.82
Also, forn m,2.1together with3.76yields
wm ym
1−R−l,mλ zm zm−1
. 3.83
Therefore,
wm−1 wm
zm−1cm−1 zm−2cm−2
λ−λ0R−l,mλ
1−R−l,mλzm/zm−11 . 3.84
LetΩ: zm−1cm−1/zm−2cm−2>0, then3.84may be rewritten as
wm−1
wm Ω−Ω
λ−λ0
zm/zm−1−1/R−l,mλ . 3.85
1 R−l,mλ
zm zm−1
q1 q2 q3
p1 p2 p3
λ
Figure 3: 1/R−l,mλ.
BySection 2, Nevanlinna resultI, sinceR−l,mλis a negative Nevanlinna function it follows that 1/R−l,mλis a positive Nevanlinna function, which has the form
1
R−l,mλ −l1
k 1
sk
λ−pk, sk>0, 3.86
bySection 2, Nevanlinna resultIII.
As beforeλ−λ0/zm/zm−1−1/R−l,mλhas expansion
fλ− p
t 1
rt
λ−qt, rt>0, 3.87
whereqt,t 1, . . . , p, corresponds to the singularities of3.85, that is, wherezm−1/zm R−l,mλ. The graph of 1/R−l,mλis as shown inFigure 3.
As before, the gradient of 1/R−l,mλatqtis positive for allt 1, . . . , p, that is
∂
∂λ 1 R−l,mλ
qt
>0, t 1, . . . , p. 3.88
Ifzdoes not obey3.76then the zeros of
λ−λ0
zm/zm−1−1/R−l,mλ 3.89
are the poles of 1/R−l,mλ, that is, thepk’s and λ λ0 where pk/λ0 for k 1, . . . , l1.
Clearly, fromFigure 3, the number ofqt’s is the same as the the number ofpk’s, thus in3.87, p l1.
Next, we examine the form offλin3.87. Asλ → ±∞it follows that 1/R−l,mλ → 1/gλh → 0. Thus
λ−λ0
zm/zm−1−1/R−l,mλ −→λ−λ0zm−1
zm . 3.90
Therefore,fλ λ−λ0zm−1/zm. Hence, wm−1
wm Ω−Ω
fλ−l1
t 1
rt λ−qt
Ω−Ω
λ−λ0zm−1 zm −l1
t 1
rt λ−qt
φλϕ−l1
t 1
t
λ−σt : Tl1,m− λ,
3.91
whereϕ: Ω Ωzm−1/zmλ0,φ: −Ωzm−1/zm<0,t: −Ωrt<0 andσt: qt fort 1, . . . , l1, which is precisely3.77.
Ifzdoes obey3.76forλ λ0thenzm−1/zm R−l,mλ0. Thus inFigure 3, one of theqt’s,t 1, . . . , pis equal toλ0and sinceλ0is less than the least eigenvalue of the boundary value problem1.1,3.76together with a boundary condition at−1as given in Theorems 3.2or3.3it follows thatq1 λ0, asλ0<pkfor allk 1, . . . , l1.
Now
λ−λ0
zm/zm−1−1/R−l,mλ
λ−λ0
1/R−l,mλ0−1/R−l,mλ R−l,mλ0R−l,mλ
R−l,mλ−R−l,mλ0
/λ−λ0
3.92
and asλ → λ0
R−l,mλ0R−l,mλ
R−l,mλ−R−l,mλ0
/λ−λ0 −→ R−l,mλ02
∂/∂λR−l,mλ
λ0
>0. 3.93
Thusλ λ0 q1is a removable singularity. Again, alternatively, we could have substituted in forR−l,mλandR−l,mλ0to illustrate that the singularity atλ λ0 q1 is removable, see Theorem 3.2. Hence the number of nonremovableqt’s is one less than the number ofpk’s, see Figure 3.
So3.87becomes
fλ−l1
t 2
rt
λ−qt, rt>0, 3.94
which may be rewritten as
fλ−l
t 1
rt λ−q
t
, rt>0, 3.95
wherern rn1andq
n qn1forn 1, . . . , l.
Now asλ → ±∞,
λ−λ0
1/R−l,mλ0−1/R−l,mλ −→R−l,mλ0λ−λ0. 3.96
So, we obtain
wm−1
wm Ω−Ω
fλ−l
t 1
rt λ−q
t
Ω−Ω
λ−λ0R−l,mλ0−l
t 1
rt λ−q
t
φλ ϕ−l
t 1
t
λ−σt : Tl,m− λ,
3.97
whereϕ: Ω Ωλ0R−l,mλ0,φ: −ΩR−l,mλ0<0,t: −Ωrt<0, andσt: q
tfor allt 1, . . . , l, that is, we obtain3.78.
4. Comparison of the Spectra
In this section, we investigate how the spectrum of the original boundary value problem compares to the spectrum of the transformed boundary value problem. This is done by considering the degree of the eigenparameter polynomial for the various eigenconditions.
Lemma 4.1. Consider the boundary value problem given by1.1forn 0, . . . , r−1 together with boundary conditions
y−1
aλb−s
k 1
ck
λ−dk y0, a >0, ck>0, 4.1
yr−1
⎡
⎣αλβ− p
j 1
γj λ−σj
⎤
⎦yr, α <0, γj<0. 4.2
Then the boundary value problem1.1,4.1,4.2hasspr1 eigenvalues. (Note that the number of unit intervals considered isr1.)
Proof. From1.1, withn 0, we obtain
y1 −c−1y−1
c0
b0 c0−λ
y0. 4.3
Substituting in fory−1from4.1yields
y1
−c−1
c0 aλb−s
k 1
ck
λ−dk
!
b0
c0−λ y0, 4.4
which may be rewritten as
y1: P01P11λ· · ·Ps11 λs1
λ−d1λ−d2· · ·λ−dsy0, 4.5
wherePi1,i 0, . . . , s1 are real constants.
Now1.1, forn 1, together with4.5results in
y2
−c0
c1
b1 c1 −λ
" P01P11λ· · ·Ps11 λs1 λ−d1λ−d2· · ·λ−ds
# y0
:
P02P12λ· · ·Ps22 λs2
λ−d1λ−d2· · ·λ−ds y0,
4.6
wherePi2,i 0, . . . , s2 are real constants.
Thus, by induction,
yr−1
P0r−1P1r−1λ· · ·Psr−1r−1 λsr−1
λ−d1λ−d2· · ·λ−ds y0, 4.7
for real constantsPir−1,i 0, . . . , sr−1. Similarly
yr
P0rP1rλ· · ·Psrr λsr λ−d1λ−d2· · ·λ−ds
y0, 4.8
for real constantsPir,i 0, . . . , sr.
Sincey0/≡0, using boundary condition4.2we obtain the following eigencondition:
P0r−1P1r−1λ· · ·Psr−1r−1 λsr−1 λ−d1λ−d2· · ·λ−ds
⎛
⎝αλβ− p
j 1
γj
λ−σj
⎞
⎠ P0rP1rλ· · ·Psrr λsr λ−d1λ−d2· · ·λ−ds
: Q0Q1λ· · ·Qp1λp1 λ−σ1λ−σ2· · ·
λ−σp! P0rP1rλ· · ·Psrr λsr λ−d1λ−d2· · ·λ−ds
,
4.9
whereQi,i 0, . . . , p1, are real constants.
Thus, the numerator is a polynomial, inλ, of orderp1sr. Note that, none of the roots of this polynomial are given bydk,k 1, . . . , sorσj,j 1, . . . , psince, from Figures1 to3, it is easy to see that none of the eigenvalues of the boundary value problem are equal to the poles of the boundary conditions. Alsoλ ±∞is not a problem as the curve of the Nevanlinna function never intersects with the horizontal or oblique asymptote. This means that there are no common factors to cancel out. Hence the eigencondition hasp1srroots giving that the boundary value problem hasp1sreigenvalues.
As a direct consequence of Theorems 2.1, 3.2, 3.3,3.4, andLemma 4.1we have the following theorem.
Theorem 4.2. For the original boundary value problem we consider twelve cases, (seeTable 1in the Appendix), each of which has s+l+m+1 eigenvalues. The corresponding transformed boundary value problem for each of the twelve cases, together with the number of eigenvalues for that transformed boundary value problem, is given inTable 1(see the appendix).
