# Eigenvalues of Vectorial Sturm-Liouville Problems with Parameter Dependent Boundary Conditions

Authors: Bilal Chanane
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We generalize theregularized sampling method introduced in 2005 by the author to
compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) to the case of vectorial SLP with parameter
dependent boundary conditions. A few problems are worked out to illustrate the effectiveness of the method
and show by the same token that we have indeed a general method capable of handling with ease very broad
classes of SLPs, whether scalar or vectorial.


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## Body

## 1. Introduction

In [1] we introduced the regularized sampling method, a method to compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) with parameter dependent boundary conditions. We subsequently used this method to compute the eigenvalues of singular and non-self-adjoint Sturm-Liouville problems. The scope of the method was further extended to include the computation of the eigenvalues of discontinuous/impulsive, nonlocal ([2] and the references therein), and two-parameter SLPs [3]. Continuing our effort we will tackle in this paper vectorial SLP with parameter dependent nonseparated boundary conditions. Vectorial Sturm-Liouville problems have been considered in [4–13] and the references therein while corresponding inverse problems appeared in [14–17].


## 2. The Characteristic Function

Consider the vectorial Sturm-Liouville problem,(1)-y′′+Q(x)y=μ2y,0<x<1A(μ)y0,μy′0,μ+B(μ)y1,μy′1,μ=0,
where Q is an n×n matrix function, A and B are real 2n×2n matrix functions of the parameter μ such that the matrix [Aμ∣B(μ)] has full rank.LetYc, Ys be the solutions of the Sturm-Liouville matrix equation -Y′′+Q(x)Y=μ2Y subject to the initial conditions Y(0,μ)=I, Y′(0,μ)=0 and Y(0,μ)=0, Y′(0,μ)=I, respectively, I being the n×n identity matrix and 0 being the n×n zero matrix.The general solution of the Sturm-Liouville equation(2)-Y′′+QxY=μ2Y
is given by Y=Yca+Ysb with arbitrary constant vectors a and b. Replacing in the boundary conditions, we get
(3)A(μ)ab+B(μ)Yc1,μa+Ys1,μbYc′1,μa+Ys′1,μb=0,A(μ)+BμYc(1,μ)Ys(1,μ)Yc′(1,μ)Ys′(1,μ)ab=0.
To have a nontrivial solution ab a necessary and sufficient condition is that F(μ)=0 where the characteristic function is
(4)Fμ=det⁡Fmatμ,
where
(5)Fmatμ=Aμ+BμYc1,μYs1,μYc′1,μYs′1,μ.The eigenvalues of (1) are the square of the zeroes of F. It is well known that the multiplicities of these eigenvalues are at most n.


## 3. Main Results

LetPWσ be the Paley-Wiener space
(6)PWσ=fentire,fμ≤CeσImμ,∫-∞∞‍fμ2dμ<∞,
and recall the celebrated Whittaker-Shannon-Kotel’nikov theorem [18].Theorem 1.
Letf∈PWσ; then
(7)fμ=∑k=-∞∞‍fkπσsinσμ-kσμ-k,
where the series converges uniformly on compact subset of C and in L2(R).It is known that, in the case of scalar Sturm-Liouville problems,y(x,μ) is an entire function of μ for each fixed x∈0,1. y(x,μ) is in a Paley-Wiener space as a function of μ for each x only in the Dirichlet case. So, we had to subtract some terms from y(x,μ) to make the difference fall in an appropriate PWσ space. We had even to subtract terms involving multiple integrals to get sharper results when it comes to computing of the eigenvalues. The regularized sampling method has been introduced recently [1] to overcome this problem; we do not have to subtract any term involving any (multiple) integration. In fact we multiplied y(x,μ)-ϕ(x,μ) and y′(x,μ)-ψ(x,μ) by an appropriate function of μ and got the eigenvalues with much greater precision at a reduced cost. Here ϕ and ψ are known simple functions.For the vectorial Sturm-Liouville problem at hand, we will use the regularized sampling method to recover the matricesYc(1,μ), Yc′(1,μ), Ys(1,μ), and Ys′(1,μ) from which we obtain F(μ), the characteristic function whose zeroes are the square roots of the sought eigenvalues of the problem.Consider the compatible vector and matrix norms given by(8)Y=max⁡i=1,…,nYi,P=max⁡i=1,…,n∑j=1n‍Pij,
where Y∈Rn, P∈Rn×n. In the following we will make use of the standard estimate.Lemma 2.
Consider(9)cos⁡⁡u≤e|Imu|,sin⁡uu≤γ01+ue|Imu|,
where γ0 is some constant (we may take γ0=1.72).To cover both cases (Y(0,μ)=I,  Y′(0,μ)=0 and Y(0,μ)=0, Y′(0,μ)=I) we will consider the following initial value problem:
(10)Y′′+μ2Y=QxY,Y0,μ=E1,Y′0,μ=E2,
where E1 and E2 are n×n matrices or n-vector. We have
(11)Yx,μ=E1cos⁡μx+E2sinμxμ+∫0x‍sinμ(x-t)μQ(t)Y(t,μ)dt.Our first result is the following theorem.Theorem 3.
Y
(
x
,
μ
) is an entire matrix function of μ for each fixed x∈(0,1] and satisfies the growth conditions
(12)Y(x,μ)≤E1+γ01+μE2eγ0∫01‍Q(t)dtexImμ≤γ1exIm⁡μ,Y(x,μ)-E1cos⁡μx+E2sinμxμ≤γ21+μexImμ≤γ2exIm⁡μ,Y′(x,μ)--μE1sinμx+E2cos⁡μx≤γ3exIm⁡μ,Y′(x,μ)+μE1sinμx-E2cos⁡μx∫0x‍cos⁡μ(x-t)Q(t)E1cos⁡μt+E2sinμtμhh-∫0x‍cos⁡μ(x-t)Q(t)E1cos⁡μt+E2sinμtμdt≤γ41+μexIm⁡μ,
for some positive constants γ1, γ2, γ3, and γ4.Proof.
From (11) and using standard arguments, we conclude that Y(x,μ) is an entire matrix function of μ for each x in (0,1]. Its derivative with respect to x,
(13)Y′x,μ=-μE1sinμx+E2cos⁡μx+∫0x‍cos⁡μx-tQtYt,μdt,
is also an entire matrix function of μ for each x in (0,1]. Going back to (11) we get at once
(14)Y(x,μ)≤E1cos⁡μx+E2sinμxμ+∫0x‍sinμx-tμx-t·x-tQt·Y(t,μ)dt≤exImμE1+γ0x1+xμE2+∫0x‍γ0x-te(x-t)ImμQt·Y(t,μ)dt≤exImμE1+γ01+μE2+exImμ∫0x‍γ0Qt·etImμYt,μdt.
Multiplying by e-xIm⁡μ, using Gronwall’s lemma, and multiplying back by exIm⁡μ we get
(15)Y(x,μ)≤E1+γ01+μE2eγ0∫01‍Q(t)dtexImμ≤γ1exImμ,
where γ1=E1+γ0E2exp⁡γ0∫01‍Q(t)dt. Now, using the above estimate in (10), we get
(16)Y(x,μ)-E1cos⁡μx+E2sinμxμ≤∫0x‍sinμx-tμx-t·x-tQt·Yt,μdt≤∫0x‍γ0e(x-t)Im⁡μ1+μ(x-t)·(x-t)Q(t)γ1etImμdt≤exIm⁡μγ0γ11+μ∫01‍Qtdt≤γ21+μexImμ≤γ2exImμ,
where γ2=γ0γ1∫01‍Q(t)dt. Likewise we have
(17)Y′(x,μ)--μE1sinμx+E2cos⁡μx≤∫0x‍cos⁡μx-t·Qt·Yt,μdt≤∫0x‍e(x-t)Im⁡μQ(t)γ1etImμdt=exIm⁡μγ1∫01‍Q(t)dt=γ3exIm⁡μ,
where γ3=γ1∫01‍Q(t)dt. As in the scalar case, Y(x,μ) is in a Paley-Wiener space only in the Dirichlet case; however, Y(x,μ)-E1cos⁡μx+E2(sinμx/μ) is. As for Y′(x,μ), it is not; nor is Y′(x,μ)--μE1sinμx+E2cos⁡μx since they are not square integrable over the reals for fixed x in (0,1]. Also,
(18)Y′(x,μ)+μE1sinμx-E2cos⁡μx-∫0x‍cos⁡μ(x-t)Q(t)E1cos⁡μt+E2sinμtμdth-∫0x‍cos⁡μ(x-t)Q(t)E1cos⁡μt+E2sinμtμdt≤∫0x‍cos⁡μx-t·Q(t)·Y(t,μ)-E1cos⁡μt+E2sinμtμdt≤∫0x‍e(x-t)Im⁡μQ(t)γ21+μetImμdt=exIm⁡μγ21+μ∫01‍Q(t)dt=γ41+μexIm⁡μ,
where γ4=γ2∫01‍Q(t)dt.We get at once the following corollaries.Corollary 4.
Y
c
(
x
,
μ
), Yc′(x,μ), Ys(x,μ), Ys′(x,μ), Y0(x,μ), and Y0′(x,μ) are entire matrix functions of μ for each fixed x∈(0,1] and satisfy the growth conditions
(19)Yc(x,μ)≤eγ0∫01‍QtdtexImμ,Yc(x,μ)-Icos⁡μx≤γ21+μexImμ≤γ2exIm⁡μ,Yc′(x,μ)+μIsinμx≤γ3exIm⁡μ,Yc′(x,μ)--μIsinμx+∫0x‍cos⁡μx-tQtcos⁡μtdt≤γ41+μexIm⁡μ,Ys(x,μ)≤γ01+μeγ0∫01‍QtdtexImμ≤γ1exIm⁡μ,Ys(x,μ)-Isinμxμ≤γ21+μexImμ≤γ2exIm⁡μ,Ys′(x,μ)-Icos⁡μx≤γ3exIm⁡μ,Ys′(x,μ)-Icos⁡μx+∫0x‍cos⁡μ(x-t)Q(t)sinμtμdt≤γ41+μexIm⁡μ,Y0(x,μ)≤D1+γ01+μD2eγ0∫01‍Q(t)dtexImμ≤γ1exIm⁡μ,Y0(x,μ)-D1cos⁡μx+D2sinμxμ≤γ21+μexImμ≤γ2exIm⁡μ,Y0′(x,μ)--μD1sinμx+D2cos⁡μx≤γ3exIm⁡μ,Y0′(x,μ)+μD1sinμx-D2cos⁡μx-∫0x‍cos⁡μ(x-t)Q(t)D1cos⁡μt+D2sinμtμdthh-∫0x‍cos⁡μ(x-t)Q(t)D1cos⁡μt+D2sinμtμdt≤γ41+μexIm⁡μ,
for some generic positive constants γ1, γ2, γ3, and γ4.Corollary 5.
The functions,(20)Yc1,μ-Icos⁡μ,Yc′1,μ--μIsinμ+∫01‍cos⁡μ1-tQtcos⁡μtdt,Ys(1,μ)-Isinμμ,Ys′1,μ-Icos⁡μ+∫01‍cos⁡μ1-tQtsinμtμdt,Y01,μ-D1cos⁡μ+D2sinμμ,Y0′1,μ--μD1sinμ+D2cos⁡μ∫01‍cos⁡μ(1-t)Q(t)D1cos⁡μt+D2sinμtμhhhhh+∫01‍cos⁡μ1-tQtD1cos⁡μt+D2sinμtμdt,
belong to the Paley-Wiener space PW˙1 as functions of μ and thus can be recovered from their samples at μk=kπ, k∈Z using the WSK series.Theorem 6.
Letθ be positive real number and m≥2 a positive integer. Consider
(21)U1(x,μ)=Yx,μ-E1cos⁡μx+E2sinμxμsin⁡θμθμm,U2(x,μ)=Y′x,μ--μE1sinμx+E2cos⁡μxsin⁡θμθμm
belong to the Paley-Wiener space PWσ where σ=x + mθ as functions of μ for each fixed x∈(0,1] for m≥1 and satisfy the growth condition U1(x,μ), U2(x,μ)≤γ5eσImμ/(1+θ|μ|)m where γ5 is some positive constant (γ5=γ0mmax⁡(γ2,γ3)).Proof.
It is enough to note thatsin(θμ)/θμ is an entire function of μ and satisfies the estimate in the above Lemma and the fact that Z(x,μ) is the product of two entire functions thus entire.Remark 7.
To avoid the first singularity ofsin(θμ)/θμ-m we will take θ<(N-m)-1.The use of the WSK theorem allows us to recoverU1(1,μ) and U2(1,μ) as
(22)U1(1,μ)=∑k∈Z‍αksinσ(μ-μk)σ(μ-μk),U21,μ=∑k∈Z‍βksinσμ-μkσμ-μk,
where αk=U1(1,μk), βk=U2(1,μk), μk=kπ/σ, and σ=1+mθ.Hence,Y(1,μ) or Y′(1,μ) can be recovered as
(23)Y1,μ=E1cos⁡μ+E2sinμμ+sin⁡θμθμ-m∑k∈Z‍αksinσμ-μkσμ-μk,Y′1,μ=-μE1sinμ+E2cos⁡μ+sin⁡θμθμ-m∑k∈Z‍αksinσμ-μkσμ-μk.
In practice, we take |k|≤N for some positive integer N, large enough, so that F(μ) can be reconstructed whose zeros are the square roots of the sought eigenvalues.Sinceμm-1U1(1,μ) and μm-1U2(1,μ) are in L2(-∞,∞), Jagerman’s result [18] is applicable and yields the following better estimate.Lemma 8 (truncation error).
LetUjN(1,μ)=∑k=-NN‍Uj(1,μk)(sinσ(μ-μk)/σ(μ-μk)) denote the truncation of Uj(1,μ), j=1,2. Then, for μ<Nπ/σ,
(24)Uj(1,μ)-UjN(1,μ)≤sinγμγ5,jππ/σm-11-4-m+1·1Nπ/σ-μ+1Nπ/σ+μ1N+1m-1,
where γ5,j=μm-1Uj1,μ2.Lemma 9.
Considerμ<Nπ/σ,
(25)Y(1,μ)-YN(1,μ),Y′(1,μ)-YN′(1,μ)≤sin⁡θμθμ-m×|sinγμ|γ5ππ/σm-11-4-m+1·1(Nπ/σ)-μ+1(Nπ/σ)+μ1N+1m-1,
where γ5=max⁡μm-1Y1,μ2,μm-1Y′1,μ2.The approximation ofY(1,μ) and Y′(1,μ) by YN(1,μ) and YN′(1,μ), respectively, induces an approximation of the characteristic function F by FN, whose zeros are the square root of the eigenvalues of the problem.Letμ¯2 denote an eigenvalue of the problem; then independent eigenfunctions associated can be obtained using basis vectors of the null space of the matrix Fmat(μ¯) as initial conditions to the differential equation -y′′+Q(x)y=μ¯2y, 0<x<1.


## 4. Numerical Examples

In this section we will illustrate the power of the regularized sampling method as applied to vectorial Sturm-Liouville problems with parameter dependent boundary conditions. We will takem=6, N=40 and a precision of 10-20 for the first three examples involving two dimensional SLPs. We will also work out two three-dimensional SLPs one of them involving parameter dependent boundary conditions. In these last two examples we take different values of N, namely, N=20,40,60,80, and 100, and take m=4 and a precision of 10-20. The reported multiplicities of the eigenvalues μ¯2 are just the dimensions of the null space of the corresponding matrices Fmat(μ¯).Example 1 (Chanane [1], 1D-version taken from fom Binding and Browne [19]).
Consider(26)-y1′′(x)=λy1(x),-y2′′(x)=λy2(x),0≤x≤1y1(0)+(λ+d)y1′(0)=0,y2(0)+(λ+d)y2′(0)=0y1(1)-λy1′(1)=0,y2(1)-λy2′(1)=0,
where d=-4π2. The first three eigenvalues were obtained as 9.730886578213082033, 88.76331625258976337, and 157.88411043863472059 putting them at about 10-18 from the exact eigenvalues. All these are double eigenvalues.Example 2.
Consider(27)-z1′′+xz1=μ2z1,-z2′′+x2z2=μ2z2,0<x<1z1(0)=0,z2(0)=0z1(1)=0,z2(1)=0.
The first four eigenvalues were obtained as10.149980317596192645, 39.426774741845613693, 88.33456043776637171, and 157.20995003768636950. Their multiplicity is two. Figure 1 illustrates the graph of the characteristic function.Figure 1
Ffor Example 2.In the next example we change the boundary conditions in Example2 to a parameter dependent one.Example 3.
Consider(28)-z1′′+xz1=μ2z1,-z2′′+x2z2=μ2z2,0<x<1z1(0)=0,z2(0)=0z1(1)+μz2(1)+μ2z1′(1)=0μ2z1(1)+z2(1)+z1′(1)+z2′(1)=0.
The first ten eigenvalues were obtained as1.8774711215942920040, 5.743710132061151193, 19.343498090220217814, 27.524136648884737570, 57.53122978141141088, 68.17260384634582088, 115.66805223522017586, 128.14561531120321356, 193.70672667853536264, and 207.68609472970380538. All these are simple eigenvalues. Figure 2 illustrates the graph of the characteristic function.Figure 2
F for Example 3.Next we consider three-dimensional vectorial SLPs, with different boundary conditions.Example 4.
Consider(29)-z1′′+x2z1=μ2z1,-z2′′+3x2z2-x2z3=μ2z2,-z3′′-x2z2+3x2z3=μ2z3,0<x<1z1(0)=0,z2(0)=0,z3(0)=0z1(1)=0,z2(1)=0,z3(1)=0.
Here, we will takem=4, N=20,40,60,80, and 100, and a precision of 10-20. Figure 3 illustrates the characteristic function FN over the range [2.8,6.9], while Figures 4 and 5 zoom into the regions containing the eigenvalues. Note that, in the range of interest [2.8,6.9], the graphs of FN, N=20,40,60,80, and 100, are on the top of each other. A 10-5 precision on μ can be obtained with just N=40. It appears clearly that in this example we have a simple eigenvalue λ1=μ12 and a double eigenvalue λ2=μ22, followed by a simple eigenvalue λ3=μ32 and a double eigenvalue λ4=μ42 (Figures 7, 8, 9, and 10). To obtain the double eigenvalues we look for the roots μ¯ of F′(μ) and then evaluate F(μ¯) which happened to be in each case of the order of 10-20. Table 1 illustrates these μ as function of N, the number of sampling points.Table 1
μ as a function of N for Example 4.
N
μ
1
μ
2
μ
3
μ
4
20
3.18255001139908084139
3.24667169525908938767
6.30074330869793149011
6.32071003783954269867
40
3.18255272015329488134
3.24667965370940216187
6.30075208373683858194
6.32073018962762953754
60
3.18255257457282246273
3.24667922487683254112
6.30075161400550151819
6.32072910805919989533
80
3.18255257724071613616
3.24667923278762738065
6.30075162260634961554
6.32072912799411873146
100
3.18255257728954451935
3.24667923293045271115
6.30075162276373835846
6.32072912835397407977Figure 3
F for Example 4.Figure 4
F around the first cluster of zeroes for Example 4.Figure 5
F for Example 5.The first fewalpha coefficients in the cardinal series expansion of U1(1,μ) are given as follows:
(30)α0=0.05070000.130-0.04470-0.04470.130,α1=-α-1=0.01650000.0451-0.01570-0.01570.0451,α2=-α-2=-0.00455000-0.01050.0035200.00352-0.0105,α3=-α-3=0.001970000.00442-0.001470-0.001470.00442.
The above data have been reported with only a few digits.Example 5.
Consider(31)-z′′+Qz=μ2z,0<x<1z(0)=0z(1)+Bz′(1)=0,
where
(32)Q=x20003x2-x20-x23x2,B=μ2μ1μμ21111.
Here, we will takem=4, N=20,40,60,80, and 100, and a precision of 10-20. Figure 6 illustrates the characteristic function FN over the range [0,8]. In this range, the graphs of FN, N=20,40,60,80, and 100, are on the top of each other. In this example the first seven (07) eigenvalues λk=μk2, k=1,…,7, are all simple. Tables 2(a) and 2(b) illustrate these μ as function of N, the number of sampling points.Table 2
μ as a function of N for Example 5.
(a)
N
μ
1
μ
2
μ
3
μ
4
20
1.89539417301926842876
2.02558758968702183901
2.29660628213819048323
4.77718462334715123614
40
1.89532715021073954015
2.02545891470303642172
2.29650089361152297043
4.77709199050137280895
60
1.89531503182513465256
2.02543500687372306198
2.29648223596348958117
4.77707495489416773295
80
1.89531699030377280216
2.02543884958899241306
2.29648526430437415553
4.77707769693838662656
100
1.89531696279196052243
2.02543879569567539948
2.29648522170989172904
4.77707765846484979002
(b)
N
μ
5
μ
6
μ
7
20
4.83834197825672007101
7.89099828962247999966
7.93058654329498644499
40
4.83814507749939589802
7.89090804120709347972
7.93039290552171137187
60
4.83810790006947131246
7.89089138476342607463
7.93035620848368882791
80
4.83811385300866362529
7.89089406045343050839
7.93036207289652547074
100
4.83811376961267015403
7.89089402293178420005
7.93036199078631128916Figure 6
F around the second cluster of zeroes for Example 4.Figure 7
F around μ1 in Example 5.Figure 8
F around μ2 and μ3 in Example 5.Figure 9
F around μ4 and μ5 in Example 5.Figure 10
F around μ6 and μ7 in Example 5.


## 5. Conclusion

In this paper we have extended the domain of application of the regularized sampling method to the case of vectorial Sturm-Liouville problems with parameter dependent boundary conditions. We have presented the theoretical foundation of the method and worked out a few examples to illustrate the method and shown by the same token that we have indeed a general method capable of handling with ease very broad classes of SLPs, whether scalar or vectorial, and providing the results at a reduced cost.



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