Home Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
Article Open Access

Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter

  • Aynur Çöl EMAIL logo
Published/Copyright: August 30, 2024
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

We consider the boundary value problem generated by a system of Dirac equations with polynomials of spectral parameter in the boundary condition. We investigate the continuity of the scattering function and provide Levinson-type formula, which shows that the increment of the scattering function’s logarithm is related to the number of eigenvalues of the boundary value problem.

Keywords: Levinson-type formula; scattering data; spectral parameter dependent boundary condition
MSC 2010: 34L25; 81U40; 34L40; 34B07; 81Q10

1 Introduction

In 1949, Levinson established a theorem for the Schrödinger equation in nonrelativistic quantum mechanics [1]. The theorem relates the number of bound states in a given angular momentum channel to the phase shift at threshold, and it is also responsible for the inverse scattering problem which consists in the recovery of potential from spectral characteristics of the boundary value problem.

Klaus [2] presented the relativistic analog of Levinson’s theorem. The author considered the Dirac equation for a particle moving in a central electrostatic potential V ( r ) . With the help of separation of variables, the following systems of equations were studied:

H κ ( c ) ψ = c 0 − 1 1 0 ψ ′ + m c 2 + V ( r ) κ ⁄ r κ ⁄ r − m c 2 + V ( r ) ψ = E ψ , ψ = ψ 1 ( r ) ψ 2 ( r )

on 0 < r < ∞ . Here, m is the mass of the particle, c is the velocity of light, E is the energy (in units where ℏ = 1 ), and κ is a nonzero integer. It is assumed that

(1) ∫ 0 ∞ ( 1 + r ) ∣ V ( r ) ∣ d r < ∞ ,

which guarantees that the differential operator H is limit point at zero (it is always limit point at infinity) so that H κ can be considered as a self-adjoint operator in a Hilbert space of vector-valued functions. The spectrum of H κ is absolutely continuous on ( − ∞ , − m c 2 ] ∪ [ m c 2 , ∞ ) and consists of at most finitely many (simple) eigenvalues in the gap [ − m c 2 , m c 2 ] .

In his work, Klaus [2] showed that there exists a deep connection between the continuous part and the discrete part of the spectrum, which is the content of Levinson’s theorem in the Schrödinger case. When the potential satisfied the condition (1), a relation between the number of bound states in the spectral gap [ − m , m ] and the variation of an appropriate phase δ κ ( − m ) − δ κ ( m ) along the continuous part of the spectrum ( − ∞ , − m ] ∪ [ m , ∞ ) was established for the Dirac equation.

Relativistic versions of Levinson’s theorem for the Dirac operator were presented in the monographs [3–9]. Also, Levinson’s theorem was studied by many authors in [10–18] and references therein. For the Sturm Liouville operator, Levinson formula was obtained in [19]. In the case of the boundary condition containing spectral parameter, Levinson-type formulas were provided in [20–23]. There were also studies on scattering problems in [24–30] and other papers.

In the present work, we consider the boundary value problem generated by the canonical system of Dirac differential equations

(2) B y ′ + m T y + Ω ( x ) y = λ y , ( 0 ≤ x < ∞ )

with the boundary condition

(3) ( α 0 + α 1 λ + α 2 λ 2 ) y 1 ( 0 ) − ( β 0 + β 1 λ + β 2 λ 2 ) y 2 ( 0 ) = 0 ,

where λ is a spectral parameter,

y = y 1 ( x ) y 2 ( x ) , B = 0 1 − 1 0 , T = 1 0 0 − 1 ,

m > 0 is the mass and the potential matrix Ω ( x ) has the canonical form

Ω ( x ) = p ( x ) q ( x ) q ( x ) − p ( x )

with p ( x ) and q ( x ) being real valued and the inequalities

(4) ∣ p ( x ) ∣ ≤ c ( 1 + x ) 2 + ε , ∣ q ( x ) ∣ ≤ c ( 1 + x ) 1 + ε

being satisfied for positive numbers c and ε .

Assume that the relations

(5) α 0 β 1 − α 1 β 0 < 0 , α 0 β 2 − α 2 β 0 = 0 , α 1 β 2 − α 2 β 1 < 0

hold for α i , β j ∈ R ( i , j = 0 , 1 , 2 ) .

The purpose of this article is to establish Levinson-type formula for a system of Dirac equations with spectral parameter in the boundary condition. In the process, we examine the continuity of the scattering function for the boundary value problem (2)–(4) and present the relation between the number of eigenvalues of the boundary value problem (2)–(4) in the interval ( − m , m ) and the variation of the argument of S ( λ ) over ( − ∞ , − m ] ∪ [ m , ∞ ) .

The rest of this article is organized as follows. In Section 2, the required theorems and lemmas for the boundary value problem (2)–(4) are provided. In Section 3, the continuity of the scattering function is investigated. Finally, Levinson-type formula is presented in Section 4.

2 Preliminaries

In this section, we mention basic tools needed and results from the work [25] that allow us to achieve this research.

It is easily seen from Theorem 1.2.1 in [24] that when Ω ( x ) satisfies condition (4), then equation (2) has a unique matrix solution f ( x , λ ) of order ( 2 × 1 ) which tends to f 0 ( x , λ ) as x → ∞ , Im λ ≥ 0 , and there exists a matrix function A ( x , t ) of order 2 such that

(6) f ( x , λ ) = f 0 ( x , λ ) + ∫ x ∞ A ( x , t ) f 0 ( t , λ ) d t .

The matrix function

f 0 ( x , λ ) = λ + m k − i e i k x

is a solution of (2) when Ω ( x ) = 0 , where k = λ 1 − m 2 λ 2 , ∣ λ ∣ > m . Furthermore, the components of the matrix kernel A ( x , t ) satisfy the inequalities

(7) ∣ A i j ∣ ≤ c 1 ( 1 + x ) ( 1 + t ) 1 + ε , i ≠ j ,

(8) ∣ A i i ∣ ≤ c 2 ( 1 + t ) 1 + ε , i = 1 , 2 .

Let us denote by

φ ( x , λ ) = φ 1 ( x , λ ) φ 2 ( x , λ )

the matrix solution of order 2 × 1 of equation (2) with the initial data

φ 1 ( 0 , λ ) = β 0 + β 1 λ + β 2 λ 2 , φ 2 ( 0 , λ ) = α 0 + α 1 λ + α 2 λ 2 .

It is evident that the solution φ ( x , λ ) satisfies the boundary condition (3).

Lemma 1

[25] For all λ in the intervals ( − ∞ , − m ) and ( m , ∞ ) , the following identity is valid:

(9) 2 i λ + m k φ ( x , λ ) E ( λ ) = f ( x , λ ) ¯ − S ( λ ) f ( x , λ ) ,

where

(10) E ( λ ) = ( α 0 + α 1 λ + α 2 λ 2 ) f 1 ( 0 , λ ) − ( β 0 + β 1 λ + β 2 λ 2 ) f 2 ( 0 , λ ) ,

(11) S ( λ ) = E ( λ ) ¯ E ( λ ) ,

and

S − 1 ( λ ) = S ( λ ) ¯ , ∣ S ( λ ) ∣ = 1 .

The function S ( λ ) defined by (11) is called the scattering function of the boundary value problem (2)–(4).

Lemma 2

[25] The function E ( λ ) is analytic in the upper half plane Im λ > 0 , continuous along the real axis except at λ = m and has only a finite number of zeros in the interval ( − m , m ) . All the zeros in ( − m , m ) are simple.

The zeros λ k , k = 1 , … , n , of the function E ( λ ) are called the singular values of the boundary value problem (2)–(4).

The numbers m k , k = 1 , … , n , are defined by using

m k − 2 ≡ ∫ 0 ∞ f * ( x , λ k ) f ( x , λ k ) d x + [ α 1 β 0 − α 0 β 1 + ( α 2 β 1 − α 1 β 2 ) ∣ λ k ∣ 2 ] ∣ α 0 + α 1 λ k + α 2 λ k 2 ∣ 2 ∣ f 2 ( 0 , λ k ) ∣ 2

and called the norming numbers of boundary value problem (2)–(4), where f * denotes the transposed vector of f ¯ .

Definition 1

[25] The collection

{ S ( λ ) , { λ k } k = 1 n , { m k } k = 1 n }

is called scattering data of the boundary value problem (2)–(4).

Theorem 1

[25] For every fixed x ≥ 0 , the kernel A ( x , t ) of solution (6) satisfies the integral equation

(12) A ( x , y ) + F ( x + y ) + ∫ x ∞ A ( x , t ) F ( t + y ) d t = 0 , y > x ,

where

F ( x + y ) = Re 1 2 π ∫ ∣ λ ∣ > m α 2 − i β 2 α 2 + i β 2 − S ( λ ) λ + m k − i − i − k λ + m e i k ( x + y ) d λ − ∑ j = 1 n m j 2 f 0 ( x , λ j ) f 0 ˜ ( y , λ j ) ,

where f 0 ˜ denotes the transposed vector of f 0 .

The integral equation (12) is called the main equation of the inverse scattering problem for the boundary value problem (2)–(4).

3 Scattering function

In this section, we present the continuity of the scattering function S ( λ ) defined by (11).

Theorem 2

S ( λ ) is the quotient of two functions so that the numerator and the denominator are analytic in the lower and upper half plane, respectively. Furthermore, the limits

lim λ → ± m S ( λ )

exist, and as λ ⟶ ± ∞

S ( λ ) = S 0 ( λ ) + O ( 1 ) ,

where S 0 ( λ ) = α 2 − i β 2 α 2 + i β 2 . Moreover, S ( λ ) is continuous in ( − ∞ , − m ] and [ m , ∞ ) .

Proof

From Lemma 1, it is evident that S ( λ ) is the quotient of two functions. Also, Lemma 2 shows that the numerator and the denominator of S ( λ ) are analytic in the lower and upper half plane, respectively (see the proof in [25]). E ( λ ) ≠ 0 for all λ in the intervals ( − ∞ , − m ) and ( m , ∞ ) . Hence, S ( λ ) is defined on ( − ∞ , − m ) and ( m , ∞ ) and continuous in these intervals. We need to show the continuity of S ( λ ) at λ = ± m . For this, we rewrite the function E ( λ ) in the form

E ( λ ) = ( α 0 + α 1 λ + α 2 λ 2 ) f 1 ( 0 , λ ) − ( β 0 + β 1 λ + β 2 λ 2 ) f 2 ( 0 , λ ) = ( α 0 + α 1 λ + α 2 λ 2 ) λ + m k + ∫ 0 ∞ A 11 ( 0 , t ) λ + m k − i A 12 ( 0 , t ) e i k t d t − ( β 0 + β 1 λ + β 2 λ 2 ) − i + ∫ 0 ∞ A 21 ( 0 , t ) λ + m k − i A 22 ( 0 , t ) e i k t d t .

Let us define the function

Ψ ( λ ) = k ⋅ E ( λ )

and let l 1 and l 2 be limits of lim λ → m k λ + m E ( λ ) and lim λ → − m E ( λ ) , respectively.

Moreover,

l 1 = lim λ → m k λ + m E ( λ ) = lim λ → m ( α 0 + α 1 λ + α 2 λ 2 ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) e i k t d t − i k λ + m ∫ 0 ∞ A 12 ( 0 , t ) e i k t d t − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ 0 ∞ A 21 ( 0 , t ) e i k t d t − i k λ + m 1 + ∫ 0 ∞ A 22 ( 0 , t ) e i k t d t = ( α 0 + α 1 m + α 2 m 2 ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) d t − ( β 0 + β 1 m + β 2 m 2 ) ∫ 0 ∞ A 21 ( 0 , t ) d t

and

Ψ ( λ ) = ( α 0 + α 1 λ + α 2 λ 2 ) ( λ + m ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) e i k t d t − i k ∫ 0 ∞ A 12 ( 0 , t ) e i k t d t − ( β 0 + β 1 λ + β 2 λ 2 ) ( λ + m ) ∫ 0 ∞ A 21 ( 0 , t ) e i k t d t − i k 1 + ∫ 0 ∞ A 22 ( 0 , t ) e i k t d t .

Hence, for the case l 1 ≠ 0 , we have

Ψ ( λ ) = 2 m l 1 + O ( 1 ) , λ ⟶ m .

Now, we shall investigate Ψ ( λ ) for the case l 1 = 0 . Therefore, we substitute x = 0 into the main equation (12) and obtain the following components of the first column

(13) A 11 ( 0 , y ) + F 11 ( y ) + ∫ 0 ∞ { A 11 ( 0 , t ) F 11 ( t + y ) + A 12 ( 0 , t ) F 21 ( t + y ) } d t = 0

and

(14) A 21 ( 0 , y ) + F 21 ( y ) + ∫ 0 ∞ { A 21 ( 0 , t ) F 11 ( t + y ) + A 22 ( 0 , t ) F 21 ( t + y ) } d t = 0 .

Integrating (13) with respect to y from z to ∞ and letting t + y = ξ , we have

(15) ∫ z ∞ A 11 ( 0 , y ) d y + ∫ z ∞ F 11 ( y ) d y + ∫ 0 ∞ A 11 ( 0 , t ) ∫ t + z ∞ F 11 ( ξ ) d ξ d t + ∫ 0 ∞ A 12 ( 0 , t ) ∫ t + z ∞ F 21 ( ξ ) d ξ d t = 0 .

Applying partial integration to the third term on the left-hand side of equality (15), we find

∫ 0 ∞ A 11 ( 0 , t ) ∫ t + z ∞ F 11 ( ξ ) d ξ d t = ∫ z ∞ F 11 ( ξ ) d ξ ∫ 0 ∞ A 11 ( 0 , y ) d y − ∫ 0 ∞ F 11 ( t + z ) ∫ t ∞ A 11 ( 0 , y ) d y d t .

Calculating the fourth term with similar process and substituting both of these integrals into equality (15), we obtain

(16) ∫ z ∞ A 11 ( 0 , y ) d y + 1 + ∫ 0 ∞ A 11 ( 0 , y ) d y ∫ z ∞ F 11 ( ξ ) d ξ − ∫ 0 ∞ F 11 ( t + z ) ∫ t ∞ A 11 ( 0 , y ) d y d t + ∫ 0 ∞ A 12 ( 0 , y ) d y ∫ z ∞ F 21 ( ξ ) d ξ − ∫ 0 ∞ F 21 ( t + z ) ∫ t ∞ A 12 ( 0 , y ) d y d t = 0 .

We now apply the same procedure to equation (14) to obtain

(17) ∫ z ∞ A 21 ( 0 , y ) d y + 1 + ∫ 0 ∞ A 22 ( 0 , y ) d y ∫ z ∞ F 21 ( ξ ) d ξ − ∫ 0 ∞ F 21 ( t + z ) ∫ t ∞ A 22 ( 0 , y ) d y d t + ∫ 0 ∞ A 21 ( 0 , y ) d y ∫ z ∞ F 11 ( ξ ) d ξ − ∫ 0 ∞ F 11 ( t + z ) ∫ t ∞ A 21 ( 0 , y ) d y d t = 0 .

Multiplying equation (16) by ( α 0 + α 1 λ + α 2 λ 2 ) and equation (17) by ( β 0 + β 1 λ + β 2 λ 2 ) , then subtracting the latter from the former, we have

( α 0 + α 1 λ + α 2 λ 2 ) 1 + ∫ 0 ∞ A 11 ( 0 , y ) d y − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ 0 ∞ A 21 ( 0 , y ) d y ∫ z ∞ F 11 ( y ) d y + ( α 0 + α 1 λ + α 2 λ 2 ) ∫ z ∞ A 11 ( 0 , y ) d y − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ z ∞ A 21 ( 0 , y ) d y − ∫ 0 ∞ F 11 ( t + z ) × ( α 0 + α 1 λ + α 2 λ 2 ) ∫ t ∞ A 11 ( 0 , y ) d y − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ t ∞ A 21 ( 0 , y ) d y d t − ∫ 0 ∞ F 21 ( t + z ) ( α 0 + α 1 λ + α 2 λ 2 ) ∫ t ∞ A 12 ( 0 , y ) d y − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ t ∞ A 22 ( 0 , y ) d y d t + ( α 0 + α 1 λ + α 2 λ 2 ) ∫ 0 ∞ A 12 ( 0 , y ) d y − ( β 0 + β 1 λ + β 2 λ 2 ) 1 + ∫ 0 ∞ A 22 ( 0 , y ) d y ∫ z ∞ F 21 ( y ) d y = 0 .

Letting λ ⟶ m and considering l 1 = 0 , we obtain

( α 0 + α 1 m + α 2 m 2 ) ∫ z ∞ A 11 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) ∫ z ∞ A 21 ( 0 , y ) d y − ∫ 0 ∞ F 11 ( t + z ) ( α 0 + α 1 m + α 2 m 2 ) ∫ t ∞ A 11 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) ∫ t ∞ A 21 ( 0 , y ) d y d t − ∫ 0 ∞ F 21 ( t + z ) ( α 0 + α 1 m + α 2 m 2 ) ∫ t ∞ A 12 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) ∫ t ∞ A 22 ( 0 , y ) d y d t + ( α 0 + α 1 m + α 2 m 2 ) ∫ 0 ∞ A 12 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) 1 + ∫ 0 ∞ A 22 ( 0 , y ) d y ∫ z ∞ F 21 ( y ) d y = 0 .

Here, we define the functions H ( z ) and G ( z ) as follows:

H ( z ) ≔ ( α 0 + α 1 m + α 2 m 2 ) ∫ z ∞ A 11 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) ∫ z ∞ A 21 ( 0 , y ) d y , G ( z ) ≔ ∫ 0 ∞ F 21 ( t + z ) ( α 0 + α 1 m + α 2 m 2 ) ∫ t ∞ A 12 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) ∫ t ∞ A 22 ( 0 , y ) d y d t − ( α 0 + α 1 m + α 2 m 2 ) ∫ 0 ∞ A 12 ( 0 , y ) d y − ( β 0 + β 1 m + β 2 m 2 ) 1 + ∫ 0 ∞ A 22 ( 0 , y ) d y ∫ z ∞ F 21 ( y ) d y .

Hence, we find an integral equation

H ( z ) − ∫ 0 ∞ F 11 ( t + z ) H ( t ) d t = G ( z ) .

H ( z ) is a bounded solution of the equation

H ( z ) − ∫ 0 ∞ F 11 ( t + z ) H ( t ) d t = 0 , 0 ≤ z < ∞ ,

and every bounded solution of this equation is summable on the half line [ 0 , ∞ ) . It means that H ( z ) ∈ L 1 ( 0 , ∞ ) (see [19] p. 211).

Returning to the function Ψ ( λ ) and by partial integration, we obtain

Ψ ( λ ) = ( λ + m ) ( α 0 + α 1 λ + α 2 λ 2 ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) d t − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ 0 ∞ A 21 ( 0 , t ) d t + i k ( α 0 + α 1 λ + α 2 λ 2 ) ( λ + m ) ∫ 0 ∞ ∫ t ∞ A 11 ( 0 , y ) d y e i k t d t − ∫ 0 ∞ A 12 ( 0 , t ) e i k t d t − i k ( β 0 + β 1 λ + β 2 λ 2 ) ( λ + m ) ∫ 0 ∞ ∫ t ∞ A 21 ( 0 , y ) d y e i k t d t − 1 + ∫ 0 ∞ A 22 ( 0 , t ) e i k t d t .

Because of the fact that l 1 = 0 , the factor of the first term of Ψ ( λ ) equals:

( α 0 + α 1 λ + α 2 λ 2 ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) d t − ( β 0 + β 1 λ + β 2 λ 2 ) ∫ 0 ∞ A 21 ( 0 , t ) d t = ( α 1 ( λ − m ) + α 2 ( λ 2 − m 2 ) ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) d t − ( β 1 ( λ − m ) + β 2 ( λ 2 − m 2 ) ) ∫ 0 ∞ A 21 ( 0 , t ) d t .

After substituting the obtained value into the function Ψ ( λ ) , we conclude that

(18) Ψ ( λ ) = λ − m K ( λ ) ^ ,

where

K ( λ ) ^ = ( λ + m ) ( α 1 λ − m + α 2 λ − m ( λ + m ) ) 1 + ∫ 0 ∞ A 11 ( 0 , t ) d t − ( β 1 λ − m + β 2 λ − m ( λ + m ) ) ∫ 0 ∞ A 21 ( 0 , t ) d t + i λ + m ( λ + m ) ∫ 0 ∞ ∫ t ∞ [ ( α 0 + α 1 λ + α 2 λ 2 ) A 11 ( 0 , y ) − ( β 0 + β 1 λ + β 2 λ 2 ) A 21 ( 0 , y ) ] d y e i k t d t − i λ + m ( α 0 + α 1 λ + α 2 λ 2 ) ∫ 0 ∞ A 12 ( 0 , t ) e i k t d t − ( β 0 + β 1 λ + β 2 λ 2 ) 1 + ∫ 0 ∞ A 22 ( 0 , t ) e i k t d t .

Hence, in the present case,

K ( m ) ^ = i ( 2 m ) 3 ⁄ 2 ∫ 0 ∞ H ( t ) d t − i ( 2 m ) 1 ⁄ 2 ( α 0 + α 1 m + α 2 m 2 ) ∫ 0 ∞ A 12 ( 0 , t ) d t − ( β 0 + β 1 m + β 2 m 2 ) × 1 + ∫ 0 ∞ A 22 ( 0 , t ) d t

and

S ( λ ) = K ( λ ) ^ ¯ K ( λ ) ^ .

Taking into account the first components of (6) and (9), we obtain

2 i ( λ + m ) φ 1 ( x , λ ) = K ( λ ) ^ { λ − m f 1 ( x , λ ) ¯ − S ( λ ) λ − m f 1 ( x , λ ) } = K ( λ ) ^ λ + m e − i k x + ∫ x ∞ [ A 11 ( x , t ) λ + m + i A 12 ( x , t ) ] e − i k t d t − S ( λ ) λ + m e i k x + ∫ x ∞ [ A 11 ( x , t ) λ + m − i A 12 ( x , t ) ] e i k t d t ,

from which it follows that K ( m ) ^ ≠ 0 , otherwise it would be φ 1 ( x , m ) = 0 and it contradicts φ 1 ( x , λ ) ≠ 0 . This shows that S ( λ ) is continuous at λ = m in the case l 1 = 0 .

Consequently, as λ ⟶ m , we have

(19) Ψ ( λ ) = 2 m l 1 + O ( 1 ) , l 1 ≠ 0 λ − m K ( m ) ^ , l 1 = 0 .

If the condition l 1 ≠ 0 holds, it is obvious that S ( λ ) is continuous at λ = m .

In the case that l 2 = lim λ → − m E ( λ ) , using a similar process, we obtain the result

(20) Ψ ( λ ) = λ + m ( c 1 + O ( 1 ) ) l 2 ≠ 0 ( λ + m ) ( c 2 + O ( 1 ) ) l 2 = 0 ,

which shows the continuity of S ( λ ) at λ = − m .

To complete the proof, using the definition of S ( λ ) , we take λ ⟶ ± ∞ and the following result is obtained

S ( λ ) = α 2 − i β 2 α 2 + i β 2 + O ( 1 ) .

The theorem is proved.□

4 Levinson-type formula

Now, after stating the above preparations, we can describe Levinson-type formula for the boundary value problem (2)–(4).

Theorem 3

The increment of the logarithm of the scattering function S ( λ ) is related to the number of eigenvalues of problems (2)–(4) through the equality

(21) n − 2 = ln S ( m + 0 ) − ln S ( ∞ ) 2 π i + ln S ( − ∞ ) − S ( − m − 0 ) 2 π i − S ( − m − 0 ) − ln S ( m + 0 ) 4 .

Proof

To achieve formula (21), we apply the argument principle to the function Ψ ( λ ) . This function is analytic in the upper half plane and continuous along the real axis. Also, for ∣ λ ∣ < m , Im λ = 0 , we have Im Ψ ( λ ) = 0 . Hence, the increment of its argument arg Ψ ( λ ) equals zero as λ runs over the curve

Γ δ , R = { λ ∣ m + δ ≤ Re λ ≤ R , Im λ = 0 ; ∣ λ ± m ∣ = δ , ∣ λ − λ j ∣ = δ , j = 1 , n ¯ , ∣ λ ∣ = R , Im λ ≥ 0 } ,

i.e.,

arg Ψ ( λ ) Γ δ , R = 0 .

Let the argument of the function Ψ ( λ ) be

arg Ψ ( λ ) Γ δ , R = μ ( λ ) .

As R ⟶ ∞ , we obtain

{ μ ( − m − δ ) − μ ( − ∞ ) } + { μ ( − m + δ ) − μ ( − m − δ ) } + { μ ( m + δ ) − μ ( m − δ ) } + { μ ( ∞ ) − μ ( m + δ ) } + ∑ j = 1 n { μ ( λ j + δ ) − μ ( λ j − δ ) } + 3 π = 0

because of that

Ψ ( λ ) = λ 3 ( α 2 − β 2 ) + O ( 1 ) , ∣ λ ∣ ⟶ ∞

for Im λ ≥ 0 . By using (19) and (20), as δ ⟶ 0 , we find that

(22) { μ ( − m − 0 ) − μ ( − ∞ ) } + { μ ( ∞ ) − μ ( m + 0 ) } − n π + 3 π = π 2 , l 1 ≠ 0 l 2 ≠ 0 , π , l 1 = 0 l 2 ≠ 0 , π , l 1 ≠ 0 l 2 = 0 , 3 π 2 , l 1 = 0 , l 2 = 0 .

On the other hand, from (19) and (20), we deduce that

S ( − m − 0 ) = − 1 , l 2 ≠ 0 , 1 , l 2 = 0 ,

and

S ( m + 0 ) = 1 , l 1 ≠ 0 , − 1 , l 1 = 0 .

Since ln S ( λ ) = − 2 i μ ( λ ) , we substitute the required values into (22) and obtain the desired result

□ ln S ( m + 0 ) − ln S ( ∞ ) 2 π i + ln S ( − ∞ ) − ln S ( − m − 0 ) 2 π i = μ ( ∞ ) − μ ( m + 0 ) π + μ ( − m − 0 ) − μ ( − ∞ ) π = n − 2 + S ( − m − 0 ) − S ( m + 0 ) 4 .

Definition 2

The formula defined by (21) is called Levinson-type formula for the boundary value problem (2)–(4).

5 Conclusion

In quantum scattering theory, Levinson’s theorem deals with the relation between the number of bound states in a spectral gap and the variation of an appropriate phase along the continuous part of the spectrum. This article presents Levinson-type formula for a system of Dirac equations with polynomials of spectral parameter in the boundary condition. In this view, the continuity of the scattering function has been investigated and the variation of its argument has been formulated. The formula demonstrates that the increment of the logarithm of S ( λ ) over ( − ∞ , − m ] ∪ [ m , ∞ ) is related to the number of eigenvalues of the boundary value problem (2)–(4) in the interval ( − m , m ) through equality (21).

Acknowledgements

The author thanks the anonymous referees for their careful reading of the manuscript and their comments.

  1. Funding information: No funds, grants, or other support was received.

  2. Author contributions: The author has accepted responsibility for the entire content of the manuscript and approved its final version.

  3. Conflict of interest: The author states that there is no conflict of interest.

  4. Data availability statement: No datasets were generated or analyzed during the current study.

  5. Ethical approval: The conducted research is not related to either human or animal use.

References

[1] N. Levinson, On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase, Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1949), no. 9, 29. Search in Google Scholar

[2] M. Klaus, On the Levinson theorem for Dirac operators, J. Math. Phys. 31 (1990), no. 1, 182–190, DOI: https://doi.org/10.1063/1.528858. 10.1063/1.528858Search in Google Scholar

[3] D. B. Hinton, M. Klaus, and J. K. Shaw, Levinson’s theorem and Titchmarsh-Weyl m(λ) theory for Dirac systems, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), no. 1–2, 173–186, DOI: https://doi.org/10.1017/S0308210500026743. 10.1017/S0308210500026743Search in Google Scholar

[4] Q. G. Lin, Levinson theorem for Dirac particles in two dimensions, Phys. Rev. A 57 (1998), no. 5, 3478–3488, DOI: https://doi.org/10.1103/PhysRevA.57.3478. 10.1103/PhysRevA.57.3478Search in Google Scholar

[5] Q. G. Lin, Levinson theorem for Dirac particles in one dimension, Eur. Phys. J. D 7 (1999), 515–524, DOI: https://doi.org/10.1007/s100530050379. 10.1007/s100530050379Search in Google Scholar

[6] Z. Q. Ma and G. J. Ni, Levinson theorem for Dirac particles, Phys. Rev. D 31 (1985), no. 6, 1482–1488, DOI: https://doi.org/10.1103/PhysRevD.31.1482. 10.1103/PhysRevD.31.1482Search in Google Scholar

[7] Z. Q. Ma, Levinson’s theorem for Dirac particles moving in a background magnetic monopole field, Phys. Rev. D 32 (1985), no. 8, 2203–2212, DOI: https://doi.org/10.1103/PhysRevD.32.2203. 10.1103/PhysRevD.32.2203Search in Google Scholar PubMed

[8] Z. Q. Ma, Levinson’s theorem for Dirac particles with a long-range potential, Phys. Rev. D 32 (1985), no. 8, 2213–2215, DOI: https://doi.org/10.1103/PhysRevD.32.2213. 10.1103/PhysRevD.32.2213Search in Google Scholar PubMed

[9] J. Piekarewicz, Levinson’s theorem for Dirac particles, Phys. Rev. C 48 (1993), no. 5, 2174–2181, DOI: https://doi.org/10.1103/PhysRevC.48.2174. 10.1103/PhysRevC.48.2174Search in Google Scholar

[10] Z. R. Iwinski, L. Rosenberg, and L. Spruch, Nodal structure of zero energy wave functions: new approach to Levinson’s theorem, Phys. Rev. A 31 (1985), no. 3, 1229–1240, DOI: https://doi.org/10.1103/PhysRevA.31.1229. 10.1103/PhysRevA.31.1229Search in Google Scholar

[11] J. M. Jauch, On the relation between scattering phase and bound states, Helv. Phys. Acta 30 (1957), no. 2–3, 143–156, DOI: https://doi.org/10.5169/seals-112810. Search in Google Scholar

[12] X. Jia, F. Nicoleau, and X. P. Wang, A new Levinson’s theorem for potentials with critical decay, Ann. Henri Poincaré 13 (2012), 41–84, DOI: https://doi.org/10.1007/s00023-011-0117-0. 10.1007/s00023-011-0117-0Search in Google Scholar

[13] J. Kellendonk and S. Richard, Levinson’s theorem for Schrödinger operators with point interaction: a topological approach, J. Phys. A: Math. Gen. 39 (2006), no. 46, 14397–14403, DOI: https://doi.org/10.1088/0305-4470/39/46/011. 10.1088/0305-4470/39/46/011Search in Google Scholar

[14] M. I. Krivoruchenko and K. S. Tyrin, Generalization of the Levinson theorem on the asymptotic value of the scattering-amplitude phase shift, Phys. Atom. Nuclei 84 (2021), 29–33, DOI: https://doi.org/10.1134/S1063778821010130. 10.1134/S1063778821010130Search in Google Scholar

[15] Z. Q. Ma, The Levinson theorem, J. Phys. A: Math. Gen. 39 (2006), no. 48, R625, DOI: https://doi.org/10.1088/0305-4470/39/48/R01. 10.1088/0305-4470/39/48/R01Search in Google Scholar

[16] A. Martin, On the validity of Levison’s theorem for non-local interactions, Nuovo Cim. 7 (1958), 607–627, DOI: https://doi.org/10.1007/BF02781566. 10.1007/BF02781566Search in Google Scholar

[17] R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, New York, 1982. 10.1007/978-3-642-88128-2Search in Google Scholar

[18] R. G. Newton, Noncentral potentials: the generalized Levinson theorem and the structure of the spectrum, J. Math. Phys. 18 (1977), 1348–1357, DOI: https://doi.org/10.1063/1.523428. 10.1063/1.523428Search in Google Scholar

[19] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser Verlag, Basel, 1986. 10.1007/978-3-0348-5485-6Search in Google Scholar

[20] Kh. R. Mamedov and N. P. Kosar, Continuity of the scattering function and Levinson type formula of a boundary-value problem, Int. J. Contemp. Math. Sciences 5 (2010), no. 4, 159–170. Search in Google Scholar

[21] S. Goktas and Kh. R. Mamedov, The Levinson-type formula for a class of Sturm-Liouville equation, Facta Univ. Ser. Math. Inform. 35 (2020), no. 4, 1219–1229, DOI: https://doi.org/10.22190/FUMI2004219G. 10.22190/FUMI2004219GSearch in Google Scholar

[22] Ö. Mızrak, Kh. R. Mamedov, and A. M. Akhtyamov, Characteristic properties of scattering data of a boundary value problem, Filomat 31 (2017), no. 12, 3945–3951, DOI: https://doi.org/10.2298/FIL1712945M. 10.2298/FIL1712945MSearch in Google Scholar

[23] T. Koprubasi and R. N. Mohapatra, An inverse scattering problem for eigenparameter-dependent discrete Dirac system with Levinson formula, Complex Var. Elliptic Equ. 69 (2024), no. 4, 695–711, DOI: https://doi.org/10.1080/17476933.2022.2159951. 10.1080/17476933.2022.2159951Search in Google Scholar

[24] M. G. Gasymov, The inverse scattering problem for a system of Dirac equations of order 2n, Trans. Moscow Math. Soc. 19 (1968), 41–119. Search in Google Scholar

[25] A. Col and Kh. R. Mamedov, On an inverse scattering problem for a class of Dirac operators with spectral parameter in the boundary condition, J. Math. Anal. Appl. 393 (2012), no. 2, 470–478, DOI: https://doi.org/10.1016/j.jmaa.2012.03.009. 10.1016/j.jmaa.2012.03.009Search in Google Scholar

[26] Kh. R. Mamedov and A. Col, On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition, Math. Methods Appl. Sci. 35 (2012), no. 14, 1712–1720, DOI: https://doi.org/10.1002/mma.2553. 10.1002/mma.2553Search in Google Scholar

[27] A. Kablan and T. Ozden, A Dirac system with transmission condition and eigenparameter in boundary condition, Abstr. Appl. Anal. 2013 (2013), no. 1, 395457, DOI: https://doi.org/10.1155/2013/395457. 10.1155/2013/395457Search in Google Scholar

[28] E. S. Panakhov, E. Yilmaz, and H. Koyunbakan, Inverse nodal problem for Dirac operator, World Appl. Sci. J. 11 (2010), no. 8, 906–911. Search in Google Scholar

[29] Y. Aygar and G. G. Ozbey, Scattering analysis of a quantum impulsive boundary value problem with spectral parameter, Hacet. J. Math. Stat. 51 (2022), no. 1, 142–155, DOI: https://doi.org/10.15672/hujms.912015. 10.15672/hujms.912015Search in Google Scholar

[30] S. Cebesoy, E. Bairamov, and Y. Aygar, Scattering problems of impulsive Schrödinger equations with matrix coefficients, Ric. Mat. 72 (2023), 399–415, DOI: https://doi.org/10.1007/s11587-022-00736-y. 10.1007/s11587-022-00736-ySearch in Google Scholar
Received: 2023-11-14
Revised: 2024-04-23
Accepted: 2024-05-11
Published Online: 2024-08-30

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0022
Keywords for this article
Levinson-type formula; scattering data; spectral parameter dependent boundary condition
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0022/html
Scroll to top button