Available at Applications and Applied

http://pvamu.edu/aam Mathematics:
Appl. Appl. Math. An International Journal
ISSN: 1932-9466 (AAM)

Vol. 13, Issue 1 (June 2018), pp. 496 – 507

The Finite Spectrum of Sturm-Liouville Operator

With δ-Interactions

1
Abdullah Kablan, 2 Mehmet Akif Çetin, & 3 Manaf Dzh. Manafov

1,2 Department of Mathematics

Faculty of Arts and Sciences
Gaziantep University
Gaziantep, Turkey
1 kablan@gantep.edu.tr, 2 makifcetin@gantep.edu.tr

3 Department of Mathematics
Faculty of Arts and Sciences
Adıyaman University
Adıyaman, Turkey
3 mmanafov@adiyaman.edu.tr

∗ Corresponding Author: Abdullah Kablan

Received: September 20, 2017; Accepted: April 20, 2018

Abstract

The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δ -
interactions. Such an equation gives us a Sturm-Liouville boundary value problem which has n
transmission conditions. We show that for any positive numbers mj (j = 0, 1, ..., n) that are related
to number of partition of the intervals between two successive interaction points, we can construct
a Sturm-Liouville equations with δ -interactions, which have exactly d eigenvalues. Where d is the
sum of mj ’s.
Keywords: Sturm-Liouville operator; finite spectrum; point interactions

MSC 2010 No.: 34B24, 47A10, 47A75

496
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 497

1. Introduction

In their article, Kong at al. have been constructed a self-adjoint and non self-adjoint Sturm-
Liouville problems with exactly n eigenvalues, Kong et al. (2001). According to this paper, for
every given positive integer n we can construct a Sturm-Liouville problem (SLP) which has ex-
actly n eigenvalues. Then, this problem has been expanded to various Sturm-Liouville problems
such as in Ao et al. (2011) and Ao et al. (2013). In recent decades, fourth order boundary value
problems with finite spectrum has been studied in Ao et al. (2014) and Bo et al. (2014). Then,
the first equivalence relation between boundary value problems with finite spectrum and matrix
eigenvalue problems was found in Kong et al. (2009). For the studies about this relations please
see Ao et al. (2012)-Kablan et al. (2016). The purpose of this paper is to study the Sturm-Liouville
equation with δ -interaction which is formally defined by
n
X
−(py 0 )0 + αj δ(x − xj )y + qy = λwy, on J, (1)
j=1

where J = (a, x1 ) ∪ (x1 , x2 ) ∪ ... ∪ (xn , b), x1 , ..., xn ∈ (a, b), with −∞ < a < b < ∞, αj ’s are real
numbers, δ(x) is the Dirac delta function and λ ∈ C is a spectral parameter. And we shall con-
struct some special type Sturm-Liouville problems with δ -interactions, which have finitely many
eigenvalues. Equation (1) comes from the time-independent one-dimensional Schrödinger equa-
tion. Schrödinger operators with point interactions in one or more dimensions are widely used in
applications to quantum and atomic physics because they can be used as exactly solvable models
in many situations Albeverio et al. (1988)-Manafov et al. (2013).
This paper consists, besides this introductory section, of three sections. Section 2 is auxiliary and
presents the statement of problem and some known results. In Section 3, we construct a Sturm-
Liouville equations with δ -interaction which has finitely many eigenvalues and finaly, Section 4 is
devoted to some examples.

2. Statement of Problem and Notations

The equation (1) is equivalent to the following many-point boundary value problem, Albeverio et
al. (1988). So we can understand problem (1) as studying the equation
−(py 0 )0 + qy = λwy, on J, (2)
and n transmission conditions
" #
y
Cj Y (xj −) = Y (xj +), Y = , j = 1, 2, ..., n, (3)
py 0
where xj ’s are inner discontinuity points and
" #
1 0
Cj = .
αj 1
Additionally, let’s consider the boundary condition of the form
AY (a) + BY (b) = 0, A, B ∈ M2 (C), (4)
498 A. Kablan et al.

where A = (aij )2×2 , B = (bij )2×2 are complex valued 2 × 2 matrices and M2 (C) denotes the set of
square matrices of order 2 over C. Here, the coefficients satisfy the minimal conditions
1
r= , q, w ∈ L(J, C), (5)
p
where L(J, C) denotes the complex valued functions which are Lebesgue integrable on J . (5) is
necessary and sufficiently condition for the uniqueness of the solution of the initial value problem,
(see Zettl (2005)).
Let u1 = y, u2 = py 0 . Then, we have the system representation of Eq. (2)
u01 = ru2 , u02 = (q − λw)u1 , on J. (6)

Definition 2.1.
By a trivial solution of Eq. (2) on some interval we mean a solution y which is identically zero and
whose quasi-derivative u2 = py 0 is also identically zero on this interval.

Lemma 2.2.
Let (5) holds and let Φ(x, λ) = [φij (x, λ)] be the fundamental matrix solution of the system (6)
determined by the initial condition Φ(a, λ) = I. Then, λ ∈ C is an eigenvalue of the Sturm-Liouville
problem with δ -interactions (1), (4) or equivalently the Sturm-Liouville problem with transmission
conditions (2)-(4) if and only if
∆(λ) = det [A + BΦ(b, λ)] = 0. (7)
Then, ∆(λ) can be written as
∆(λ) = det(A) + det(B) + h11 φ11 (b, λ) + h12 φ12 (b, λ) (8)
+h21 φ21 (b, λ) + h22 φ22 (b, λ),
where
" # " #
h11 h12 a22 b11 − a12 b21 a11 b21 − a21 b11
H= = .
h21 h22 a22 b12 − a12 b22 a11 b22 − a21 b12

Proof:
Let’s consider the linear algebra system
[A + BΦ(b, λ)] C = 0 (9)

and assume that ∆(λ) = 0. Then, the system (9) has a nontrivial vector solution. If we solve the
following initial value problem
" # " #
1
0 y
Y0 = p , Y = on J, Y (a) = C, (10)
q − λw 0 py 0

we obtain Y (b) = Φ(b, λ)Y (a) and [A + BΦ(b, λ)] Y (a) = 0. So we conclude that y which is the top
component of Y is an eigenfunction of the problem (2)-(3) and λ is an eigenvalue of this problem.
Conversely, if λ is an eigenvalue corresponding to eigenfunction y , then Y defined in (10) satisfies
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 499

Y (b) = Φ(b, λ)Y (a) and consequently [A + BΦ(b, λ)] Y (a) = 0. Since Y (a) is an eigenfunction, it
can never be zero so we have that det [A + BΦ(b, λ)] = 0.
On the other hand, for any A = (aij ), B = (bij ) ∈ M2 (C), we know that

det(A + B) = det(A) + det(B) + P (A, B),

where P (A, B) denotes the sum of the possible products of the elements belonging to different
rows and different columns in matrices A and B. So we have

∆(λ) = det [A + BΦ(b, λ)]

= det(A) + det(BΦ(b, λ)) + P (A, BΦ(b, λ)).

Since Φ(a, λ) = I, then det(Φ(b, λ)) = 1, and P (A, BΦ(b, λ)) can be written in the form

P (A, BΦ(b, λ)) = h11 φ11 (b, λ) + h12 φ12 (b, λ) + h21 φ21 (b, λ) + h22 φ22 (b, λ),

where h11 , h12 , h21 and h22 are constants which depend only on the matrices A and B. Then, we
can conclude that (8) is followed. 

Definition 2.3.
The SLP with transmission conditions (2)-(4), or equivalently (6), (3), (4) is said to be degenerate
if in (8) either ∆(λ) ≡ 0 for all λ ∈ C or ∆(λ) 6= 0 for any λ ∈ C.

3. The finite spectrum of SLP’s with δ−interactions

Throughout this section we assume (5) holds and there exists a partition of subintervals of J
a = x0 = x00 < x01 < x02 < ... < x0,2m0 +1 = x1 ,
x1 = x10 < x11 < x12 < ... < x1,2m1 +1 = x2 ,
.. (11)
.
xn−1 = xn−1,0 < xn−1,1 < xn−1,2 < ... < xn−1,2mn−1 +1 = xn ,
xn = xn0 < xn1 < xn2 < ... < xn,2mn +1 = xn+1 = b,

for some integers mj , j = 0, 1, ..., n. Then, for each j ∈ {0, 1, ..., n} we suppose that
xj,2k+1
Z
1
r = = 0 on (xj,2k , xj,2k+1 ), w 6= 0, k = 0, 1, ..., mj , (12)
p
xj,2k

and
xj,2k+2
Z
q = w = 0 on (xj,2k+1 , xj,2k+2 ), r 6= 0, k = 0, 1, ..., mj − 1. (13)
xj,2k+1

Further, we assign some notations we will use later. For each j ∈ {0, 1, ..., n} and given (11)-(13),
500 A. Kablan et al.

let
xj,2k+2
Z
rjk = r, k = 0, 1, ..., mj − 1, (14)
xj,2k+1
xj,2k+1
Z xj,2k+1
Z
qjk = q, wjk = w, k = 0, 1, ..., mj .
xj,2k xj,2k

Before stating the main theorem of this paper, we determine the structure of the principal funda-
mental matrix of the system (6).

Lemma 3.1.
Let (5) and (11)-(13) hold. Let Φ(x, λ) = [φst (x, λ)] be the fundamental matrix solution of the
system (6) determined by the initial condition Φ(x00 , λ) = I (here Φ(x00 , λ) = Φ(x00 +, λ) denote
the right limit at point x00 ) for each λ ∈ C. Then, we have that
" #
1 0
Φ(x01 , λ) = , (15)
q00 − λw00 1

" #
1 + (q00 − λw00 )r00 r00
Φ(x03 , λ) = , (16)
φ21 (x03 , λ) 1 + (q01 − λw01 )r00

where
φ21 (x03 , λ) = (q00 − λw00 ) + (q01 − λw01 ) + (q00 − λw00 )(q01 − λw01 )r00 .

Then, in general, for k = 1, 2, ..., m0 ,

" #
1 r0,k−1
Φ(x0,2k+1 , λ) = Φ(x0,2k−1 , λ). (17)
q0k − λw0k 1 + (q0k − λw0k )r0,k−1

Proof:
Observe from (6) that u1 is constant on each subinterval of (x0 , x1 ) where r is identically zero and
u2 is constant on each subinterval of (x0 , x1 ) where both q and w are identically zero. So we obtain
the result from repeated applications of (6). 

Lemma 3.2.
h i
j
Let (5) and (11)-(13) hold. Let Ψj (x, λ) = ψst (x, λ) be the fundamental matrix solution of the
system (6) determined by the initial condition Ψj (xj , λ) = I for each j ∈ {1, ..., n} (here Ψj (xj , λ) =
Ψj (xj +, λ) denote the right limit at point xj ) for each λ ∈ C. Then, for each j ∈ {1, 2, ..., n} we
have that
" #
1 0
Ψj (xj1 , λ) = , (18)
qj0 − λwj0 1
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 501

" #
1 + (qj0 − λwj0 )rj0 rj0
Ψj (xj3 , λ) = j , (19)
ψ21 (xj3 , λ) 1 + (qj1 − λwj1 )rj0

where
j
ψ21 (xj3 , λ) = (qj0 − λwj0 ) + (qj1 − λwj1 ) + (qj0 − λwj0 )(qj1 − λwj1 )rj0 .

Then, in general, for k = 1, 2, ..., mj ,

" #
1 rj,k−1
Ψj (xj,2k+1 , λ) = Ψj (xj,2k−1 , λ). (20)
qjk − λwjk 1 + (qjk − λwjk )rj,k−1

Proof:
For each j ∈ {1, 2, ..., n} on the intervals (xj , xj+1 ) since the proof is similar to the proof of Lemma
3.1 we ommited it. 

Lemma 3.3.
Let (5) and (11)-(13) hold. Let Φ(x, λ) = [φst (x, λ)] be the fundamental matrix
h
solution
i
of the
j
system (6) determined by the initial condition Φ(x00 , λ) = I , and Ψj (x, λ) = ψst (x, λ) be the
fundamental matrix solution of the system (6) determined by the initial condition Ψj (xj , λ) = I,
for each j ∈ {1, ..., n} and λ ∈ C. Then, we have that
n
Y
Φ(b, λ) = Ψn−j (xn−j+1 , λ)Cn−j (21)
j=0

where C0 = I and Ψj (xj+1 , λ) = Ψj (xj+1 −, λ) denotes the left limit at point xj+1 for j = 1, 2, ..., n.

Proof:
In this proof for the sake of simplicity we will show Φ(x, λ) with Ψ0 (x, λ). From the transmission
condition (3) for j = 1 and the initial condition we have that
Φ(x1 +, λ) = C1 Ψ0 (x1 −, λ).

Additionally, from the definition of the fundamental matrix solution we can write that
Φ(x1 +, λ) = (Ψ1 (x2 , λ))−1 Φ(x2 −, λ).

Hence,
Φ(x2 −, λ) = Ψ1 (x2 , λ)C1 Ψ0 (x1 , λ). (22)
Now plugging (22) into the transmission condition (3) for j = 2 and using the initial condition and
definition of the fundamental matrix solution again, we arrive at the following equality
Φ(x3 −, λ) = Ψ2 (x3 , λ)C2 Ψ1 (x2 , λ)C1 Ψ0 (x1 , λ).

After repeating these processes we obtain the result. 

The structure of Φ given in Lemma 3.1 and mathematical induction yield the following.
502 A. Kablan et al.

Corollary 3.4.
For the fundamental matrix Φ we have that

φ11 (b, λ) = R11 λd + φe11 (λ), (23)

φ12 (b, λ) = R12 λd−1 + φe12 (λ), (24)

φ21 (b, λ) = R21 λd+1 + φe21 (λ), (25)

φ22 (b, λ) = R22 λd + φe22 (λ), (26)

where
n
X
d= mj , (27)
j=0

and R11 , R12 , R21 and R22 are related to α, for each j ∈ {0, 1, ..., n} rjk , k = 0, 1, ..., mj − 1; wjk and
qjk , k = 0, 1, ..., mj . φe11 (λ), φe12 (λ), φe21 (λ) and φe22 (λ) are functions of λ, in which the degrees of λ
are smaller than d, d − 1, d + 1 and d respectively.

Now we construct Sturm-Liouville problems with δ -interactions which have exactly d eigenvalues
for each d ∈ N.

Theorem 3.5.
For each j = 0, 1, ..., n, let mj ∈ N and let (5) and (11)-(13) hold. Let H = (hij )2×2 be defined as
in Lemma 2.2 and d be defined as in (27), then:
(1) If h21 6= 0, then the SLP with δ -interactions (1) has exactly d + 1 eigenvalues λk , k = 0, 1, ..., d.
(2) If h21 = 0, h11 6= 0, and h22 6= 0, then the SLP with δ -interactions (1) has exactly d eigenvalues
λk , k = 0, 1, ..., d − 1.

(3) If h21 = h11 = h22 = 0, but h12 6= 0, then the SLP with δ -interactions (1) has exactly d −
1 eigenvalues λk , k = 0, 1, ..., d − 2.

(4) If none of the above conditions holds, then the SLP with δ -interactions (1) either has
l eigenvalues for l ∈ {1, 2, ..., d − 2} or is degenerate.

Proof:
We prove the case (1), and the other cases can be proved in the same way. From (12) that the
degrees of φ11 (b, λ), φ12 (b, λ), φ21 (b, λ), and φ22 (b, λ) in λ are d, d − 1, d + 1, and d respectively.
Thus when h21 6= 0, we can conclude from Corollary 3.4 amd Lemma 2.2 that the degree of the
characteristic polynomial function ∆(λ) is d + 1, hence from the fundamental theorem of algebra
we find that ∆(λ) has exactly d + 1 roots. Then, the case (1) is achieved. 
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 503

4. Examples

We now work out a simple examples to illustrate the above study.

Example 4.1.
Consider the SLP with δ -interactions on J = (−6, −3) ∪ (−3, 2) ∪ (2, 5),
−(py 0 )0 + (2δ(x + 3) + δ(x − 2))y + qy = λwy. (28)
We know that, this equation is equivalent to the following SLP
−(py 0 )0 + qy = λwy (29)
with transmission conditions
(
y(−3−) − y(−3+) = 0
(30)
2y(−3−) + py 0 (−3−) − py 0 (−3+) = 0
and
(
y(2−) − y(2+) = 0
(31)
y(2−) + py 0 (2−) − py 0 (2+) = 0.

Then, let’s consider the boundary conditions

(
y(−6) = 0
(32)
y(5) = 0.

Let choose m0 = 1, m1 = 2 and m2 = 1 and p, q , w are piecewise constant functions defined as

follows:
  

 ∞, (−6, −5) 
 4, (−6, −5) 
 1, (−6, −5)
1
  
2 , (−5, −4) 0, (−5, −4) 0, (−5, −4)

 
 


 
 

∞, (−4, −3) 1, (−4, −3) 2, (−4, −3)

 
 


 
 

1
  
∞, (−3, −2) 2 , (−3, −2) 1, (−3, −2)

 
 


 
 

 1  
 4 , (−2, −1)  0, (−2, −1)  0, (−2, −1)

 
 

p(x) = ∞, (−1, 0) q(x) = 2, (−1, 0) w(x) = 3, (−1, 0)
  



 1, (0, 1) 


 0, (0, 1) 


 0, (0, 1)
1
∞, (1, 2) 1, (1, 2) 2 , (1, 2)

 
 


 
 

 1 , (2, 3)  1 , (2, 3)
  
 ∞, (2, 3)
  

 
 4 
 8
1
 3 , (3, 4) 0, (3, 4) 0, (3, 4)

 
 

 
 


 ∞, (4, 5) 
 3, (4, 5) 
 1, (4, 5)

From condition (32)

" # " #
10 00
A= , B= (33)
00 10

and from transmission conditions (30) and (31)

" # " #
10 10
C1 = , C2 = .
21 11
504 A. Kablan et al.

It follows from (21) that

Φ(5, λ) = Ψ2 (5, λ)C2 Ψ1 (2, λ)C1 Ψ0 (−3, λ). (34)

By using the matrices (33) in (8) and taking account (34) we arrive at

∆(λ) = φ12 (5, λ)

 2 2
 1 2 1
= ψ11 (5, λ) + ψ12 (5, λ) ψ11 (2, λ) + ψ12 (5, λ) ψ21 (2, λ)
 2 2
 1 2 1
0
+2 ψ11 (5, λ) + ψ12 (5, λ) ψ12 (2, λ) + 2ψ12 (5, λ) ψ22 (2, λ) ψ12 (−3, λ)
 2 2
 1 2 1
0
+ ψ11 (5, λ) + ψ12 (5, λ) ψ12 (2, λ) + ψ12 (5, λ) ψ22 (2, λ) ψ22 (−3, λ)

After some long calculations we find that

2223 2 8277 2161
∆(λ) = −135λ3 + λ − λ+ = 0.
2 4 2
As a result the SLP with δ -interactions (28) has exactly m0 + m1 + m2 − 1 = d − 1 = 3 eigenvalues
which are

λ1 = 0.95658, λ2 = 1.43138, λ3 = 5.84537.

Example 4.2.
Consider the SLP with δ -interactions on J = (−3, 0) ∪ (0, 4) ∪ (4, 9) ∪ (9, 12),

−(py 0 )0 + (δ(x − 0) + δ(x − 4) + δ(x − 9))y + qy = λwy. (35)

As in the first example, this equation is equivalent to the following SLP

−(py 0 )0 + qy = λwy (36)

with transmission conditions

(
y(0−) − y(0+) = 0
, (37)
y(0−) + py 0 (0−) − py 0 (0+) = 0

(
y(4−) − y(4+) = 0
(38)
y(4−) + py 0 (4−) − py 0 (4+) = 0

and
(
y(9−) − y(9+) = 0
(39)
y(9−) + py 0 (9−) − py 0 (9+) = 0.

Then, we can consider the following boundary conditions

(
py 0 (−3) + py 0 (12) = 0
(40)
py 0 (−3) = 0.
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 505

By selecting m0 = 1, m1 = 1, m2 = 2 and m3 = 1; let’s define the piecewise constant functions p,

q , w are as follows:
  
1


 ∞, (−3, −2) 

 1, (−3, −2) 

 2 , (−3, −2)
1, (−2, −1) 0, (−2, −1) 0, (−2, −1)

 
 


 
 

  



 ∞, (−1, 0) 


 2, (−1, 0) 


 1, (−1, 0)
1 1
∞, (0, 2) 4 , (0, 2) 2 , (0, 2)

 
 


 
 

  



 2, (2, 3) 


 0, (2, 3) 


 0, (2, 3)
∞, (3, 4) 4, (3, 4) 2, (3, 4)

 
 


 
 


 ∞, (4, 5) 
 1, (4, 5) 
 2, (4, 5)
p(x) = 1 q(x) = w(x) =
2, (5, 6) 0, (5, 6) 0, (5, 6)

 
 

  
∞, (6, 7) 3, (6, 7) 5, (6, 7)

 
 


 
 

  


 3, (7, 8) 

 0, (7, 8) 

 0, (7, 8)
  
∞, (8, 9) 0, (8, 9) 1, (8, 9)

 
 


 
 

  1 



 ∞ (9, 10) 


 2 , (9, 10)



 3, (9, 10)
1
 3 (10, 11) 0, (10, 11) 0, (10, 11)

 
 

 
 


 ∞ (11, 12) 
 1, (11, 12)  1 , (11, 12)

5

From the boundary condition (40)

" # " #
01 01
A= , B= (41)
01 00

and from the transmission conditions (37), (38) and (39)

" #
10
C1 = C2 = C3 = .
11

It follows from (21) that

Φ(12, λ) = Ψ3 (12, λ)C3 Ψ2 (9, λ)C2 Ψ1 (4, λ)C1 Ψ0 (0, λ). (42)
By using the matrices (41) in (8) and taking account (42) we arrive at
∆(λ) = φ21 (12, λ)
0 0
= (E + F ) ψ11 (0, λ) + F ψ21 (0, λ).
Here,
3 3
  2 2

E= ψ21 (12, λ) + ψ22 (12, λ)
ψ11 (9, λ) + ψ12 (9, λ)
3 2 3 2
1
+ψ22 (12, λ)ψ21 (9, λ) + ψ22 (12, λ)ψ22 (9, λ) ψ11 (4, λ)
 3 3
 2 3 2
1
+ ψ21 (12, λ) + ψ22 (12, λ) ψ12 (9, λ) + ψ22 (12, λ)ψ22 (9, λ) ψ21 (4, λ)
and
3 3
  2 2

F = ψ21 (12, λ) + ψ22 (12, λ) ψ11 (9, λ) + ψ12 (9, λ)
3 2 3 2
1
+ψ22 (12, λ)ψ21 (9, λ) + ψ22 (12, λ)ψ22 (9, λ) ψ12 (4, λ)
 3 3
 2 3 2
1
+ ψ21 (12, λ) + ψ22 (12, λ) ψ12 (9, λ) + ψ22 (12, λ)ψ22 (9, λ) ψ22 (4, λ).
506 A. Kablan et al.

After the end of lengthy calculations we find that

1484 5 125389 4 340565 3 291595 2 195309 8168
∆(λ) = 16λ6 − λ + λ − λ + λ − λ+ = 0.
5 60 48 24 20 3
Consequently the SLP with δ -interactions (35) has exactly m0 + m1 + m2 + m3 + 1 = d + 1 = 6
eigenvalues which are
λ1 = 0.56528, λ2 = 1.66858, λ3 = 1.93014,
λ4 = 2.93511, λ5 = 4.75851, λ6 = 6.69238.

5. Conclusion

In this paper we have enlarged the scope of the Sturm-Liouville problems with finite spectrum
which was devised initially for second and fourth order problems. We have extended the concept “
finite spectrum” to the Sturm-Liouville operator with δ -interactions. We have presented an example
to illustrate the discussion above.

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