The equiconvergence of the eigenfunction expansion for a singular version of one-dimensional Schrodinger operator with explosive factor

This paper is devoted to prove the equiconvergence formula of the eigenfunction expansion for some version of Schrodinger operator with explosive factor. The analysis relies on asymptotic calculation and complex integration. The paper is of great interest for the community working in the area.

(2000) Mathematics Subject Classification

34B05; 43B24; 43L10; 47E05

Consider the Dirichlet problem

where q(x) is a non-negative real function belonging to L₁[0, π], λ is a spectral parameter, and ρ(x) is of the form

In [1], the author studied the asymptotic formulas of the eigenvalues, and eigenfunctions of problem (1.1)-(1.2) and proved that the eigenfunctions are orthogonal with weight function ρ(x). In [2], the author also studied the eigenfunction expansion of the problem(1.1)-(1.2). The calculation of the trace formula for the eigenvalues of the problem(1.1)-(1.2) is to appear. We mention here the basic definition and results from [1] that are needed in the progress of this work. Let φ(x, λ), ψ(x, λ) be the solutions of the problem (1.1)-(1.2) with the boundary conditions φ (0, λ) = 0, φ'(0, λ) = 1, ψ(π, λ) = 0, ψ'(π, λ) = 1 and let W(λ) = φ(x, λ)ψ'(x, λ) - ψ(x, λ)φ'(x, λ) be the Wronskian of the two linearly independent solutions φ(x, λ), ψ(x, λ). It is known that W is independent of x so that for x = a let W(λ) = Ψ(λ), the eigenvalues of (1.1)-(1.2) coincide with the roots of the equation Ψ(λ) = 0, which are simple. It is easy to see that the roots of Ψ(λ) = 0 are simple. The function

is called the Green's function of the Dirichlet problem (1.1)-(1.2). This function satisfies for λ = λ_k the relation

where λ_k are the eigenvalues of the Dirichlet problem (1.1)-(1.2) and a_k ≠ 0, where are the normalization numbers of the eigenfunctions of the same problem (1.1)-(1.2). We consider now the Dirichlet problem (1.1)-(1.2) in the simple form of q(x) ≡ 0. For q(x) = 0, the Dirichlet problem (1.1)-(1.2) takes the form

Let the eigenfunctions of the problem (1.6) be characterized by the index "o," i.e., φ_o(x, λ) and ψ_o(x, λ) are the solutions of the problem (1.6) in cases of ρ(x) = 1 and ρ(x) = -1, respectively, where

From (1.7), we notice that φ_o(x, λ) ψ_o(x, λ) are defined on parts of the interval [0, π], and these formulas must be extended to all intervals [0, π] to enable us to study the Green's function R(x, ξ, λ) in case of q(x) ≡ 0. The following lemma study this extension

Lemma 1.1 The solutions φ_o(x, λ) and ψ_o(x, λ) have the following asymptotic formulas

Proof: The fundamental system of solutions of the equation -y″ = s²y, (0 ≤ x ≤ a) is y₁(x, s) = sin sx, y₂(x, s) = cos sx. Similarly, the fundamental system of the equation y″ = s²y, (a < x ≤ π) is z₁(x, s) = sinh s(π - x), z₂(x, s) = cosh s(π - x). So that the solutions φ_o(x, λ) and ψ_o(x, λ), over [0, π], can be written in the forms

The constants c_i,i = 1, 2, 3, 4 are calculated from the continuity of φ_o(x, λ) and ψ_o(x, λ) together with their first derivatives at the point x = a, from which it can be easily seen that

Substituting (1.12) into (1.10), we get (1.8). In a similar way, we calculate the constants c₃, c₄ where

Substituting (1.12) and (1.13) into (1.10) and (1.11), respectively, we get the required relations (1.8) and (1.9)

The Green's function plays an important role in studying the equiconvergence theorem, so that, in addition to R(x, ξ, λ), we must study the corresponding Green's function for q(x) ≡ 0. Let R_o(x, ξ, λ) be the Green's function of problem (1.6), which is defined by

where the function

satisfies the following inequality on Γ_n, which is defined by (2.21)

Following [2], we state some basic asymptotic relations that are useful in the discussion. The solutions φ(x, λ) and ψ(x, λ) of the Dirichlet problem (1.1)-(1.2) have the following asymptotic formula

where

As we introduce in (1.4), the function R(x, ξ, λ) is the Green's function of the problem (1.1)-(1.2), and R_o(x, ξ, λ) is the corresponding Green's function of the problem (1.6). In the following lemma, we prove an important asymptotic relation for the Green's function

Lemma 2.2 For q(x) ∈ L₁(0, π) and by the help of the asymptotic formulas (2.4), (2.5) for φ(x, λ) and ψ(x, λ), respectively, the Green's function R(x, ξ, λ) satisfies the relation

where r(x, ξ, λ), λ ∈ Γ_n, n → ∞, satisfies

Proof: From (2.4) and (2.5), the function

takes the form

or

The function Ψ_o(λ) is given by (2.2). for x ≤ ξ, we discuss three possible cases:

(i) 0 ≤ x ≤ ξ ≤ a (ii) a ≤ x ≤ ξ ≤ π (iii) 0 ≤ x ≤ a ≤ ξ ≤ π.

The case (i) 0 ≤ x ≤ ξ ≤ a

From (1.4) and using (2.4) and (2.5), we have

Using (2.9), (2.10), and (2.3), we have

So that from (2.1), for 0 ≤ x ≤ ξ ≤ a, we have

The case (ii) a ≤ x ≤ ξ ≤ π.

Again, from (1.4) and using (2.4) and (2.5), we have

Using (2.9), (2.10), and (2.3), we have

So that from (2.1), for a ≤ x ≤ ξ ≤ π, we have

The case (iii) 0 ≤ x ≤ a ≤ ξ ≤ π.

From (1.4) and using (2.4) and (2.5), we have

Using (2.9), (2.10), and (2.3), we have

So that from (2.1), for a ≤ x ≤ ξ ≤ π, we have

The asymptotic formulas of R(x, ξ, λ) in case of ξ ≤ x remains to be evaluated and this, in turn, consists of three cases

(i*) 0 ≤ ξ ≤ x ≤ a (ii*) a ≤ ξ ≤ x ≤ π (iii*) 0 ≤ ξ ≤ a ≤ x ≤ π.

The case (i*) 0 ≤ ξ ≤ x ≤ a from (1.4) and using (2.4) and (2.5), we have

Using (2.9), (2.10), and (2.3), we have

So that from (2.1), for a ≤ ξ ≤ x ≤ a, we have

The case (ii*) a ≤ ξ ≤ x ≤ π from (1.4) and using (2.4) and (2.5), we have

Using (2.9), (2.10), and (2.3), we have

So that from (2.1), for a ≤ ξ ≤ x ≤ π, we have

The case (iii*) 0 ≤ ξ ≤ x ≤ a ≤ x ≤ π from (1.4) and using (2.4) and (2.5), we have

Using (2.9), (2.10), and (2.3), we have

So that from (2.1), for a ≤ ξ ≤ x ≤ a, we have

Now from (2.11) and (2.14), we have

also, from (2.12) and (2.15), we have

As a result of the last discussion from (2.13), (2.16), (2.17), and (2.18), we deduce that R(x, ξ, λ) obeys the asymptotic relation

where

We remind here that the main purpose of this paper is to prove the equiconvergence of the eigenfunction expansion of the Dirichlet problem (1.1)-(1.2). We introduce the following notations, let Δ_n,f(x) denotes the nth partial sum

where, from [1], . It should be noted here, from [2], that as n → ∞, the series (2.20) converges uniformly to a function f(x) ∈ L₂(0, π, ρ(x)). Let also be the corresponding nth partial sum as (2.20), for the Dirichlet problem (1.1)-(1.2) in case of q(x) ≡ 0. The equiconvergence of the eigenfunction expansion means that the difference uniformly converges to zero as n → ∞, x ∈ [0, π]. In the following theorem, we prove the equiconvergence theorem of the expansions . This means that the two expansions have the same condition of convergence. Following [1], the contour Γ_n is defined by

Denote by the upper half of the contour Γ_n, Ims ≥ 0, and let L_n be the contour, in λ-domain, formed from by the mapping λ = s². From (1.4), it is obvious that the poles of R(x, ξ, λ) are the roots of the function Ψ(s), which is the spectrum of the problem (1.1)-(1.2).

Theorem 2.1 Under the validity of lemma 1.1 and lemma 2.2, the following relation of equiconvergence holds true

Proof: Multiply both sides of (2.7) by ρ(ξ) f (ξ) and then integrating from 0 to π, we have

where f(x) ∈ L₂[0, π, ρ(x)]. We multiply the last equation by and then integrating over the contour L_n in the λ-domain, we have

From equation (1.5), we have the following

Applying Cauchy residues formula to the first integral of (2.23) and using (2.24), we have

Similarly, we carry out the same procedure to the second integral of (2.23) and we get an expression analogous to (2.25)

So that from (2.25), (2.26), and (2.23), we get

from which it follows that

The last Equation (2.27) is an essential relation in the proof of the theorem, because the theorem is established if we prove that tends to zero uniformly, x ∈ [0, π]. We use the same technique as in [3] We have

where M₁ and M₂ are constants.

We treat now the integral in (2.30). Let δ > 0 be a sufficiently small number and let λ = s², so that, for x, ξ ∈ [0, a], we have

This means that

where C₁, C₂, and C₃ are independent of x, n and δ. In a similar way, we estimate the second integral in (2.30) in the form

where , and are independent of x, n, and δ. Substituting (2.30) and (2.31) into (2.28) and using (2.29), we have

where A,B, and C are constants independent of x, n, and δ. We apply now the property of absolute continuity of Lesbuge integral to the function f(x) ∈ L₁[0, π].

∀ ϵ > 0, ∃ δ > 0 is sufficiently small such that ∫_|x-ξ|≤δ |f(ξ)|dξ ≤ ϵ, where ϵ is independent of x (the set {ξ : |x - ξ| ≤ δ} is measurable). Fixing δ in (2.32), there exists N such that for all and e^-δn < ϵ, so that (2.32) takes the form

Since ϵ is sufficiently small as we please, it follows that as n → ∞, uniformly with respect to x ∈ [0, π], which completes the proof.

It should be noted here that, the theorem of equiconvergence of the eigenfunction expansion is one of interesting analytical problem that arising in the field of spectral analysis of differential operators, see [4-6]. In [3], the author studied the equiconvergence theorem of the problem

There are many differences between problems (3.34)-(3.35) and the present one (1.1)-(1.2), and the differences are as follows:

1- The boundary conditions of (3.35) is separated boundary conditions, whereas (1.2) is the Dirichlet-Dirichlet condition

2- The eigenfunctions of (3.34)-(3.35) is given by

and

3- The contour of integration is of the form

4- The remainder function r(x, ξ, λ) admits the following inequality for λ ∈ Γ_n, n → ∞.

Although there are four differences between the two problems, we find that the proof of the equiconvergence formula as n → ∞ is similar. So as long as the proof of the equiconvergence relation is carried out by means of the contour integration, we obtain the uniform convergence of the series (2.20)

The authors declare that they have no competing interests.

The two authors typed read and approved the final manuscript also they contributed to each part of this work equally.

We are indebted to an anonymous referee for a detailed reading of the manuscript and useful comments and suggestions, which helped us improve this work. This work was supported by the research center of Alexandria University.

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