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On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential

  • Elif Başkaya EMAIL logo
Published/Copyright: February 14, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, Sturm-Liouville problem with one boundary condition including an eigenparameter is considered, and the asymptotic expansion of its eigenparameter is calculated. The problem also has a symmetric single-well potential, which is an important function in quantum mechanics.

Keywords: Sturm-Liouville problem; symmetric single-well potential; eigenvalue parameter in the boundary condition; asymptotic expansions
MSC 2010: 34B30; 34L15

1 Introduction

In this study, we deal with the following equation:

(1) y ″ ( t ) + [ λ − q ( t ) ] y ( t ) = 0 , t ∈ [ 0 , a ] ,

where λ is a real parameter and q ( t ) is a real-valued, continuous function. This equation is known as the regular Sturm-Liouville equation. We consider the equation with the following boundary value conditions:

(2) a 1 y ( 0 ) − a 2 y ′ ( 0 ) = λ [ a 1 ′ y ( 0 ) − a 2 ′ y ′ ( 0 ) ]

and

(3) y ( a ) cos β + y ′ ( a ) sin β = 0 ,

where a 1 , a 2 , a 1 ′ , and a 2 ′ are real constants and β ∈ [ 0 , π ) . Equations (1)–(3) is different from the usual regular Sturm-Liouville problem because the eigenvalue parameter λ is held in the boundary condition at zero. Such problems often arise from physical problems, quantum mechanics, and geophysics. Fulton gives more than a hundred references in [1] and [2] (see also [3]), so his works serve as a historical guide. It is also shown by Walter [4] that this problem is a self-adjoint problem if the relation

(4) δ = a 1 ′ a 1 a 2 ′ a 2 > 0

holds. There are a lot of studies with Sturm-Liouville problems where the spectral parameter appears in the boundary conditions, see, e.g., [5–12].

Besides, equation (1) is equal to one-dimensional Schrödinger equation, and especially in recent years, since quantum mechanics has gained importance, there have been a lot of studies on the eigenvalues of Hill’s equation and Schrödinger’ s operator with symmetric single-well potential, such as anharmonic oscillator. The eigenvalues of these equations represent excitation energy, and eigenfunctions are named as wavefunctions in physics. A symmetric single-well potential on [ 0 , a ] is defined as symmetric with respect to the midpoint a ∕ 2 and nonincreasing on [ 0 , a ∕ 2 ] . The eigenvalue problems with symmetric single-well potential can be found in [13–18].

The purpose of this article is to obtain asymptotic approximations for eigenvalues λ n of equations (1)–(3) with symmetric single-well potential q ( t ) such that condition (4) is satisfied. We remark that a symmetric single-well potential on [ 0 , a ] means that a continuous function q ( t ) is symmetric on [ 0 , a ] and nonincreasing on [ 0 , a ∕ 2 ] , so we have q ( t ) = q ( a − t ) mathematically. We assume, without loss of generality, that q ( t ) has a mean value of zero.

2 The method

First, we note that q ′ ( t ) exists since a monotone function on an interval I is differentiable almost everywhere on I [19]. After then, let us consider the following lemma:

Lemma 2.1

The eigenvalues λ n of equations (1)–(3) satisfy as λ → ∞

(i) a 2 ′ ≠ 0 and β ≠ 0

( n + 1 ) π = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − tan − 1 a 1 − a 2 [ r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) ] − λ [ a 1 ′ − a 2 ′ ( r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) ) ] ( a 2 − λ a 2 ′ ) [ r 2 ( 0 , λ ) + ρ 2 ( 0 , λ ) ] − tan − 1 cos β + [ r 1 ( a , λ ) + ρ 1 ( a , λ ) ] sin β sin β [ r 2 ( a , λ ) + ρ 2 ( a , λ ) ] + O ( λ − 3 ∕ 2 ) ,

(ii) a 2 ′ ≠ 0 and β = 0

( 2 n + 3 ) π 2 = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − tan − 1 a 1 − a 2 [ r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) ] − λ [ a 1 ′ − a 2 ′ ( r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) ) ] ( a 2 − λ a 2 ′ ) [ r 2 ( 0 , λ ) + ρ 2 ( 0 , λ ) ] + O ( λ − 3 ∕ 2 ) ,

where r 1 ( t , λ ) + ρ 1 ( t , λ ) and r 2 ( t , λ ) + ρ 2 ( t , λ ) are defined as follows:

(5) r 1 ( t , λ ) + ρ 1 ( t , λ ) = 1 2 λ − 1 ∕ 2 ∫ 0 t q ′ ( x ) sin 2 λ 1 ∕ 2 ( t − x ) d x − 1 2 λ − 1 ∫ 0 t q ′ ( x ) ∫ x t q ( s ) d s cos 2 λ 1 ∕ 2 ( t − x ) d x + 1 4 λ − 1 ∫ 0 t q 2 ( x ) cos 2 λ 1 ∕ 2 ( t − x ) d x + O ( λ − 3 ∕ 2 ) ,

and

(6) r 2 ( t , λ ) + ρ 2 ( t , λ ) = λ 1 ∕ 2 − 1 2 λ − 1 ∕ 2 q ( t ) + 1 2 λ − 1 ∕ 2 ∫ 0 t q ′ ( x ) cos 2 λ 1 ∕ 2 ( t − x ) d x + 1 2 λ − 1 ∫ 0 t q ′ ( x ) ∫ x t q ( s ) d s sin 2 λ 1 ∕ 2 ( t − x ) d x − 1 4 λ − 1 ∫ 0 t q 2 ( x ) sin 2 λ 1 ∕ 2 ( t − x ) d x + O ( λ − 3 ∕ 2 ) .

Lemma 2.2

The eigenvalues λ n of equations (1)–(3) satisfy as λ → ∞

(i) a 2 ′ = 0 and β ≠ 0

( n + 1 ) π = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − cot − 1 a 2 [ r 2 ( 0 , λ ) + ρ 2 ( 0 , λ ) ] a 1 − a 2 [ r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) ] − λ a 1 ′ − tan − 1 cos β + [ r 1 ( a , λ ) + ρ 1 ( a , λ ) ] sin β sin β [ r 2 ( a , λ ) + ρ 2 ( a , λ ) ] + O ( λ − 3 ∕ 2 ) ,

(ii) a 2 ′ = 0 and β = 0

( 2 n + 3 ) π 2 = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − cot − 1 a 2 [ r 2 ( 0 , λ ) + ρ 2 ( 0 , λ ) ] a 1 − a 2 [ r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) ] − λ a 1 ′ + O ( λ − 3 ∕ 2 ) .

Proof

If we rearrange the main theorems of [6] in line with our aim for N = 2 , we prove the lemmas, easily.□

Lemmas 2.1 and 2.2 are important in literature, because they give good approximations of the asymptotic estimates of the eigenvalues with the continuous potential such that its derivative exists and is integrable. We note that our method is based on Lemmas 2.1 and 2.2.

3 Main results

In this article, we obtain the following asymptotic approximations for eigenvalues λ n of equations (1)–(3) with symmetric single-well potential q ( t ) by using Lemmas 2.1 and 2.2:

Theorem 3.1

Let q ( t ) be symmetric in equation (1). Then, the eigenvalues λ n of (1)–(3) satisfy as n → ∞

(i) a 2 ′ ≠ 0 and β ≠ 0

λ n 1 ∕ 2 = ( n + 1 ) π a + 1 ( n + 1 ) π a 1 ′ a 2 ′ + cot β − a 2 ( n + 1 ) 2 π 2 ∫ 0 a ∕ 2 q ′ ( x ) sin 2 ( n + 1 ) π a x d x + O ( n − 3 ) ,

(ii) a 2 ′ ≠ 0 and β = 0

λ n 1 ∕ 2 = ( 2 n + 3 ) π 2 a + 2 ( 2 n + 3 ) π a 1 ′ a 2 ′ + O ( n − 3 ) ,

(iii) a 2 ′ = 0 and β ≠ 0

λ n 1 ∕ 2 = ( 2 n + 3 ) π 2 a + 2 ( 2 n + 3 ) π a 2 a 1 ′ + cot β + 2 a ( 2 n + 3 ) 2 π 2 ∫ 0 a ∕ 2 q ′ ( x ) sin ( 2 n + 3 ) π a x d x + O ( n − 3 ) ,

(iv) a 2 ′ = 0 and β = 0

λ n 1 ∕ 2 = ( n + 2 ) π a + 1 ( n + 2 ) π a 2 a 1 ′ + a 2 ( n + 2 ) 2 π 2 ∫ 0 a ∕ 2 q ′ ( x ) sin 2 ( n + 2 ) π a x d x + O ( n − 3 ) .

Proof

(i) We calculate the terms in Lemmas 2.1(i). First, from equations (5) and (6), we find that

r 1 ( 0 , λ ) + ρ 1 ( 0 , λ ) = O ( λ − 3 ∕ 2 ) ,

r 2 ( 0 , λ ) + ρ 2 ( 0 , λ ) = λ 1 ∕ 2 − 1 2 λ − 1 ∕ 2 q ( 0 ) + O ( λ − 3 ∕ 2 ) ,

r 1 ( a , λ ) + ρ 1 ( a , λ ) = 1 2 λ − 1 ∕ 2 ∫ 0 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x − 1 2 λ − 1 ∫ 0 a q ′ ( x ) ∫ x a q ( s ) d s cos 2 λ 1 ∕ 2 ( a − x ) d x + 1 4 λ − 1 ∫ 0 a q 2 ( x ) cos 2 λ 1 ∕ 2 ( a − x ) d x + O ( λ − 3 ∕ 2 ) ,

and

r 2 ( a , λ ) + ρ 2 ( a , λ ) = λ 1 ∕ 2 − 1 2 λ − 1 ∕ 2 q ( a ) + 1 2 λ − 1 ∕ 2 ∫ 0 a q ′ ( x ) cos 2 λ 1 ∕ 2 ( a − x ) d x + 1 2 λ − 1 ∫ 0 a q ′ ( x ) ∫ x a q ( s ) d s sin 2 λ 1 ∕ 2 ( a − x ) d x − 1 4 λ − 1 ∫ 0 a q 2 ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x + O ( λ − 3 ∕ 2 ) .

Therefore, if we define

(7) Ω ≔ a 1 − λ a 1 ′ + O ( λ − 1 ∕ 2 ) − λ 3 ∕ 2 a 2 ′ + λ 1 ∕ 2 a 2 + 1 2 λ 1 ∕ 2 a 2 ′ q ( 0 ) − 1 2 λ − 1 ∕ 2 a 2 q ( 0 ) + O ( λ − 1 ∕ 2 )

and

(8) η ≔ cos β + sin β 1 2 λ − 1 ∕ 2 A 2 − 1 2 λ − 1 B 1 + 1 4 λ − 1 C 1 + O ( λ − 3 ∕ 2 ) sin β λ 1 ∕ 2 − 1 2 λ − 1 ∕ 2 q ( a ) + 1 2 λ − 1 ∕ 2 A 1 + 1 2 λ − 1 B 2 − 1 4 λ − 1 C 2 + O ( λ − 3 ∕ 2 ) ,

where

(9) A 1 ≔ ∫ 0 a q ′ ( x ) cos ( 2 λ 1 ∕ 2 ( a − x ) ) d x ,

A 2 ≔ ∫ 0 a q ′ ( x ) sin ( 2 λ 1 ∕ 2 ( a − x ) ) d x ,

B 1 ≔ ∫ 0 a q ′ ( x ) ∫ x a q ( s ) d s cos ( 2 λ 1 ∕ 2 ( a − x ) ) d x ,

B 2 ≔ ∫ 0 a q ′ ( x ) ∫ x a q ( s ) d s sin ( 2 λ 1 ∕ 2 ( a − x ) ) d x ,

C 1 ≔ ∫ 0 a q 2 ( x ) cos ( 2 λ 1 ∕ 2 ( a − x ) ) d x ,

C 2 ≔ ∫ 0 a q 2 ( x ) sin ( 2 λ 1 ∕ 2 ( a − x ) ) d x ,

we can rearrange Lemmas 2.1(i) as follows:

(10) ( n + 1 ) π = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − tan − 1 ( Ω ) − tan − 1 ( η ) + O ( λ − 3 ∕ 2 ) .

We will calculate the asymptotic eigenvalues from the last equation. By using series expansion for Ω , we obtain

Ω = − λ a 1 ′ + a 1 + O ( λ − 1 ∕ 2 ) − λ 3 ∕ 2 a 2 ′ 1 − a 2 a 2 ′ λ − 1 − 1 2 λ − 1 q ( 0 ) + O ( λ − 2 ) = λ − 1 ∕ 2 a 1 ′ a 2 ′ − λ − 3 ∕ 2 a 1 a 2 ′ + O ( λ − 2 ) 1 + λ − 1 a 2 a 2 ′ + 1 2 λ − 1 q ( 0 ) + O ( λ − 2 ) = λ − 1 ∕ 2 a 1 ′ a 2 ′ + O ( λ − 3 ∕ 2 ) .

From the last equation and inverse tangent expansion tan − 1 ( Ω ) = Ω − Ω 3 3 + ⋯ , we find

(11) tan − 1 ( Ω ) = λ − 1 ∕ 2 a 1 ′ a 2 ′ + O ( λ − 3 ∕ 2 ) .

Similarly, we derive series expansion for η , and find that

η = cot β + 1 2 λ − 1 ∕ 2 A 2 − 1 2 λ − 1 B 1 + 1 4 λ − 1 C 1 + O ( λ − 3 ∕ 2 ) λ 1 ∕ 2 1 − 1 2 λ − 1 q ( a ) + 1 2 λ − 1 A 1 + 1 2 λ − 3 ∕ 2 B 2 − 1 4 λ − 3 ∕ 2 C 2 + O ( λ − 2 ) = λ − 1 ∕ 2 cot β + 1 2 λ − 1 A 2 − 1 2 λ − 3 ∕ 2 B 1 + 1 4 λ − 3 ∕ 2 C 1 + O ( λ − 2 ) × 1 + 1 2 λ − 1 q ( a ) − 1 2 λ − 1 A 1 − 1 2 λ − 3 ∕ 2 B 2 + 1 4 λ − 3 ∕ 2 C 2 + O ( λ − 2 ) = λ − 1 ∕ 2 cot β + 1 2 λ − 1 A 2 + O ( λ − 3 ∕ 2 ) .

From the last equality, equation (9), and inverse tangent expansion, we obtain that

tan − 1 ( η ) = λ − 1 ∕ 2 cot β + 1 2 λ − 1 ∫ 0 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x + O ( λ − 3 ∕ 2 ) .

We study with symmetric single-well potential q ( t ) . In this case, we have

∫ a ∕ 2 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x = − ∫ a ∕ 2 0 q ′ ( a − u ) sin 2 λ 1 ∕ 2 u d u = ∫ a ∕ 2 0 q ′ ( u ) sin 2 λ 1 ∕ 2 u d u .

The last equality holds because q ( t ) is symmetric and q ′ ( t ) exists, so q ′ ( t ) = − q ′ ( a − t ) . Then,

∫ 0 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x = ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x + ∫ a ∕ 2 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x = ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x − ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 x d x = sin 2 λ 1 ∕ 2 a ∫ 0 a ∕ 2 q ′ ( x ) cos 2 λ 1 ∕ 2 x d x − cos 2 λ 1 ∕ 2 a ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 x d x − ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 x d x

so

(12) ∫ 0 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x = sin 2 λ 1 ∕ 2 a ∫ 0 a ∕ 2 q ′ ( x ) cos 2 λ 1 ∕ 2 x d x − [ 1 + cos 2 λ 1 ∕ 2 a ] ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 x d x .

If we use the last equation in tan − 1 ( η ) , we see that

(13) tan − 1 ( η ) = λ − 1 ∕ 2 cot β + 1 2 λ − 1 sin 2 λ 1 ∕ 2 a ∫ 0 a ∕ 2 q ′ ( x ) cos 2 λ 1 ∕ 2 x d x − 1 2 λ − 1 [ 1 + cos 2 λ 1 ∕ 2 a ] ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 x d x .

Now, we should compute ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t to calculate equation (10). From equation (6), we have

(14) ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t = λ 1 ∕ 2 ∫ 0 a 1 d t − 1 2 λ − 1 ∕ 2 ∫ 0 a q ( t ) d t + 1 2 λ − 1 ∕ 2 I 1 + 1 2 λ − 1 I 2 − 1 4 λ − 1 I 3 + O ( λ − 3 ∕ 2 ) ,

where

I 1 ≔ ∫ 0 a ∫ 0 t q ′ ( x ) cos 2 λ 1 ∕ 2 ( t − x ) d x d t , I 2 ≔ ∫ 0 a ∫ 0 t q ′ ( x ) ∫ x t q ( s ) d s sin 2 λ 1 ∕ 2 ( t − x ) d x d t ,

and

I 3 ≔ ∫ 0 a ∫ 0 t q 2 ( x ) sin 2 λ 1 ∕ 2 ( t − x ) d x d t .

The second term on the right side of equation (14) is zero, because q ( t ) has a mean value of zero. Now, we apply Leibniz formula for integrals I 1 , I 2 , and I 3 as follows:

(15) I 1 = 1 2 λ 1 ∕ 2 ∫ 0 t q ′ ( x ) sin 2 λ 1 ∕ 2 ( t − x ) d x t = 0 a = 1 2 λ − 1 ∕ 2 ∫ 0 a q ′ ( x ) sin 2 λ 1 ∕ 2 ( a − x ) d x ,

(16) I 2 = − 1 2 λ 1 ∕ 2 ∫ 0 t q ′ ( x ) ∫ x t q ( s ) d s cos 2 λ 1 ∕ 2 ( t − x ) d x t = 0 a + 1 2 λ 1 ∕ 2 ∫ 0 t q ( t ) q ′ ( x ) cos 2 λ 1 ∕ 2 ( t − x ) d x t = 0 a = − 1 2 λ − 1 ∕ 2 ∫ 0 a q ′ ( x ) ∫ x a q ( s ) d s cos 2 λ 1 ∕ 2 ( a − x ) d x + 1 2 λ − 1 ∕ 2 q ( a ) ∫ 0 a q ′ ( x ) cos 2 λ 1 ∕ 2 ( a − x ) d x ,

and

(17) I 3 = − 1 2 λ 1 ∕ 2 ∫ 0 t q 2 ( x ) cos 2 λ 1 ∕ 2 ( t − x ) d x t = 0 a + 1 2 λ 1 ∕ 2 q 2 ( t ) t = 0 a = − 1 2 λ − 1 ∕ 2 ∫ 0 a q 2 ( x ) cos 2 λ 1 ∕ 2 ( a − x ) d x .

The last equality holds because q ( t ) = q ( a − t ) . Therefore, the fourth and fifth terms on the right side of (14) obtain into error term O ( λ − 3 ∕ 2 ) because of equations (16) and (17). So by using equations (12) and (15), we rewrite equation (14) as follows:

(18) ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t = λ 1 ∕ 2 a + 1 4 λ − 1 sin 2 λ 1 ∕ 2 a ∫ 0 a ∕ 2 q ′ ( x ) cos 2 λ 1 ∕ 2 x d x − 1 4 λ − 1 [ 1 + cos 2 λ 1 ∕ 2 a ] ∫ 0 a ∕ 2 q ′ ( x ) sin 2 λ 1 ∕ 2 x d x + O ( λ − 3 ∕ 2 ) .

Finally, substituting equations (11), (13), and (18) into equation (10) and using reversion, we prove the theorem.

(ii) Similar to (i), we can arrange Lemmas 2.1(ii) as follows:

( 2 n + 3 ) π 2 = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − tan − 1 ( Ω ) + O ( λ − 3 ∕ 2 ) ,

where Ω is defined by equation (7). Theorem 3.1(ii) follows from the substitution of equations (11) and (18) into the last equation and using reversion.

(iii) We can reformulate Lemmas 2.2(i) as follows:

(19) ( n + 1 ) π = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − cot − 1 ( δ ) − tan − 1 ( η ) + O ( λ − 3 ∕ 2 ) ,

where η is defined by equation (8) and δ is defined by

(20) δ ≔ λ 1 ∕ 2 a 2 − 1 2 λ − 1 ∕ 2 a 2 q ( 0 ) + O ( λ − 3 ∕ 2 ) a 1 − λ a 1 ′ + O ( λ − 3 ∕ 2 ) .

By using series expansion to δ , we obtain

δ = λ 1 ∕ 2 a 2 − 1 2 λ − 1 ∕ 2 a 2 q ( 0 ) + O ( λ − 3 ∕ 2 ) − λ a 1 ′ 1 − λ − 1 a 1 a 1 ′ + O ( λ − 5 ∕ 2 ) = − λ − 1 ∕ 2 a 2 a 1 ′ + 1 2 λ − 3 ∕ 2 a 2 a 1 ′ q ( 0 ) + O ( λ − 5 ∕ 2 ) × 1 + λ − 1 a 1 a 1 ′ + λ − 2 a 1 2 ( a 1 ′ ) 2 + O ( λ − 5 ∕ 2 ) = − λ − 1 ∕ 2 a 2 a 1 ′ + O ( λ − 3 ∕ 2 ) .

From the last equation and inverse cotangent expansion cot − 1 ( δ ) = π 2 − δ + δ 3 3 + ⋯ , we write that

(21) cot − 1 ( δ ) = π 2 + λ − 1 ∕ 2 a 2 a 1 ′ + O ( λ − 3 ∕ 2 ) .

Substituting equations (13), (18), and (21) into equation (19), we prove the theorem.

(iv) We can reformulate Lemmas 2.2(ii) as follows:

( 2 n + 3 ) π 2 = ∫ 0 a [ r 2 ( t , λ ) + ρ 2 ( t , λ ) ] d t − cot − 1 ( δ ) + O ( λ − 3 ∕ 2 ) ,

where δ is defined by equation (20). Theorem 3.1(iv) follows from substitution of equations (18) and (21) into the last equation and using reversion.□

Example 1

Let us consider the following equation. This eigenvalue equation is named as anharmonic oscillator in physics:

y ″ ( t ) + [ λ − q ( t ) ] y ( t ) = 0 , t ∈ [ 0 , π ) ,

where q ( t ) = 1 4 t − π 2 4 + 1 2 t − π 2 2 . This potential is symmetric single-well. Since we accepted q ( t ) has mean value of zero for the sake of brevity, we should rearrange the anharmonic oscillator potential so that its mean value is zero. Therefore, q ( t ) can be calculated as follows:

q ( t ) = 1 4 t − π 2 4 + 1 2 t − π 2 2 − π 2 24 − π 4 320 .

In this case, by calculating the integral terms in Theorem 3.1, we obtain the following as n → ∞ :

(i) if a 2 ′ ≠ 0 and β ≠ 0

λ n 1 ∕ 2 = n + 1 + 1 ( n + 1 ) π a 1 ′ a 2 ′ + cot β + π 32 ( n + 1 ) 5 [ ( π 2 + 4 ) n ( n + 2 ) + π 2 − 2 ] + O ( n − 3 ) ,

(ii) if a 2 ′ ≠ 0 and β = 0

λ n 1 ∕ 2 = ( 2 n + 3 ) 2 + 2 ( 2 n + 3 ) π a 1 ′ a 2 ′ + O ( n − 3 ) ,

(iii) if a 2 ′ = 0 and β ≠ 0

λ n 1 ∕ 2 = ( 2 n + 3 ) 2 + 2 ( 2 n + 3 ) π a 2 a 1 ′ + cot β + 2 8 ( 2 n + 3 ) 6 × { 8 ( 4 n 2 + 12 n + 3 ) ( − 1 ) n + 1 − π ( 2 n + 3 ) [ 4 ( π 2 + 4 ) n ( n + 3 ) + 9 π 2 + 12 ] } + O ( n − 3 ) ,

(iv) if a 2 ′ = 0 and β = 0

λ n 1 ∕ 2 = n + 2 + 1 ( n + 2 ) π a 2 a 1 ′ − π 32 ( n + 2 ) 5 [ ( π 2 + 4 ) n ( n + 4 ) + 4 π 2 + 10 ] + O ( n − 3 ) .

Example 2

The Morse potential is a commonly accepted model for the description of a covalent bond like that between hydrogen and oxygen atoms. One can write the Morse potential in a compact form [20]:

q ( t ) = q 0 1 2 ( − 1 + A cosh α t ) 2 + B sinh α t .

[20] studies for A > 1 , B = 0 so that the potential is symmetric single-well in this situation. Hence, we can accept the Morse potential on [ 0 , 1 ] as follows:

q ( t ) = 1 2 − 1 + 2 cosh t − 1 2 2 .

Since we accepted that q ( t ) has mean value of zero for the sake of brevity, we should rearrange the Morse potential so that its mean value is zero. Therefore, q ( t ) can be calculated as follows:

q ( t ) = 1 2 − 1 + 2 cosh t − 1 2 2 + e − 1 8 e sinh 1 2 − e 2 − 3 e + 1 2 .

For this potential, by calculating the integral terms in Theorem 3.1, the asymptotic eigenvalues of equations (1)–(3) are given easily.

4 Conclusions

The asymptotic approximations for eigenvalues λ n of equations (1)–(3) with symmetric single-well potential q ( t ) are obtained in this work. These approximations for eigenvalues have been obtained with better error terms than previous works in the literature. When we applied the method from the beginning with a symmetric single potential, we obtained our results in Theorem 3.1 with better error terms than the results obtained by substituting the symmetric potential directly in the results in Lemmas 2.1 and 2.2. This is the importance of our results. Also, as a the result of the Theorem 3.1, we can say that, if we study with symmetric single well potential for (1.1)–(1.3), it is enough to observe the half interval instead of the whole interval of the problem. Because our result Theorem 3.1 can present the asymptotic eigenvalues on the half interval.

Acknowledgement

I would like to express my deepest gratitude to the referees and the handling editor for their valuable comments.

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

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Received: 2023-03-20
Revised: 2023-09-22
Accepted: 2023-11-14
Published Online: 2024-02-14

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

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

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https://doi.org/10.1515/dema-2023-0129
Keywords for this article
Sturm-Liouville problem; symmetric single-well potential; eigenvalue parameter in the boundary condition; asymptotic expansions
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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
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  8. Spectral collocation method for convection-diffusion equation
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  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
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  21. Results on solutions of several systems of the product type complex partial differential difference equations
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  38. Some aspects of normal curve on smooth surface under isometry
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  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
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  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
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  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
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  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
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  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
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  76. Review Articles
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  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
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  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
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  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
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  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
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  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
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