Real and non-real eigenvalues of regular indefinite Sturm–Liouville problems

Yingchun Zhao; Mandula Buren

doi:10.1515/math-2025-0200

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Real and non-real eigenvalues of regular indefinite Sturm–Liouville problems

Yingchun Zhao and Mandula Buren

Published/Copyright: October 31, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this paper, we study the regular Sturm–Liouville problems with discontinuous boundary conditions and a weight function which changes sign. We show that the algebraic and geometric multiplicities of the real eigenvalues are the same for some classes of problems. And then the sufficient condition for the existence and exact number of non-real eigenvalues for the problems is obtained.

Keywords: Sturm–Liouville problems; discontinuous boundary conditions; non-real eigenvalues; indefinite weight function

MSC 2020: 34B20; 34B24

1 Introduction

Consider the boundary value problem

(1.1) − ( p y ′ ) ′ + q y = λ w y o n J = ( a , b ) , λ ∈ C , − ∞ < a < b < ∞ ,

(1.2) A Y ( a ) + B Y ( b ) = 0 , Y = y ( p y ′ ) ,

with coefficients p, q, w and matrices A , B ∈ M 2 ( C ) satisfying

(1.3) 1 p , q , | w | ∈ L 1 ( J , R ) , p > 0 a . e . on J ,

(1.4) r a n k ( A : B ) = 2 and A E A * = B E B * , E = 0 − 1 1 0 .

Here L 1 ( J , R ) denotes the real valued Lebesgue integrable functions on J and we use the notation M 2 ( C ) to denote the 2 × 2 matrices over C , the complex numbers.

In 1918, Richardson [1] showed that when w changes sign there may be non-real eigenvalues. We will refer to “weight” functions w which change sign on J and to the corresponding boundary value problems as “indefinite”. Which indefinite problems have non-real eigenvalues? How many? These questions are still open problems today. The eigenvalue problems of the indefinite Sturm–Liouville operators have been studied by many authors in recent years, see [2], [3], [4], [5].

Recently there has been a lot of interest in the literature about Sturm–Liouville problems with discontinuous boundary conditions specified at an interior point of the underlying interval J. Such conditions are known by various names including: transmission conditions [6], interface conditions [7], discontinuous conditions [8], multi-point conditions [9],10], point interactions (in the physics literature) [11],12], conditions on trees graphs or networks [13], [14], [15]. For an informative survey of such problems arising in applications, including an extensive bibliography and historical notes, see [14],15]. These problems are not covered by the classical theory, since it is well known that all solutions y and their quasi-derivatives (py′) are continuous on J in the classical case. Thus, in particular, this theory does not produce eigenfunctions satisfying discontinuous boundary conditions.

In this paper, we study real and non-real eigenvalues of boundary value problems consisting of the regular Sturm–Liouville equation

(1.5) − ( p y ′ ) ′ + q y = λ w y , λ ∈ C , o n J = ( a , c ) ∪ ( c , b )

with coefficients p, q, w satisfying the conditions

(1.6) 1 p , q , w ∈ L 1 ( J , R ) , p > 0 a . e . o n J , w c h a n g e s sign o n J ,

together with discontinuous boundary conditions

(1.7) A Y ( a ) + B Y ( b ) = 0 , C Y ( c − ) + D Y ( c + ) = 0 , a < c < b ,

and for h , k ∈ R , h > 0 , k > 0 , the matrices A , B , C , D ∈ M 2 ( C ) satisfy the following conditions

(1.8) h A E A * = k B E B * , r a n k ( A : B ) = 2 , h C E C * = k D E D * , r a n k ( C : D ) = 2 .

Remark 1.

“w changes sign on J” in (1.6) means that there are three possibilities for sign changes in weight function w as follows:

w is positive (or negative) on (a, c) and negative (or positive) on (c, b);
w changes sign on one of the two intervals (a, c) and (c, b) when the sign of w is unchanged on the other interval;
w changes sign on both intervals (a, c) and (c, b).

Remark 2.

When C = −D = I, the identity matrix, and h = k, the boundary conditions (1.7)– (1.8) reduce to the conditions (1.2)– (1.4).

Remark 3.

Note that the conditions (1.8) do not depend on h and k if the boundary conditions (1.7) are separated. If one of these boudary conditions is coupled then there is a dependence on h and k.

Remark 4.

Here and below, when we consider two problems which have the same coefficients p, q on the same interval J, the same boundary condition, one with weight function w which changes sign and the other with weight function |w|, we refer to the former as the w problem and to the latter as the |w| problem.

Our approach is to apply the 2-interval theory developed by Everitt and Zettl [16] and its extensions developed by Mukhtarov and Yakubov [6] and Wang et al. [17] to the intervals J ₁ = (a, c) and J ₂ = (c, b) with a < c < b. We also apply the two parameter spectral theory and the “eigencurve” method to study the existence and nonexistence of non-real eigenvalues of the problems. The main focus of the present paper is to find sufficient conditions for the existence of non-real eigenvalues and determine the exact number of non-real eigenvalues for some classes of problems. These results can be viewed as a partial answer to the open problem number X in the list of open problems in [18], p. 300].

The paper is organized as follows: following this Introduction we study the algebraic and geometric multiplicities of the real eigenvalues in Section 2 and the existence and exact number of non-real eigenvalues in Section 3.

2 Algebraic and geometric multiplicities of the real eigenvalues

In this section we discuss the equality of algebraic and geometric multiplicities of the real eigenvalues.

Let w _r = w on J _r (r = 1, 2) and H _r = L ²(J _r, |w _r|), r = 1, 2. Denoting elements of H _u in bold face type: f = {f ₁, f ₂} with f ₁ ∈ H ₁, f ₂ ∈ H ₂, the inner product in H _u is given by

f , g = h ∫ J 1 f 1 g 1 ̄ w 1 + k ∫ J 2 f 2 g 2 ̄ w 2 , h > 0 , k > 0 .

To “transfer” results from the right-definite theory with weight function |w| to corresponding problems for the indefinite weight function w, we use results and methods of operator theory in the Krein space K = (H _u, [⋅, ⋅]). This is the space of all of functions from H _u, but with the indefinite inner product

[ f , g ] = h ∫ J 1 f 1 g 1 ̄ w 1 + k ∫ J 2 f 2 g 2 ̄ w 2 .

Recall that a self-adjoint operator A in the Krein space K has n ( ∈ N 0 ) negative squares, if the Hermitian form [A f, g] (f, g ∈ D(A)) has n negative squares, that is, there exists an n-dimensional subspace M in D(A) such that [A f, f] < 0, if f ∈ M and f ≠ {0, 0}, but no (n + 1)-dimensional subspace with this property [2].

Definition 1.

Let A be a linear operator in the Krein space K. The set

N ( A ) = { f ∈ D ( A ) | A f = 0 }

is a subspace of K, termed the null space of A. If for a number λ ∈ C the operator A − λI is not invertible, we say that λ is an eigenvalue of A, the subspace N ( A − λ I ) is called the eigenspace associated to λ, and the subspace

M λ = ⋃ j = 1 ∞ N ( ( A − λ I ) j )

is called the algebraic eigenspace. The dimension of M λ is called the algebraic multiplicity of λ as an eigenvalue of A. The eigenvalue λ is said to be semi-simple, if M λ = N ( A − λ I ) . In particular, if M λ = 1 , eigenvalue λ is said to be simple.

Remark 5.

Below, let A be the operator generated by the w problem (1.5), (1.7)– (1.8) and let S be the operator generated by the corresponding |w| problem. It is well known that the operator A is self-adjoint in the Krein space K, since the operator S is self-adjoint in the Hilbert space H = ( H u , ⋅ , ⋅ ) .

Lemma 1.

Consider the w problem (1.5)–(1.8) and the corresponding |w| problem. Assume that 0 is not an eigenvalue of the w problem and the corresponding |w| problem has n(≥1) negative simple eigenvalues. If λ 0 ∈ R is a real eigenvalue of the w problem and y ₀ is the corresponding eigenfunction, then y ₀ is linearly independent of the eigenfunctions corresponding to the negative eigenvalues of the corresponding |w| problem.

Proof.

Let φ ₁, φ ₂, …, φ _n are the eigenfunctions corresponding to the negative eigenvalues ξ ₁, ξ ₂, …, ξ _n of the operator S. We assume that y ₀, φ ₁, φ ₂, …, φ _n are linearly dependent. So there exist nonzero constants c ₁, c ₂, …, c _m (1 ≤ m ≤ n) such that

y 0 = ∑ i = 1 m c i φ k i , 1 ≤ k 1 < … < k i < … < k m ≤ n .

Then we obtain that

( A − λ 0 I ) y 0 = ∑ i = 1 m c i [ ξ k i sgn w φ k i − λ 0 φ k i ] = 0 ,

i.e.,

∑ i = 1 m c i [ ξ k i − λ 0 ] φ k i ( t ) = 0 , when w ( t ) > 0 , ∑ i = 1 m c i [ ξ k i + λ 0 ] φ k i ( t ) = 0 , when w ( t ) < 0 .

Since φ ₁, φ ₂, …, φ _n are the corresponding eigenfunctions to the eigenvalues

ξ 1 , ξ 2 , … , ξ n ,

then we have that

∑ i = 1 m c i [ ξ k i − λ 0 ] ξ k i j φ k i ( t ) = 0 , when w ( t ) > 0 , j = 1,2 , … , m − 1 , ∑ i = 1 m c i [ ξ k i + λ 0 ] ξ k i j φ k i ( t ) = 0 , when w ( t ) < 0 , j = 1,2 , … , m − 1 .

It is well known that

1 1 … 1 ξ k 1 ξ k 2 … ξ k m … … … … ξ k 1 m − 1 ξ k 2 m − 1 … ξ k m m − 1 ≠ 0 ,

and then if λ 0 ≠ ξ k i ( λ 0 ≠ − ξ k i ) ( i = 1,2 , … , m ) , we obtain that

φ k i ( t ) = 0 , when w ( t ) > 0 ( φ k i ( t ) = 0 , when w ( t ) < 0 ) , i = 1,2 , … , m .

On the other hand, the number of zeros or the oscillation count of the eigenfunction φ k i in the interval J is finite. So there must exist some k i 1 , k i 2 , 1 ≤ i 1 , i 2 ≤ m such that λ 0 = ξ k i 1 , λ 0 = − ξ k i 2 . Evidently, these two equations do not hold simultaneously. Hence, y ₀ is linearly independent of φ ₁, φ ₂, …, φ _n. □

Assumption 1.

We assume that the |w| problem has one negative simple eigenvalues, that is

ξ 1 < 0 < ξ 2 ,

and φ j = { φ j 1 , φ j 2 } φ j , φ j = 1 , j = 1,2 are the eigenfunctions corresponding to the ξ _j. Let

α = ξ 2 ξ 2 − ξ 1 , β 1 = h ∫ J 12 | φ 11 | 2 | w 1 | + k ∫ J 22 | φ 12 | 2 | w 2 | 2 , β 2 = h ∫ J 11 | φ 11 | 2 | w 1 | + k ∫ J 21 | φ 12 | 2 | w 2 | 2 ,

where J _r1 = {t ∈ J _r|w _r(t) > 0}, J _r2 = {t ∈ J _r|w _r(t) < 0}, r = 1, 2.

Theorem 1.

Consider the w problem (1.5)–(1.8) and the corresponding |w| problem. If the Assumption 1 holds and

(2.1) min { 2 β 1 , 2 β 2 } ≤ 9 16 α < α < max { 2 β 1 , 2 β 2 } ≤ 25 16 α ,

then the algebraic and geometric multiplicities of the real eigenvalues of the w problem are the same.

Proof.

Let λ 0 ∈ R be a real eigenvalue of the operator A and let M λ 0 , G λ 0 be the corresponding algebraic eigenspace and geometric eigenspace, respectively. Assume that the algebraic multiplicity of λ ₀ is greater than the geometric multiplicity of λ ₀, i.e. dim M λ 0 > dim G λ 0 . Hence, there must exist an nonzero element y 0 ∈ M λ 0 such that

( A − λ 0 ) 2 y 0 = 0 , ( A − λ 0 ) y 0 ≠ 0 .

Let y 0 ̃ = ( A − λ 0 ) y 0 ( A − λ 0 ) y 0 , ( A − λ 0 ) y 0 . It is obvious that y 0 ̃ is an eigenfunction corresponding to the eigenvalue λ ₀ of the operator A and we know that

(2.2) y 0 ̃ , y 0 ̃ = [ y 0 , ( A − λ 0 I ) y 0 ̃ ] ( A − λ 0 ) y 0 , ( A − λ 0 ) y 0 = [ y 0 , 0 ] ( A − λ 0 ) y 0 , ( A − λ 0 ) y 0 = 0 , y 0 ̃ , y 0 ̃ = 1 .

It follows from Lemma 1 that y 0 ̃ is linearly independent of φ ₁. Let

φ 0 = y 0 ̃ − y 0 ̃ , φ 1 φ 1 , F = s p a n { φ 0 , φ 1 } , y = c 0 φ 0 + c 1 φ 1 , c 0 , c 1 ∈ C , y , y = 1 .

Since φ 0 , φ 1 = 0 , and we have

y ∈ F , | c 1 | 2 + | c 0 | 2 φ 0 , φ 0 = 1 , 0 < φ 0 , φ 0 = 1 − y 0 ̃ , φ 1 2 .

And then we obtain that

S y , y = c 0 S φ 0 + c 1 S φ 1 , c 0 φ 0 + c 1 φ 1 = | c 1 | 2 ξ 1 + | c 0 | 2 S φ 0 , φ 0 ≤ S φ 0 , φ 0 φ 0 , φ 0 = − y 0 ̃ , φ 1 2 ξ 1 1 − y 0 ̃ , φ 1 2 .

So we have

ξ 2 ≤ sup { S y , y : y ∈ F , y , y = 1 } ≤ − y 0 ̃ , φ 1 2 ξ 1 1 − y 0 ̃ , φ 1 2 .

Therefore

(2.3) ξ 2 ≤ y 0 ̃ , φ 1 2 [ ξ 2 − ξ 1 ] .

Since λ 0 [ y 0 ̃ , φ 1 ] = ξ 1 y 0 ̃ , φ 1 , we have from (2.2) that

(2.4) ( λ 0 − ξ 1 ) 2 y 0 ̃ , φ 1 2 = 4 λ 0 2 h ∫ J 12 y 01 ̃ φ 11 ̄ | w 1 | + k ∫ J 22 y 02 ̃ φ 12 ̄ | w 2 | 2 ≤ 2 λ 0 2 h ∫ J 12 | φ 11 | 2 | w 1 | + k ∫ J 22 | φ 12 | 2 | w 2 | 2 ≤ 2 λ 0 2 β 1 ,

(2.5) ( λ 0 + ξ 1 ) 2 y 0 ̃ , φ 1 2 = 4 λ 0 2 h ∫ J 11 y 01 ̃ φ 11 ̄ | w 1 | + k ∫ J 21 y 02 ̃ φ 12 ̄ | w 2 | 2 ≤ 2 λ 0 2 h ∫ J 11 | φ 11 | 2 | w 1 | + k ∫ J 21 | φ 12 | 2 | w 2 | 2 ≤ 2 λ 0 2 β 2 .

From (2.3)–(2.5) we have

(2.6) α ( λ 0 − ξ 1 ) 2 ≤ 2 λ 0 2 β 1 ,

(2.7) α ( λ 0 + ξ 1 ) 2 ≤ 2 λ 0 2 β 2 .

If 2β ₁ < α < 2β ₂, and we obtain that

λ 0 ∈ α + 2 β 1 α α − 2 β 1 ξ 1 , α − 2 β 1 α α − 2 β 1 ξ 1 ∩ − ∞ , − α − 2 β 2 α α − 2 β 2 ξ 1 ,

if 2β ₂ < α < 2β ₁, and we obtain that

λ 0 ∈ − α + 2 β 2 α α − 2 β 2 ξ 1 , − α − 2 β 2 α α − 2 β 2 ξ 1 ∩ α + 2 β 1 α α − 2 β 1 ξ 1 , + ∞ .

But from (2.1), we know that

− α − 2 β 2 α α − 2 β 2 ξ 1 < α + 2 β 1 α α − 2 β 1 ξ 1 .

It indicates that the operator A has no such real eigenvalue λ ₀ that the corresponding eigenfunction y ₀ satisfies λ ₀[y ₀, y ₀] = 0. Hence the algebraic and geometric multiplicities of the real eigenvalues of the w problem are the same. □

3 Existence and exact number of the non-real eigenvalues

The main objective of this section is to study the existence and number of the non-real eigenvalues.

Definition 2.

An element x ∈ K is called positive (negative, neutral, respectively) if [x, x] > 0 ([x, x] < 0, [x, x] = 0, respectively). A subspace M of K is called positive (negative) if all its nonzero elements are positive (negative). A subspace M of K is called neutral if [x, x] = 0 for all x ∈ M.

The signature signM of a subspace M of K is a triplet signM = (k ₋, k ₀, k ₊), where k ₊ and k ₋ denote respectively the dimension of a maximal positive and maximal negative subspace of M, and k ₀ is the dimension of the isotropic subspace of M, that is, the set M ∩ M ^[⊥] of all the elements of M, where M ^[⊥] = {x ∈ K|[x, y] = 0, y ∈ M}.

Lemma 2.

Let A be a self-adjoint operator in the Krein space K, assume that ρ(A) is nonempty and that A has k negative squares. Then the non-real spectrum of A consists of at most k pairs { μ i , μ ̄ i } , μ i ∈ C + , of eigenvalues with finite-dimensional algebraic eigenspaces. Furthermore, we denote for an eigenvalue λ of A the signature of the indefinite inner product [., .] on the algebraic eigenspace by {k ₋(λ), k ₀(λ), k ₊(λ)}, and then

(3.1) ∑ λ ∈ σ p ( A ) ⋂ ( − ∞ , 0 ) ( k + ( λ ) + k 0 ( λ ) ) + ∑ λ ∈ σ p ( A ) ⋂ ( 0 , ∞ ) ( k − ( λ ) + k 0 ( λ ) ) + ∑ i k 0 ( μ i ) ≤ k ,

and if 0 ∉ σ _p(A), then equality holds in (3.1).

Proof.

See Theorem 3.1 in [2]. □

Theorem 2.

Consider the w problem (1.5)–(1.8) and the corresponding |w| problem. If the Assumption 1 holds and

(3.2) min { 4 β 1 , 4 β 2 } ≤ 25 64 α < α < max { 4 β 1 , 4 β 2 } ≤ 9 64 α ,

then the w problem has exactly 2 non-real eigenvalues.

Proof.

Let λ 0 ∈ R be a real eigenvalue of the operator A and y ₀ be the corresponding eigenfunction. Assume that

λ 0 [ y 0 , y 0 ] ≤ 0 , y 0 , y 0 = 1 .

It follows from Lemma 1 that y ₀ is linearly independent of φ ₁. Let

φ 0 = y 0 − y 0 , φ 1 φ 1 , F = s p a n { φ 0 , φ 1 } , y = c 0 φ 0 + c 1 φ 1 , c 0 , c 1 ∈ C , y , y = 1 .

Since φ 0 , φ 1 = 0 , and we have

y ∈ F , | c 1 | 2 + | c 0 | 2 φ 0 , φ 0 = 1 , 0 < φ 0 , φ 0 = 1 − y 0 , φ 1 2 .

And then we obtain that

S y , y = c 0 S φ 0 + c 1 S φ 1 , c 0 φ 0 + c 1 φ 1 = | c 1 | 2 ξ 1 + | c 0 | 2 S φ 0 , φ 0 ≤ S φ 0 , φ 0 φ 0 , φ 0 = λ 0 [ y 0 , y 0 ] − y 0 , φ 1 2 ξ 1 1 − y 0 , φ 1 2 ≤ − y 0 , φ 1 2 ξ 1 1 − y 0 , φ 1 2 .

So we have

ξ 2 ≤ sup { S y , y : y ∈ F , y , y = 1 } ≤ − y 0 , φ 1 2 ξ 1 1 − y 0 , φ 1 2 .

Therefore

(3.3) ξ 2 ≤ y 0 , φ 1 2 [ ξ 2 − ξ 1 ] .

Since λ 0 [ y 0 , φ 1 ] = ξ 1 y 0 , φ 1 , from (2.2) we obtain that

(3.4) ( λ 0 − ξ 1 ) 2 y 0 , φ 1 2 = 4 λ 0 2 h ∫ J 12 y 01 φ 11 ̄ | w 1 | + k ∫ J 22 y 02 φ 12 ̄ | w 2 | 2 ≤ 4 λ 0 2 h ∫ J 12 | φ 11 | 2 | w 1 | + k ∫ J 22 | φ 12 | 2 | w 2 | 2 ≤ 4 λ 0 2 β 1 ,

(3.5) ( λ 0 + ξ 1 ) 2 y 0 , φ 1 2 = 4 λ 0 2 h ∫ J 11 y 01 φ 11 ̄ | w 1 | + k ∫ J 21 y 02 φ 12 ̄ | w 2 | 2 ≤ 4 λ 0 2 h ∫ J 11 | φ 11 | 2 | w 1 | + k ∫ J 21 | φ 12 | 2 | w 2 | 2 ≤ 4 λ 0 2 β 2 .

We obtain from (3.3)–(3.5) that

(3.6) α ( λ 0 − ξ 1 ) 2 ≤ 4 λ 0 2 β 1 ,

(3.7) α ( λ 0 + ξ 1 ) 2 ≤ 4 λ 0 2 β 2 .

If 4β ₁ < α < 4β ₂, we obtain that

λ 0 ∈ α + 4 β 1 α α − 4 β 1 ξ 1 , α − 4 β 1 α α − 4 β 1 ξ 1 ∩ − ∞ , − α − 4 β 2 α α − 4 β 2 ξ 1 ,

if 4β ₂ < α < 4β ₁, we obtain that

λ 0 ∈ − α + 4 β 2 α α − 4 β 2 ξ 1 , − α − 4 β 2 α α − 4 β 2 ξ 1 ∩ α + 4 β 1 α α − 4 β 1 ξ 1 , + ∞ .

But from (3.2), we know that

− α − 4 β 2 α α − 4 β 2 ξ 1 < α + 4 β 1 α α − 4 β 1 ξ 1 .

It indicates that the operator A has no such real eigenvalue λ ₀ and the corresponding eigenfunction y ₀ satisfies λ ₀[y ₀, y ₀] ≤ 0. Then it follows directly from Lemma 2 and Theorem 1 that the w problem has exactly 2 non-real eigenvalues. □

Corresponding author: Yingchun Zhao, College of Mathematics Science, Inner Mongolia Normal University, Hohhot, 010021, China, E-mail: zhyc116@126.com

Funding source: National Nature Science Foundation of China

Award Identifier / Grant number: Grant No.12162003

Funding source: Natural Science Foundation of Inner Mongolia

Award Identifier / Grant number: Grant No.2024LHMS01008

Funding source: Research Foundation for Advanced Talents of Inner Mongolia Normal University

Award Identifier / Grant number: Grant No.2025YJRC072 and Grant No.2025YJRC073

Research ethics: Not applied.
Informed consent: Not applied.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. All authors contributed equally in this work.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflicts of interest.
Research funding: The work of the first and second authors is supported by the National Nature Science Foundation of China (Grant No.12162003), the Natural Science Foundation of Inner Mongolia (Grant No.2024LHMS01008), and the Research Foundation for Advanced Talents of Inner Mongolia Normal University (Grant No.2025YJRC072 and Grant No.2025YJRC073).
Data availability: No data sets were generated or analyzed during the current study.

References

[1] R. G. D. Richardson, Theorems of oscillation for two linear differential equations of second order with two parameters, Trans. Amer. Math. Soc. 13 (1912), no. 1, 22–34, https://doi.org/10.1090/S0002-9947-1912-1500902-8.Search in Google Scholar

[2] J. Behrndt and C. Trunk, On the negative squares of indefinite Sturm–Liouville operators, J. Differential Equations 238 (2007), no. 2, 491–519, https://doi.org/10.1016/j.jde.2007.01.026.Search in Google Scholar

[3] J. G. Qi, B. Xie, and S. Z. Chen, The upper and lower bounds on non-real eigenvalues of indefinite Sturm–Liouville problems, Proc. Amer. Math. Soc. 144 (2016), no. 2, 547–559, https://doi.org/10.1090/proc/12854.Search in Google Scholar

[4] J. Behrndt, P. Schmitz, G. Teschl, and C. Trunk, Perturbation and spectral theory for singular indefinite Sturm–Liouville operators, J. Differential Equations 405 (2024), 151–178, https://doi.org/10.1016/j.jde.2024.05.043.Search in Google Scholar

[5] X. X. Han and S. Fu, Nonreal eigenvalues of singular indefinite Sturm–Liouville problems, Rocky Mountain J. Math. 54 (2024), no. 5, 1345–1357, https://doi.org/10.1216/rmj.2024.54.1345.Search in Google Scholar

[6] O. S. Mukhtarov and S. Yakubov, Problems for differential equations with transmission conditions, Appl. Anal. 81 (2002), no. 5, 1033–1064, https://doi.org/10.1080/000368102761599733.Search in Google Scholar

[7] Q. Kong and Q. R. Wang, Using time scales to study multi-interval Sturm–Liouville problems with interface conditions, Results Math. 63 (2013), 451–465, https://doi.org/10.1007/s00025-011-0208-8.Search in Google Scholar

[8] C.-T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 (2008), no. 1, 266–272, https://doi.org/10.1016/j.jmaa.2008.05.097.Search in Google Scholar

[9] L. Kong, Q. Kong, M. K. Kwong, and J. S. W. Wong, Linear Sturm–Liouville problems with multi-point boundary conditions, Math. Nachr. 286 (2013), nos. 11–12, 1167–1179, https://doi.org/10.1002/mana.201200187.Search in Google Scholar

[10] A. Zettl, The lack of self-adjointness in three point boundary value problems, Proc. Amer. Math. Soc. 17 (1966), no. 2, 368–371, https://doi.org/10.1090/S0002-9939-1966-0190422-9.Search in Google Scholar

[11] D. Buschmann, G. Stolz, and J. Weidmann, One-dimensional Schrödinger operators with point interactions, J. Reine Angew. Math. 467 (1995), no. 467, 169–186, https://doi.org/10.1515/crll.1995.467.169.Search in Google Scholar

[12] C. S. Christ and G. Stolz, Spectral theory of one-dimensional Schrödinger operators with point interactions, J. Math. Anal. Appl. 184 (1994), no. 3, 491–516, https://doi.org/10.1006/jmaa.1994.1218.Search in Google Scholar

[13] M. Schmied, R. Sims, and T. Teschl, On the absolutely continuous spectrum of Sturm–Liouville operators with applications to radial quantum trees, Oper. Matrices 2 (2008), no. 3, 417–434, https://doi.org/10.7153/oam-02-25.Search in Google Scholar

[14] Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks (geometric graphs), J. Math. Sci. 119 (2004), no. 6, 691–718, https://doi.org/10.1023/B:JOTH.0000012752.77290.fa.Search in Google Scholar

[15] Y. V. Pokornyi and V. L. Pryadiev, The qualitative Sturm–Liouville theory on spacial networks, J. Math. Sci. 119 (2004), no. 6, 788–835, https://doi.org/10.1023/B:JOTH.0000012756.25200.56.Search in Google Scholar

[16] W. N. Everitt and A. Zettl, Sturm–Liouville differential operators in direct sum spaces, Rocky Mountain J. Math. 16 (1986), no. 3, 497–516, https://doi.org/10.2307/44237009.Search in Google Scholar

[17] A. P. Wang, J. Sun, and A. Zettl, Two-interval Sturm–Liouville operators in modified Hilbert spaces, J. Math. Anal. Appl. 328 (2007), no. 1, 390–399, https://doi.org/10.1016/j.jmaa.2006.05.058.Search in Google Scholar

[18] A. Zettl, Sturm–Liouville Theory, American Mathematical Society, Providence, 2005.Search in Google Scholar

Received: 2025-02-04

Accepted: 2025-08-19

Published Online: 2025-10-31

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Keywords for this article

Sturm–Liouville problems; discontinuous boundary conditions; non-real eigenvalues; indefinite weight function

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