doi:10.1155/2010/947058
Research Article
Transformations of Difference Equations I
Sonja Currie and Anne D. Love
School of Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050, Johannesburg, South Africa
Correspondence should be addressed to Sonja Currie,[email protected]
Received 13 April 2010; Revised 27 July 2010; Accepted 29 July 2010
Academic Editor: Mariella Cecchi
Copyrightq2010 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.
We consider a general weighted second-order difference equation. Two transformations are studied which transform the given equation into another weighted second order difference equation of the same type, these are based on the Crum transformation. We also show how Dirichlet and non-Dirichlet boundary conditions transform as well as how the spectra and norming constants are affected.
1. Introduction
Our interest in this topic arose from the work done on transformations and factorisations of continuousas opposed to discreteSturm-Liouville boundary value problems by, amongst others, Binding et al., notably1,2. We make use of similar ideas to those discussed in3–5
to study the transformations of difference equations.
In this paper, we consider a weighted second-order difference equation of the form
ly:−cnyn1 bnyn−cn−1yn−1 cnλyn, 1.1
wherecn>0 represents a weight function andbna potential function.
given by permuting the factors S and Q or the factors P and R, that is, by QS or RP, respectively, as in the continuous case. Applying this transformation must then result in a transformed equation of exactly the same type as the original equation. In order to ensure this, we require that the original difference equation which we consider has the form given in1.1. In particular the weight,cn, also determines the dependence on the off-diagonal elements. We note that the more general equation
cnyn1−bnyn cn−1yn−1 −anλyn, 1.2
can be factorised asSQ, however, reversing the factors that is, findingQSdoes not necessarily result in a transformed equation of the same type as 1.2. The importance of obtaining a transformed equation of exactly the same form as the original equation, is that ultimately we willin a sequel to the current paperuse these transformations to establish a hierarchy of boundary value problems with 1.1 and various boundary conditions; see 4 for the differential equations case. Initially we transform, in this paper, non-Dirichlet boundary conditions to Dirichlet boundary conditions and back again. In the sequel to this paper, amongst other things, non-Dirichlet boundary conditions are transformed to boundary conditions which depend affinely on the eigenparameter λ and vice versa. At all times, it is possible to keep track of how the eigenvalues of the various transformed boundary value problems relate to the eigenvalues of the original boundary value problem.
The transformations given in Theorems2.1and3.1are almost isospectral. In particular, depending on which transformation is applied at a specific point in the hierarchy, we either lose the least eigenvalue or gain an eigenvalue below the least eigenvalue. It should be noted that if we apply the two transformations of Sections2and3successively the resulting boundary value problem has precisely the same spectrum as the boundary value problem we began with. In fact, for a suitable choice of the solutionznof1.1, withλless than the least eigenvalue of the boundary value problem fixed,Corollary 3.3gives that applying the transformation given inTheorem 2.1followed by the transformation given inTheorem 3.1
yields a boundary value problem which is exactly the same as the original boundary value problem, that is, the same difference equation, boundary conditions, and hence spectrum.
It should be noted that the work6, Chapter 11of Teschl, on spectral and inverse spectral theory of Jacobi operators, provides a factorisation of a second-order difference equation, where the factors are adjoints of each another. It is easy to show that the factors given in this paper are not adjoints of each other, making our work distinct from that of Teschl’s.
Difference equations, difference operators, and results concerning the existence and construction of their solutions have been discussed in7,8. Difference equations occur in a variety of settings, especially where there are recursive computations. As such they have applications in electrical circuit analysis, dynamical systems, statistics, and many other fields. More specifically, from Atkinson 9, we obtained the following three physical applications of the difference equation1.1. Firstly, we have the vibrating string. The string is taken to be weightless and bears m particles p0, . . . , pm−1 at the points say x0, . . . , xm−1 with masses c0, . . . , cm−1 and distances between them given by xr1 −xr 1/cr,
r 0, . . . , m−2. Beyondcm−1 the string extends to a length 1/cm−1 and beyond
c0to a length 1/c−1. The string is stretched to unit tension. Ifsnis the displacement of the particle pn at time t, the restoring forces on it due to the tension of the string are
we can find the second-order differential equation of motion for the particles. We require solutions to be of the formsn yncosωt, whereynis the amplitude of oscillation of the particlepn. Solving for ynthen reduces to solving a difference equation of the form
1.1. Imposing various boundary conditions forces the string to be pinned down at one end, both ends, or at a particular particle, see Atkinson9for details. Secondly, there is an equivalent scenario in electrical network theory. In this case, thecnare inductances, 1/cn capacitances, and thesnare loop currents in successive meshes. The third application of the three-term difference equation1.1is in Markov processes, in particular, birth and death processes and random walks. Although the above three applications are somewhat restricted due to the imposed relationship between the weight and the off-diagonal elements, they are nonetheless interesting.
There is also an obvious connection between the three-term difference equation and orthogonal polynomials; see10. Although, not the focus of this paper, one can investigate which orthogonal polynomials satisfy the three-term recurrence relation given by1.1and establish the properties of those polynomials. In Atkinson9, the link between the norming constants and the orthogonality of polynomials obeying a three-term recurrence relation is given. Hence the necessity for showing how the norming constants are transformed under the transformations given in Theorems2.1and 3.1. As expected, from the continuous case, we find that the nth new norming constant is justλn −λ0 multiplied by the original nth norming constant or 1/λn−λ0multiplied by the originalnth norming constant depending on which transformation is used.
The paper is set out as follows.
InSection 2, we transform1.1with non-Dirichlet boundary conditions at both ends to an equation of the same form but with Dirichlet boundary conditions at both ends. We prove that the spectrum of the new boundary value problem is the same as that of the original boundary value problem but with one eigenvalue less, namely, the least eigenvalue.
In Section 3, we again consider an equation of the form 1.1, but with Dirichlet boundary conditions at both ends. We assume that we have a strictly positive solution,zn, to1.1forλλ0withλ0less than the least eigenvalue of the given boundary value problem. We can then transform the given boundary value problem to one consisting of an equation of the same type but with specified non-Dirichlet boundary conditions at the ends. The spectrum of the transformed boundary value problem has one extra eigenvalue, in particular
λ0.
The transformation inSection 2followed by the transformation inSection 3, gives in general, an isospectral transformation of the weighted second-order difference equation of the form1.1with non-Dirichlet boundary conditions. However, for a particular choice of
znthis results in the original boundary value problem being recovered.
In the final section, we show that the process outlined in Sections 2 and 3 can be reversed.
2. Transformation 1
2.1. Transformation of the Equation
Consider the second-order difference equation1.1, which may be rewritten as
wheren0, . . . , m−1. Denote byλ0the least eigenvalue of1.1with boundary conditions
hy−1 y0 0, Hym−1 ym 0, 2.2
wherehandHare constants; see9. We wish to find a factorisation of the formal operator,
lyn:−yn1
bn
cn−λ0
yn−ccn−1
n yn−1 λ−λ0yn, 2.3
forn 0, . . . , m−1, such thatl SQ, whereSandQare both first order formal difference operators.
Theorem 2.1. Letu0nbe a solution of1.1corresponding toλλ0and define the formal difference
operators
Syn:yn−yn−1u0n−1cn−1
u0ncn , n
0, . . . , m,
Qyn:yn1−ynu0n1
u0n , n−
1, . . . , m−1.
2.4
Then formallylyn SQyn,n0, . . . , m−1and the so-called transformed operator is given by
lyn QSyn,n0, . . . , m−1. Hence the transformed equation is
cnyn1−bnyn cn−1yn−1 −cnλyn, n0, . . . , m−2, 2.5
where
cn u0ncn
u0n1 >
0, n−1, . . . , m−1, 2.6
bn
u
0ncn
u0n1cn1−
cn−1u0n−1
cnu0n
bn
cn−λ0 u
0ncn
u0n1, n
0, . . . , m−1.
2.7
Proof. By the definition ofSandQ, we have that
SQyn S
yn1−u0n1
u0n yn
yn1−u0n1yn
u0n −
yn− u0n
u0n−1yn− 1
u
0n−1cn−1
u0ncn .
Using2.3, substituting in foru0n1and cancelling terms, gives
SQyn yn1− 1
u0n
−λ0u0n−cn− 1
cn u0n−1 cbnnu0n
yn
−ynu0n−1cn−1
u0ncn yn−
1cn−1
cn
yn1− bn
cn−λ0
yn ccn−n1yn−1
−λ−λ0yn, n0, . . . , m−1.
2.9
HencelSQ.
Now, settingQyn yn,n−1, . . . , m−1, gives
QSyn QSQyn −Qλ−λ0yn
−λ−λ0yn, n0, . . . , m−1,
2.10
which is the required transformed equation. To findl, we need to determineQSyn. Firstly,
Syn yn−yn−1u0n−1cn−1
u0ncn , n
0, . . . , m−1, 2.11
thus forn0, . . . , m−2,
QSynyn1
−yn u0ncn
u0n1cn1−
yn−yn−1u0n−1cn−1
u0ncn u
0n1
u0n
yn1−yn
u
0ncn
u0n1cn1
u0n1
u0n
yn−1 u
0n−1cn−1u0n1
u0ncnu0n
.
By multiplying byu0ncn/u0n1, this may be rewritten as
u0ncn
u0n1yn 1−
u
0ncn
u0n1cn1−
cn−1u0n−1
cnu0n
bn
cn −λ0 u
0ncn
u0n1yn
u0n−1cn−1
u0n yn−1 −λ−λ0yn
u0ncn
u0n1.
2.13
Thus we obtain2.5.
2.2. Transformation of the Boundary Conditions
We now show how the non-Dirichlet boundary conditions2.2are transformed underQ. By the boundary conditions2.2yis defined forn−1, . . . , m.
Theorem 2.2. The mapping y → y given by yn yn 1− ynu0n 1/u0n, n
−1, . . . , m−1, where u0 is an eigenfunction to the least eigenvalue λ0 of 1.1,2.2, transforms
yobeying boundary conditions2.2toyobeying Dirichlet boundary conditions of the form
y−1 0, ym−1 0. 2.14
Proof. Sinceyn yn1−ynu0n1/u0n, we get that
y−1 y0−y−1 u00
u0−1
−hy−1−y−1−h
0.
2.15
Hence as y obeys the non-Dirichlet boundary condition hy−1 y0 0, y obeys the Dirichlet boundary condition,y−1 0.
Similarly, for the second boundary condition,
ym−1 ym−ym−1 u0m
u0m−1
−Hym−1−ym−1−H
0.
2.16
We call2.14the transformed boundary conditions.
Combining the above results we obtain the following corollary.
The spectrum of the transformed boundary value problem2.5,2.14is the same as that of 1.1,
2.2, except for the least eigenvalue,λ0, which has been removed.
Proof. Theorems2.1and2.2prove that the mappingy→ytransforms eigenfunctions of1.1,
2.2to eigenfunctionsor possibly the zero solutionof 2.5,2.14. The boundary value problem1.1,2.2 has meigenvalues which are real and distinct and the corresponding eigenfunctionsu0n, . . . , um−1nare linearly independent when considered forn0, . . . , m− 1; see11for the case of vector difference equations of which the above is a special case. In particular, ifλ0 < λ1 < · · · < λm−1 are the eigenvalues of 1.1, 2.2 with eigenfunctions
u0, . . . , um−1, thenu0 ≡0 andu1, . . . ,um−1are eigenfunctions of2.5,2.14with eigenvalues
λ1, . . . , λm−1. By a simple computation it can be shown thatu1, . . . ,um−1/≡0. Since the interval of the transformed boundary value problem is precisely one shorter than the original interval,
2.5,2.14has one less eigenvalue. Henceλ1, . . . , λm−1constitute all the eigenvalues of2.5,
2.14.
2.3. Transformation of the Norming Constants
Letλ0<· · ·< λm−1be the eigenvalues of1.1with boundary conditions2.2andy0, . . . , ym−1 be associated eigenfunctions normalised byyn0 1. We prove, in this subsection, that under the mapping given inTheorem 2.2, the new norming constant is 1/λn−λ0times the original norming constant.
Lemma 2.4. Letρndenote the norming constants of 1.1and be defined by
ρn: m−1
j0
−cj
ynj
yn0 2
m−1
j0
−cjynj2. 2.17
Ifτnis defined by
τn: m−2
j0
−cjynj2, 2.18
then, foru0an eigenfunction forλλ0normalised byu00 1,
λn−λ0ρnτn−c−1uu0−1 00 yn0
2c−1y
n−1yn0−ynm−12cm−1uu0m 0m−1
ynm−1ynmcm−1.
Proof. Substituting in forynjandcj,n1, . . . , m−1, we have that
Then, using the definition ofρn, we obtain that
Now,
−bm−1−λ0cm−1ynm−12
−cm−1ynmynm−1−cm−2ynm−2ynm−1
−cm−1λn−λ0ynm−12.
2.22
Therefore,
λn−λ0ρnτn−c−1u0− 1
u00 yn
02c−1yn−1yn0 cm−2uu0m−2 0m−1ynm−
12
−cm−2ynm−2ynm−1−cm−1λn−λ0ynm−12.
2.23
Using1.1to substitute in forcm−2u0m−2andcm−2ynm−2gives
λn−λ0ρnτn−c−1uu0−1 00 yn
02c−1yn−1yn0
ynm−12
−cm−1λ0bm−1−cm−1uu0m 0m−1
−ynm−1−cm−1λnynm−1 bm−1ynm−1−cm−1ynm
−cm−1λn−λ0ynm−12
τn−c−1uu0−1 00 yn
02c−1yn−1yn0−ynm−12cm−1 u0m
u0m−1
ynm−1ynmcm−1.
2.24
Theorem 2.5. If ρn, as defined in Lemma 2.4, are the norming constants of 1.1 with boundary
conditions2.2and
ρn: m−2
j0
−cj
ynj
yn0 2
2.25
are the norming constants of 2.5with boundary conditions2.14, then
ρn λn−λ0ρn. 2.26
Proof. The boundary conditions2.2together withLemma 2.4give
Now by2.14,y−1 0, and thus
y0 y1−u01
u00y
0 y1−u01 −λ−λ0. 2.28
Therefore,
λn−λ0ρnyn02 m−2
j0
−cj
ynj
yn0 2
λn−λ02 m−2
j0
−cj
ynj
yn0 2
. 2.29
Thus we have that
ρn λn−λ0ρn. 2.30
3. Transformation 2
3.1. Transformation of the Equation
Consider 2.5, where n 0, . . . , m−2 and yn, n −1, . . . , m−1, obeys the boundary conditions2.14.
Letznbe a solution of2.5withλλ0 such thatzn>0 for alln−1, . . . , m−1, whereλ0is less than the least eigenvalue of2.5,2.14.
We want to factorise the operatorlz, where
lzyn −yn1
bn
cn −λ0
yn−ccn−1
n yn−1 λ−λ0yn, 3.1
forn 0, . . . , m−2,such thatlz PR, whereP andRare both formal first order difference operators.
Theorem 3.1. Let
Pyn:yn1−ynzn−1cn−1
zncn , n0, . . . , m−2,
Ryn:yn−yn−1 zn
zn−1, n0, . . . , m−1.
3.2
ThenlzPRandyn Rynis a solution of the transformed equationRPy−λ−λ0ygiving,
forn1, . . . , m−2,
where, forn0, . . . , m−1,
giving thatyis a solution of the transformed equation.
This implies that
zn−1cn−1
zn yn1−
zn−
1cn−1
zncn
zn
zn−1 zn−
1cn−1
zn yn
znz−2n−cn−2
1 yn−1 −λ−λ0yn
zn−1cn−1
zn .
3.8
3.2. Transformation of the Boundary Conditions
At present, yn is defined forn 0, . . . , m−1. We extend the definition of yn to n
−1, . . . , mby forcing the boundary conditions
hy−1 y0 0, Hym−1 ym 0, 3.9
where
h:
c0
c−1
b0
c0 −
z1
z0−
b0
c0 −1
,
H: bm−2
cm−2−
bm−1
cm−1−
zm−2cm−2
zm−1cm−1.
3.10
Here we takec−1 c−1.
Theorem 3.2. The mappingy→ygiven byyn yn−yn−1zn/zn−1,n0, . . . , m− 1, wherezn is as previously defined (in the beginning of the section), transformsy which obeys boundary conditions2.14toywhich obeys the non-Dirichlet boundary conditions3.9andyis a solution oflyn λcnynforn0, . . . , m−1.
Proof. By the construction ofhandHit follows that the boundary conditions3.9are obeyed byy.
We now show thatyis a solution to the extended problem. FromTheorem 3.1we need only prove thatlyn λcnynforn0 andnm−1. Forn0, from3.3with3.9, we have that
c0y1 c−1 −y
0
h
b0−c0λy0. 3.11
Also the mapping, forn0, gives
y0 y0−y−1 z0
Thus using2.14, we obtain thaty0 y0. So we now have
Next, using the mapping atn1, we obtain that
Rearranging the terms above results in
Subtracting3.15from3.16yields
Combining Theorems3.1and3.2we obtain the corollary below.
Corollary 3.3. Letznbe a solution of 2.5forλλ0, whereλ0is less than the least eigenvalue of
2.5,2.14, such thatzn >0forn −1, . . . , m−1. Then we can transform the given equation,
2.5, to an equation of the same type,3.3with a specified non-Dirichlet boundary condition,3.9, at either the initial or end point. The spectrum of the transformed boundary value problem3.3,3.9 is the same as that of 2.5,2.14except for one additional eigenvalue, namely,λ0.
Proof. Theorems 3.1and 3.2prove that the mappingy → y, transforms eigenfunctions of
2.5,2.14to eigenfunctions of3.3,3.9. In particular ifλ1<· · ·< λm−1are the eigenvalues of 2.5, 2.14,n −1, . . . , m−1, with eigenfunctionsu1, . . . ,um−1, then z,u1, . . . ,um−1 are eigenfunctions of3.3,3.9,n−1, . . . , m, with eigenvaluesλ0, λ1, . . . , λm−1. Since the index set of the transformed boundary value problem is precisely one larger than the original,3.3,
3.9has one more eigenvalue. Henceλ0, λ1, . . . , λm−1 constitute all the eigenvalues of3.3,
3.9.
Corollary 3.4. The transformation of 1.1, 2.2 to2.5, 2.14 and then to 3.3, 3.9 is an isospectral transformation. That is, the spectrum of 1.1,2.2is the same as the spectrum of 3.3,
3.9.
We now show that for a suitable choice of znthe transformation of1.1,2.2 to
2.5,2.14and then to3.3,3.9results in the original boundary value problem.
Without loss of generality, by a shift of the spectrum, it may be assumed that the least eigenvalue,λ0, of1.1,2.2isλ0 0. Furthermore, letu0nbe an eigenfunction to1.1,
which, when we substitute in forz,c, andb, simplifies to zero. Obviously the right-hand side of2.5is equal to 0 forλλ00. Thusznis a solution of2.5forλλ00, whereλ00 is less than the least eigenvalue of2.5,2.14.
Substituting for zn, zn − 1 and cn− 1, in the equation for cn, we obtain immediately thatcn cnforn0, . . . , m−1 and by assumptionc−1 c−1.
Next, a similar substitution into the equation forbnyields
Sincezis a solution of2.5forλ00, we have, forn0,
z1
z0−
b0
c0
z−1c−1
z0c0 0. 3.23
Putting this into the equation forhabove yields
h−cc−−1
1. 3.24
Substituting in forc−1gives
h −u00
u0−1,
3.25
so by2.2hh.
Using precisely the same method, it can be easily shown thatHH.
Hence, as claimed, we have proved the following result.
Corollary 3.6. The transformation of 1.1, 2.2 to 2.5, 2.14 and then to 3.3, 3.9 with zn:1/u0ncnresults in the original boundary value problem.
3.3. Transformation of the Norming Constants
Assume that we have the following normalisation:y0 1. A result analogous to that in
Theorem 2.5is obtained.
Lemma 3.7. As before letρndenote the norming constants of 2.5and be defined by
ρn m−2
j0
−cj
ynj
yn0 2
m−2
j0
−cjynj2. 3.26
Ifτnis defined by
τn m−1
j0
−cjynj2, 3.27
then,
λn−λ0ρnτnc−1zz−01yn−12−c−1yn−1yn0 ynm2cm−1zzmm−1
−ynm−1ynmcm−1.
Proof. Substituting in forcandy, we obtain that
Then, using the definition ofρn, we obtain that
λn−λ0ρn conditions2.14and
are the norming constants of 3.3with boundary conditions3.9, then
ρn λ 1
n−λ0ρn. 3.32
Proof. Using the boundary conditions2.14together withLemma 3.7, we obtain that
λn−λ0ρnτn. 3.33
Now,
y0 y0 1. 3.34
Therefore,
λn−λ0ρnyn02 m−1
j0
−cj
ynj
yn0 2
m−1
j0
−cj
ynj
yn0 2
. 3.35
Thus,
ρn λ 1
n−λ0ρn. 3.36
4. Conclusion
To conclude, we illustrate how the process may be done the other way around. To do this we start by transforming a second-order difference equation with Dirichlet boundary conditions at both ends to a second-order difference equation of the same type with non-Dirichlet boundary conditions at both ends and then transform this back to the original boundary value problem.
Considervnsuch thatvnsatisfies2.5and2.14. The mappingv→v, given by
vn vn−vn−1 zn
zn−1, n0, . . . , m−1, 4.1
can be extended to includev−1andvmby forcing3.9. Hereznis a solution of2.5
forλ λ0 0, withλ0 less than the least eigenvalue of2.5,2.14such thatzn > 0 for alln −1, . . . , m−1. The mapping v → v then gives thatvnsatisfies3.3and3.9. So
3.3,3.9has the same spectrum as2.5,2.14except that one eigenvalue has been added, namely,λλ00.
Now the mappingv→vgiven by
vn vn1−vnu0n1
u0n , n−
whereu0nis an eigenfunction of3.3,3.9corresponding to the eigenvalueλ00 yields
cnvn1−bnvn cn−1vn−1 −cnλvn, n0, . . . , m−2, 4.3
where
cn u0ncn
u0n1 > 0,
bn
u0ncn
u0n1cn1−
cn−1u0n−1
cnu0n
bn
cn−λ0
u0ncn
u0n1,
4.4
with boundary conditions
v−1 0, vm−1 0. 4.5
Thus this boundary value problem invhas the same spectrum as that of3.3,3.9but with one eigenvalue removed, namely,λλ00.
Lemma 4.1. Letu0n: 1/zn−1cn−1, whereznis a solution of 2.5withλ λ0 0,
whereλ0is less than the least eigenvalue of2.5,2.14, such thatzn>0for alln−1, . . . , m−1.
Thenu0nis an eigenfunction of3.3,3.9corresponding to the eigenvalueλ00, where we define
u00vialu01 0andu0−1 −u00/h.
Proof. By construction, we have that
hu0−1 u00 0. 4.6
Also
H bm−2
cm−2−
bm−1
cm−1−
zm−2
zm−1
cm−2
cm−1. 4.7
By substituting in forbm−1, cm−1andcm−2, we obtain
H bm−2
cm−2−
zm−2
zm−1
cm−2
cm−1−
zm−1
zm−2−
zm−3cm−3
zm−2cm−2. 4.8
Thus
H− zzmm−−2
1
cm−2
Hence substituting the expressions foru0m−1, u0m, andHinto the above equation yields
Hu0m−1 u0m 0. 4.10
Thusu0nobeys the boundary conditions3.9.
Next, we show thatu0nsolves3.3. Substituting in foru0n, we obtain that, for
n1, . . . , m−1,
cnu0n1−bnu0ncn−1u0n−1 zncncn−
bn
zn−1cn−1
cn−1
zn−2cn−2.
4.11
Using the expressions forcandb, we obtain by direct substitution that
cnu0n1−bnu0n cn−1u0n−1 0, n1, . . . , m−1. 4.12
Now, if we examine the right-hand side of3.3, we immediately see that it is equal to 0 for
λ00.
We now show thatu0nsolves3.3forn 0 as well, that islu00 0. By3.3for
λ00,
lu00 −c0u01 b0u00−c−1u0−1. 4.13
Usingu0−1 −u00/h,
lu00 −c0u01
b0 c−1
h
u00. 4.14
Now,lu01 0 gives
u00 − c
1u02 b1u01
c0 . 4.15
Thus,
lu00 −c0u01
b0
c0
−c1u02 b1u01
c−1
hc0
−c1u02 b1u01
.
Substituting in forcn, bn, n0,1, andu0n, n1,2, we obtain that
giving by a straightforward calculation
lu00 0. 4.23
Theorem 4.2. The boundary value problem4.3with boundary conditions
v−1 0, vm−1 0, 4.24
is the same as the original boundary value problem forvn, that is,2.5and2.14, whereu0nis
as inLemma 4.1.
Proof. All we need to show is thatcn cnandbn bn. Substituting in foru0 gives directly that
cn cn. 4.25
Now,
bn
u0ncn
u0n1cn1−
cn−1u0n−1
cnu0n
bn
cn −λ0
u0ncn
u0n1, 4.26
which sincecn cnandcn u0ncn/u0n1gives
bn
cn
cn1−
cn−1
cn
zn−1cn−1
zncn
zn
zn−1−λ0
cn. 4.27
Using the expression forc, we obtain that
bn cnzn1
cnzn −
cn−1zn
cn−1zn−1
zn−1cn−1
zncn
zn
zn−1−λ0
cn. 4.28
Butznobeys2.5forλλ00 thus
bn
bnzn
zncn
cn bn. 4.29
Acknowledgments
The authors would like to thank Professor Bruce A. Watson for his ideas, guidance, and assistance. This paper was supported by NRF Grant nos. TTK2007040500005 and FA2007041200006.
References
1 P. A. Binding, P. J. Browne, and B. A. Watson, “Spectral isomorphisms between generalized Sturm-Liouville problems,” inLinear Operators and Matrices, Operator Theory: Advances and Applications, pp. 135–152, Birkh¨auser, Basel, Switzerland, 2001.
3 P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with reducible boundary conditions,”Proceedings of the Edinburgh Mathematical Society, vol. 49, no. 3, pp. 593–608, 2006.
4 P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II,”Journal of Computational and Applied Mathematics, vol. 148, no. 1, pp. 147–168, 2002.
5 P. A. Binding, P. J. Browne, and B. A. Watson, “Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions,” The Bulletin of the London Mathematical Society, vol. 33, no. 6, pp. 749–757, 2001.
6 G. Teschl,Jacobi Operators and Completely Integrable Nonlinear Lattices, vol. 72 ofMathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2000.
7 K.S. Miller,Linear Difference Equations, W. A. Benjamin, 1968.
8 K. S. Miller,An Introduction to the Calculus of Finite Differences and Difference Equations, Dover, New York, NY, USA, 1966.
9 F. V. Atkinson,Discrete and Continuous Boundary Value Problems, Academic Press, New York, NY, USA, 1964.
10 W. Gautschi,Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, NY, USA, 2004.