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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 828176, 16 pages
Research Article

The Eigenfunction Expansion for a Dirichlet Problem with Explosive Factor

1Department of Mathematics, Faculty of Education, Alexandria University, Alexanderia 21526, Egypt
2Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo, Egypt

Received 10 February 2011; Accepted 2 May 2011

Academic Editor: PavelΒ DrΓ‘bek

Copyright Β© 2011 Zaki F. A. El-Raheem and A. H. Nasser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We prove the eigenfunction expansion formula for a Dirichlet problem with explosive factor by two ways, first by standard method and second by proving a convergence in some metric space 𝐿2(0,πœ‹;𝜌(π‘₯)).

1. Introduction

The solutions of many problems of mathematical physics are reduced to the spectral investigation of a differential operator or finding the expansion of arbitrary function, in series or integration, in terms of the eigenfunctions of a differential operator. The differential operator is called regular if it its domain is finite and its coefficients are continuous; otherwise it is called singular differential operator. The Sturm-Liouville theory occupies a central position of the spectral theory of regular operator. During the development of quantum mechanics, there was an increase in the interest of spectral theory of singular operator on which we will restrict our attention. The first basic role in the development of spectral theory of singular operator dates back to E. Ch. Tetchmarsh [1]. He gave a new approach in the spectral theory of singular differential operator of the second order by using contour integration. Also Levitan [2] gave a new method, he obtained the eigenfunction expansion in infinite interval by taking limit of a regular case. In the last time about twenty five or so years, due to the needs of mathematical physics, in particular, quantum mechanics, the question of solving various spectral problems with explosive factor has been arisen. These appeared also in the study of geophysics and electromagnetic field, see Alemov [3]. The spectral theory of differential operators with explosive factor is studied by A. N. Tekhanov, M. G. Krien, M. G. Gasimov, and others. In this paper, we find the eigenfunction expansion formula and prove its convergence for following version of a Dirichlet problem (1.2), (1.3). The introduction of the weight function 𝜌(π‘₯) (1.4) as Β± signs causes many analytical difficulties, see [4], because the problem is to be treated as two separated problems and so the formula of eigenfunctions expansion is written as two sums. In [5] the author considered the weight function of the form ξƒ―π›ΌπœŒ(π‘₯)=2;Im𝛼≠0,0≀π‘₯β‰€π‘Ž<πœ‹,1;π‘Ž<π‘₯β‰€πœ‹,(1.1) and the spectrum was both continuous and discreet so that the formula of eigenfunctions expansion obtained there was written as a summation and integration. We must notice that the result of this paper is a starting point in solving the inverse spectral problem which will be investigated later on.

Consider the following Dirichlet problem: βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=πœ†πœŒ(π‘₯)𝑦,0≀π‘₯β‰€πœ‹,(1.2)𝑦(0)=0,𝑦(πœ‹)=0,(1.3) where the nonnegative real function π‘ž(π‘₯) has a second piecewise integrable derivatives on [0,πœ‹], πœ† is a spectral parameter, and the weight function or the explosive factor 𝜌(π‘₯) is of the form ξƒ―πœŒ(π‘₯)=1,0≀π‘₯β‰€π‘Ž<πœ‹,βˆ’1,π‘Ž<π‘₯β‰€πœ‹.(1.4) In [4] the author proved that the eigenvalues πœ†Β±π‘›,𝑛=0,1,2,…, of problem (1.2)-(1.3) are real and the corresponding eigenfunctions πœ‘(π‘₯,πœ†Β±π‘›),πœ“(π‘₯,πœ†Β±π‘›) are orthogonal with weight function 𝜌(π‘₯). We prove, here, the reality of these eigenfunctions under the condition that π‘ž(π‘₯) is real. Indeed, let πœ‘(π‘₯,πœ†) be the solution of the differential equation (1.2), π‘₯∈(0,π‘Ž) which satisfies the conditionsπœ‘(0,πœ†)=0,πœ‘ξ…ž(0,πœ†)=1(1.5) so that βˆ’πœ‘ξ…žξ…ž(π‘₯,πœ†)+π‘ž(π‘₯)πœ‘(π‘₯,πœ†)=πœ†πœ‘(π‘₯,πœ†),(1.6) taking the complex conjugate we have βˆ’πœ‘ξ…žξ…ž(π‘₯,πœ†)+π‘ž(π‘₯)πœ‘(π‘₯,πœ†)=πœ†πœ‘(π‘₯,πœ†).(1.7) By the aid of the uniqueness theorem, we have πœ‘(π‘₯,πœ†)=πœ‘(π‘₯,πœ†). In a similar way, we can see that πœ“(π‘₯,πœ†)=πœ“(π‘₯,πœ†) where πœ“(π‘₯,πœ†) is the solution of (1.2), π‘₯∈(π‘Ž,πœ‹),β€‰β€‰πœ“(πœ‹,πœ†)=0,β€‰β€‰πœ“β€²(πœ‹,πœ†)=1, that is, the eigenfunctions of the problem (1.2)-(1.3) are real. As we know from [4], the eigenvalues of problem (1.2)-(1.3) coincide with the roots of the function Ξ¨(πœ†)=0, where Ξ¨(πœ†) is the Wronskian of the two solutions πœ‘(π‘₯,πœ†), πœ“(π‘₯,πœ†) of (1.2)-(1.3)[]Ξ¨(πœ†)=π‘Šπœ‘(π‘Ž,πœ†),πœ“(π‘Ž,πœ†)=0.(1.8) In the following lemma, under the reality of π‘ž(π‘₯), we prove the simplicity of the eigenvalues, that is, we prove that the roots of (1.8) are simple, in other cases for π‘ž(π‘₯) is complex the roots of (1.8) may not be simple.

Lemma 1.1. Under the conditions stated in the introduction with respect to the problem (1.2)-(1.3), the eigenvalues of the problem (1.2)-(1.3) are simple.

Proof. We prove that Μ‡Ξ¨(πœ†)β‰ 0 where the dot means differentiation with respect to πœ†. Let πœ‘(π‘₯,πœ†) be the solution of the problem βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=πœ†π‘¦,(0≀π‘₯β‰€π‘Ž),𝑦(0)=0,π‘¦ξ…ž(0)=1,(1.9) and let πœ“(π‘₯,πœ†) be the solution of the problem βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=βˆ’πœ†π‘¦,(π‘Ž<π‘₯β‰€πœ‹),𝑦(πœ‹)=0,π‘¦ξ…ž(πœ‹)=1.(1.10) From (1.9), we have βˆ’πœ‘ξ…žξ…ž(π‘₯,πœ†)+π‘ž(π‘₯)πœ‘(π‘₯,πœ†)=πœ†πœ‘(π‘₯,πœ†).(1.11) Differentiating (1.11) with respect to πœ†, we have βˆ’Μ‡πœ‘ξ…žξ…ž(π‘₯,πœ†)+π‘ž(π‘₯)Μ‡πœ‘(π‘₯,πœ†)=πœ†Μ‡πœ‘(π‘₯,πœ†)+πœ‘(π‘₯,πœ†).(1.12) Multiplying (1.11) by Μ‡πœ‘(π‘₯,πœ†) and (1.12) by πœ‘(π‘₯,πœ†) and then subtracting the two results we have (𝑑/𝑑π‘₯)[Μ‡πœ‘πœ‘ξ…žβˆ’Μ‡πœ‘ξ…žπœ‘]=πœ‘2(π‘₯,πœ†) from which, by integrating with respect to π‘₯, from 0 to π‘Ž and using the conditions in (1.10), we obtain πœ‘ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έξ€·Μ‡πœ‘π‘Ž,πœ†π‘›ξ€Έβˆ’Μ‡πœ‘ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έπœ‘ξ€·π‘Ž,πœ†π‘›ξ€Έ=ξ€œπ‘Ž0πœ‘2ξ€·π‘₯,πœ†π‘›ξ€Έπ‘‘π‘₯.(1.13) In a similar way, from (1.10), we can write βˆ’πœ“ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έξ€·Μ‡πœ“π‘Ž,πœ†π‘›ξ€Έ+Μ‡πœ“ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έπœ“ξ€·π‘Ž,πœ†π‘›ξ€Έξ€œ=βˆ’πœ‹π‘Žπœ“2ξ€·π‘₯,πœ†π‘›ξ€Έπ‘‘π‘₯.(1.14) From (1.8), we have πœ‘ξ€·π‘Ž,πœ†π‘›ξ€Έπœ“ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έ=πœ‘ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έπœ“ξ€·π‘Ž,πœ†π‘›ξ€Έ.(1.15) From which we have πœ‘ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έ=πœ‘ξ€·π‘Ž,πœ†π‘›ξ€Έπœ“ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έπœ“ξ€·π‘Ž,πœ†π‘›ξ€Έ,πœ“ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έ=πœ‘ξ…žξ€·π‘Ž,πœ†π‘›ξ€Έπœ“ξ€·π‘Ž,πœ†π‘›ξ€Έπœ‘ξ€·π‘Ž,πœ†π‘›ξ€Έ.(1.16) Substituting from (1.16) into (1.13) and (1.14), we can see by adding that Μ‡Ξ¨ξ€·πœ†π‘›ξ€Έ=π‘Ÿπ‘›π‘Žπ‘›,(1.17) where π‘Žπ‘›=ξ€œπ‘Ž0πœ‘2ξ€·π‘₯,πœ†π‘›ξ€Έ1𝑑π‘₯βˆ’π‘Ÿ2π‘›ξ€œπœ‹π‘Žπœ“2ξ€·π‘₯,πœ†π‘›ξ€Έπ‘‘π‘₯,π‘Ÿπ‘›=πœ“ξ€·π‘Ž,πœ†π‘›ξ€Έπœ‘ξ€·π‘Ž,πœ†π‘›ξ€Έ,(1.18) and the numbers π‘Žπ‘› are the normalization numbers of the eigenfunctions of problem (1.2)-(1.3). Following [4], we have π‘Ÿπ‘›β‰ 0 and π‘Žπ‘›β‰ 0 which complete the proof of lemma.

2. The Function 𝑅(π‘₯,πœ‰,πœ†)

We introduce the function 𝑅(π‘₯,πœ‰,πœ†) by 𝑅(π‘₯,πœ‰,πœ†)=βˆ’1ξƒ―Ξ¨(πœ†)πœ‘(π‘₯,πœ†)πœ“(πœ‰,πœ†),π‘₯β‰€πœ‰,πœ‘(πœ‰,πœ†)πœ“(π‘₯,πœ†),πœ‰β‰€π‘₯,(2.1) which is called the Green’s function of the nonhomogenous Dirichlet problem βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=πœ†πœŒ(π‘₯)𝑦+𝜌(π‘₯)𝑓(π‘₯),0≀π‘₯β‰€πœ‹,𝑦(0)=𝑦(πœ‹)=0,(2.2) where 𝑓(π‘₯)∈𝐿1(0,πœ‹) and 𝜌(π‘₯) is defined by (1.4). The function 𝑅(π‘₯,πœ‰,πœ†) is, also, called the kernel of the resolvent π‘…πœ†=(π΄βˆ’πœ†πΌ)βˆ’1, where π΄β‰‘βˆ’(𝑑2/𝑑π‘₯2)+π‘ž(π‘₯),𝐷(𝐴)={𝑦(π‘₯)βˆΆβˆƒπ‘¦ξ…žξ…ž,𝑦(0)=𝑦(πœ‹)=0}. In the following lemmas, we prove some essential properties of 𝑅(π‘₯,πœ‰,πœ†) which are useful in the forthcoming study of the eigenfunction expansion of the problem (1.2)-(1.3)

Lemma 2.1. Let 𝑓(π‘₯) be any function belonging to 𝐿1(0,πœ‹), then the function ξ€œπ‘¦(π‘₯,πœ†)=πœ‹0𝑅(π‘₯,πœ‰,πœ†)𝑓(πœ‰)𝜌(πœ‰)π‘‘πœ‰(2.3) is the solution of the nonhomogenous Dirichlet problem (2.2).

Proof. First we show that (2.3) satisfies the boundary conditions of (2.2). From (2.2) by using (1.9) and (1.10), we, respectively, have 𝑦(0)=βˆ’1ξ€œΞ¨(πœ†)πœ‹0πœ‘(0,πœ†)πœ“(πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=0,𝑦(πœ‹)=βˆ’1ξ€œΞ¨(πœ†)πœ‹0πœ‘(πœ‰,πœ†)πœ“(πœ‹,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=0.(2.4) Secondly, we calculate the solution of (2.2) by the method of variation of parameters. We seek a solution of the nonhomogenous problem (2.2) in the form 𝑦(π‘₯,πœ†)=𝐢1πœ‘(π‘₯,πœ†)+𝐢2πœ“(π‘₯,πœ†),(2.5) where πœ‘(π‘₯,πœ†),πœ“(π‘₯,πœ†) are given together with their asymptotic formulas in [4]. By using the standard calculation, we find 𝐢1(π‘₯,πœ†)=βˆ’1ξ€œΞ¨(πœ†)πœ‹π‘₯𝐢𝑓(πœ‰)𝜌(πœ‰)πœ“(πœ‰,πœ†)π‘‘πœ‰,2(π‘₯,πœ†)=βˆ’1ξ€œΞ¨(πœ†)π‘₯0𝑓(πœ‰)𝜌(πœ‰)πœ‘(πœ‰,πœ†)π‘‘πœ‰.(2.6) Substituting from (2.6) into (2.5) and keeping in mind (2.1), we get the required formula (2.3).

Lemma 2.2. Under the conditions of Lemma 2.1, the function 𝑅(π‘₯,πœ‰,πœ†) satisfies the following formula: 𝑅(π‘₯,πœ‰,πœ†)=βˆ’1πœ†βˆ’πœ†π‘›π‘Žπ‘›πœ‘ξ€·π‘₯,πœ†π‘›ξ€Έπœ‘ξ€·πœ‰,πœ†π‘›ξ€Έ+𝑅1(π‘₯,πœ‰,πœ†),(2.7) where 𝑅1(π‘₯,πœ‰,πœ†) is regular in the neighborhood of πœ†=πœ†π‘› and π‘Žπ‘›=βˆ«πœ‹0𝜌(πœ‰)πœ‘2(π‘₯,πœ†π‘›)𝑑π‘₯.

Proof. So long as, from Lemma 1.1, the roots of the function Ξ¨(πœ†) are simple; it follows that the poles of the function 𝑅(π‘₯,πœ‰,πœ†) are simple. So that 𝑅(π‘₯,πœ‰,πœ†) can be represented in the form []𝑅(π‘₯,πœ‰,πœ†)=Res𝑅(π‘₯,πœ‰,πœ†)πœ†βˆ’πœ†π‘›+𝑅1(π‘₯,πœ‰,πœ†)(2.8) from (2.1); for π‘₯β‰€πœ‰, we have Resπœ†=πœ†π‘›[]πœ‘ξ€·π‘…(π‘₯,πœ‰,πœ†)=βˆ’π‘₯,πœ†π‘›ξ€Έπœ“ξ€·πœ‰,πœ†π‘›ξ€ΈΜ‡Ξ¨ξ€·πœ†π‘›ξ€Έ.(2.9) From (1.17) and (1.18), the relation (2.9) takes the form Resπœ†=πœ†π‘›[]πœ‘ξ€·π‘…(π‘₯,πœ‰,πœ†)=βˆ’π‘₯,πœ†π‘›ξ€Έπœ‘ξ€·πœ‰,πœ†π‘›ξ€Έπ‘Žπ‘›.(2.10) Formula (2.7) is obtained by substituting from (2.10) into (2.8) we deduce the formula (2.7). We notice that in case of πœ‰β‰€π‘₯, in a similar way. in order to prove the convergence of the eigenfunction expansion of the Dirichlet problem (1.2)-(1.3), we must write an equality for the function 𝑅(π‘₯,πœ‰,πœ†) and this, in turn, needs to extend the asymptotic formulas of πœ‘(π‘₯,πœ†),πœ“(π‘₯,πœ†) over all the interval [0,πœ‹]. In [4], the asymptotic formulas for πœ‘(π‘₯,πœ†),β€‰β€‰πœ“(π‘₯,πœ†) were deduced for π‘₯∈(0,π‘Ž) and (π‘Ž,πœ‹), respectively. In the following lemma, we write this asymptotic formulas over all [0,πœ‹] for both πœ‘(π‘₯,πœ†) and πœ“(π‘₯,πœ†).

Lemma 2.3. The solutions πœ‘(π‘₯,πœ†) and πœ“(π‘₯,πœ†) of the Dirichlet problem (1.2)-(1.3) have the following asymptotic formula: ⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©πœ‘(π‘₯,πœ†)=sin𝑠π‘₯𝑠𝑒+β—‹|Im𝑠|π‘₯||𝑠2||ξ‚Ά,0≀π‘₯β‰€π‘Ž,𝛽(π‘₯)[],𝑒𝑠𝛽(π‘Ž)sinπ‘ π‘Žcosh𝑠(π‘Žβˆ’π‘₯)βˆ’cosπ‘ π‘Žsinh𝑠(π‘Žβˆ’π‘₯)+β—‹|Im𝑠|π‘Ž+|β„œπ‘ |(π‘Žβˆ’π‘₯)||𝑠2||ξ‚ΆβŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩,π‘Ž<π‘₯β‰€πœ‹,(2.11)πœ“(π‘₯,πœ†)=𝛼(π‘₯)[],𝑒𝑠𝛼(π‘Ž)cos𝑠(π‘₯βˆ’π‘Ž)sinh𝑠(πœ‹βˆ’π‘Ž)βˆ’sin𝑠(π‘₯βˆ’π‘Ž)cosh𝑠(πœ‹βˆ’π‘Ž)+β—‹|Im𝑠|(π‘₯βˆ’π‘Ž)+|β„œπ‘ |(πœ‹βˆ’π‘Ž)||𝑠2||ξ‚Ά,0≀π‘₯β‰€π‘Ž,sinh𝑠(πœ‹βˆ’π‘₯)𝑠𝑒+β—‹|β„œπ‘ |(πœ‹βˆ’π‘₯)||𝑠2||ξ‚Ά,π‘Žβ‰€π‘₯β‰€πœ‹,(2.12) where 1𝛼(π‘₯)=2ξ€œπ‘₯01π‘ž(𝑑)𝑑𝑑,𝛽(π‘₯)=2ξ€œπœ‹π‘₯π‘ž(𝑑)𝑑𝑑,πœ†=𝑠2.(2.13)

Proof. Following [4], the solutions πœ‘(π‘₯,πœ†) and πœ“(π‘₯,πœ†) of the Dirichlet problem (1.2)-(1.3) have the representation πœ‘(π‘₯,πœ†)=sin𝑠π‘₯𝑠𝑒+β—‹|ℑ𝑠|π‘₯||𝑠2||ξ‚Ά[],,π‘₯∈0,π‘Ž(2.14)πœ“(π‘₯,πœ†)=sinh𝑠(πœ‹βˆ’π‘₯)𝑠𝑒+β—‹|β„œπ‘ |(πœ‹βˆ’π‘₯)||𝑠2||ξ‚Ά[].,π‘₯βˆˆπ‘Ž,πœ‹(2.15) We can see, from [6], that the solution 𝑦(π‘₯,𝑠) of the equation βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=𝑠2𝑦0≀π‘₯β‰€π‘Ž,  𝑦(0)=0 has the representation 𝑦(π‘₯,𝑠)=𝑒𝑖𝑠π‘₯𝛼(π‘₯)𝑠1+○𝑠2,π‘¦ξ…ž(π‘₯,𝑠)=𝑒𝑖𝑠π‘₯1𝑖𝛼(π‘₯)+○𝑠,(2.16) and the solution 𝑧(π‘₯,𝑠) of the equation βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=βˆ’π‘ 2𝑦,π‘Ž<π‘₯β‰€πœ‹,𝑧(πœ‹)=0 has the representation 𝑧(π‘₯,𝑠)=𝑒𝑠(πœ‹βˆ’π‘₯)𝛽(π‘₯)𝑠1+○𝑠2,π‘§ξ…ž(π‘₯,𝑠)=𝑒𝑠(πœ‹βˆ’π‘₯)1βˆ’π›½(π‘₯)+○𝑠,(2.17) where 𝛼(π‘₯) and 𝛽(π‘₯) are given by (2.13), so that, πœ‘(π‘₯,πœ†) can be extended to (π‘Ž,πœ‹) in terms of the two linearly independent solutions 𝑧(π‘₯,𝑠),  𝑧(π‘₯,βˆ’π‘ ) as πœ‘(π‘₯,πœ†)=π‘š1𝑧(π‘₯,𝑠)+π‘š2𝑧(π‘₯,βˆ’π‘ ).(2.18) From the continuity of πœ‘(π‘₯,πœ†) at π‘₯=π‘Ž and by using the asymptotic relations, (2.14), of πœ‘(π‘₯,πœ†) and (2.17) of 𝑧(π‘₯,𝑠), the constants π‘š1, π‘š2 are calculated in the form π‘š1=𝑒𝑠(π‘Žβˆ’π‘₯)𝛽(π‘₯)ξ‚Έ2𝛽(π‘Ž)sinπ‘ π‘Žβˆ’cosπ‘ π‘Žπ‘ ξ‚΅π‘’+β—‹|Im𝑠|π‘Žπ‘ 2,π‘šξ‚Άξ‚Ή2=𝑒𝑠(πœ‹βˆ’π‘Ž)1𝑒2𝛽(π‘Ž)βˆ’sinπ‘ π‘Žβˆ’cosπ‘ π‘Ž+β—‹|Im𝑠|π‘Žπ‘ .ξ‚Άξ‚Ή(2.19) Substituting from (2.19) into (2.18), we have, for π‘₯∈(π‘Ž,πœ‹]πœ‘(π‘₯,πœ†)=𝛽(π‘₯)[]𝑒𝑠𝛽(π‘Ž)sinπ‘ π‘Žcosh𝑠(π‘Žβˆ’π‘₯)βˆ’cosπ‘ π‘Žsinh𝑠(π‘Žβˆ’π‘₯)+β—‹|Im𝑠|π‘Ž+|Res|(π‘Žβˆ’π‘₯)||𝑠2||ξ‚Ά.(2.20) From (2.20) together with (2.14), the relation (2.11) is followed. The proof of the relation (2.12) is quite similar to the proof of (2.11). Indeed πœ“(π‘₯,πœ†),π‘₯∈(π‘Ž,πœ‹] is a linear combination of the two linearly independent solutions 𝑦(π‘₯,𝑠),  𝑦(π‘₯,βˆ’π‘ ) as πœ“(π‘₯,πœ†)=π‘˜1𝑦(π‘₯,𝑠)+π‘˜2𝑦(π‘₯,βˆ’π‘ ), where π‘˜1=π‘’βˆ’π‘–π‘ π‘Ž[]𝑒2𝛼(π‘Ž)sinh𝑠(πœ‹βˆ’π‘Ž)+𝑖cosh𝑠(πœ‹βˆ’π‘Ž)+β—‹|Res|(πœ‹βˆ’π‘Ž)ξ‚Ά,π‘˜|𝑠|2=π‘’π‘–π‘ π‘Ž[]𝑒2𝛼(π‘Ž)βˆ’sinh𝑠(πœ‹βˆ’π‘Ž)+𝑖cosh𝑠(πœ‹βˆ’π‘Ž)+β—‹|Res|(πœ‹βˆ’π‘Ž)|ξ‚Ά,𝑠|(2.21) and by using the asymptotic formulas (2.16) of 𝑦(π‘₯,±𝑠) and (2.15) of πœ“(π‘₯,πœ†), we have, for π‘₯∈[0,π‘Ž], πœ“(π‘₯,πœ†)=𝛼(π‘₯)[]𝑒𝑠𝛼(π‘Ž)cos𝑠(π‘₯βˆ’π‘Ž)sinh𝑠(πœ‹βˆ’π‘Ž)βˆ’sin𝑠(π‘₯βˆ’π‘Ž)cosh𝑠(πœ‹βˆ’π‘Ž)+β—‹|Im𝑠|(π‘₯βˆ’π‘Ž)+|Res|(πœ‹βˆ’π‘Ž)||𝑠2||ξ‚Ά.(2.22) The relation (2.20) together with (2.14) and the relation (2.22) together with (2.15) complete the proof of lemma.

The following inequality proves an inequality satisfied by 𝑅(π‘₯,πœ‰,πœ†).

Lemma 2.4. Under the conditions of Lemma 2.3, the resolvent 𝑅(π‘₯,πœ‰,πœ†) satisfies the following inequality: π‘…βŽ§βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©β—‹ξ‚΅π‘’(π‘₯,πœ‰,πœ†)=βˆ’|Im𝑠||2π‘Žβˆ’π‘₯βˆ’πœ‰|ξ‚Ά[],○𝑒|𝑠|,π‘₯,πœ‰βˆˆ0,π‘Žβˆ’|Res||π‘₯+πœ‰βˆ’2π‘Ž|ξ‚Ά[],○𝑒|𝑠|,π‘₯,πœ‰βˆˆπ‘Ž,πœ‹βˆ’|Im𝑠|(π‘Žβˆ’π‘₯)βˆ’|Res|(πœ‰βˆ’π‘Ž)○𝑒|𝑠|,0≀π‘₯β‰€π‘Ž<πœ‰β‰€πœ‹,βˆ’|Im𝑠|(π‘Žβˆ’πœ‰)βˆ’|Res|(π‘₯βˆ’π‘Ž)ξ‚Ά|𝑠|,0β‰€πœ‰β‰€π‘Ž<π‘₯β‰€πœ‹.(2.23)

Proof. From (2.11) and (2.12), we have ⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©β—‹ξ‚΅π‘’πœ‘(π‘₯,πœ†)=βˆ’|Im𝑠|π‘₯○𝑒|𝑠|,0≀π‘₯β‰€π‘Ž,βˆ’|Im𝑠|π‘Ž+|Res|(π‘Žβˆ’π‘₯)ξ‚ΆβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©β—‹ξ‚΅π‘’|𝑠|,π‘Ž<π‘₯β‰€πœ‹,(2.24)πœ“(π‘₯,πœ†)=βˆ’|Im𝑠|(π‘₯βˆ’π‘Ž)+|Res|(πœ‹βˆ’π‘Ž)○𝑒|𝑠|,0≀π‘₯β‰€π‘Ž,βˆ’|Res|(πœ‹βˆ’π‘₯)ξ‚Ά|𝑠|,π‘Ž<π‘₯β‰€πœ‹.(2.25) It can be easily seen that, for π‘ βˆˆΞ“π‘›, we have 𝑒Ψ(πœ†)β‰₯𝐢|Im𝑠|π‘Ž+|Res|(πœ‹βˆ’π‘Ž)|𝑠|,π‘ βˆˆΞ“π‘›,(2.26) where 𝑠Γ𝑛 is the quadratic contour, as defined in [4] Γ𝑛=ξ‚»||||β‰€πœ‹Resπ‘Žξ‚€1π‘›βˆ’4+πœ‹,||||β‰€πœ‹2π‘ŽIm𝑠1πœ‹βˆ’π‘Žπ‘›βˆ’4+πœ‹ξ‚Ό2(πœ‹βˆ’π‘Ž).(2.27) From (2.1) we have six possibilities, three of which for π‘₯β‰€πœ‰ and the other three for πœ‰β‰€π‘₯. Now for πœ‰β‰€π‘₯ we have the following situation: (i) 0≀π‘₯β‰€πœ‰β‰€π‘Ž, (ii) π‘Ž<π‘₯β‰€πœ‰β‰€πœ‹, and (iii) 0≀π‘₯β‰€π‘Žβ‰€πœ‰β‰€πœ‹. In cases (i), (ii), and (iii) by direct substitution from (2.24), (2.25), (2.26) into the first branch of (2.1), we obtain 𝑒(i)𝑅(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Im𝑠||2π‘Žβˆ’π‘₯βˆ’πœ‰|𝑒|𝑠|,0≀π‘₯β‰€πœ‰β‰€π‘Ž,(2.28)(ii)𝑅(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Res||π‘₯+πœ‰βˆ’2π‘Ž||𝑒𝑠|,π‘Žβ‰€π‘₯β‰€πœ‰β‰€πœ‹,(2.29)(iii)𝑅(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Im𝑠|(π‘Žβˆ’π‘₯)βˆ’|Res|(πœ‰βˆ’π‘Ž)ξ‚Ά|𝑠|,0≀π‘₯β‰€π‘Ž<πœ‰β‰€πœ‹.(2.30)In the case of πœ‰β‰₯π‘₯, we discuss (i*) 0β‰€πœ‰β‰€π‘₯β‰€π‘Ž, (ii*) π‘Žβ‰€πœ‰β‰€π‘₯β‰€πœ‹, and (iii*) 0β‰€πœ‰β‰€π‘Žβ‰€π‘₯β‰€πœ‹.
Again by substituting (2.24), (2.25), and (2.26) into the second branch of (2.1), we get ξ€·iβˆ—ξ€Έξ‚΅π‘’π‘…(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Im𝑠||2π‘Žβˆ’π‘₯βˆ’πœ‰|ξ‚Άξ€·|𝑠|,0β‰€πœ‰β‰€π‘₯β‰€π‘Ž,(2.31)iiβˆ—ξ€Έξ‚΅π‘’π‘…(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Res||π‘₯+πœ‰βˆ’2π‘Ž||𝑠|,π‘Žβ‰€πœ‰β‰€π‘₯β‰€πœ‹,(2.32)iiiβˆ—ξ€Έξ‚΅π‘’π‘…(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Im𝑠|(π‘Žβˆ’πœ‰)βˆ’|Res|(π‘₯βˆ’π‘Ž)ξ‚Ά|𝑠|,0β‰€πœ‰β‰€π‘Ž<π‘₯β‰€πœ‹.(2.33) From (2.28) and (2.31), we have 𝑒𝑅(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Ims||2π‘Žβˆ’π‘₯βˆ’πœ‰|ξ‚Ά[]|𝑠|,π‘₯,πœ‰βˆˆ0,π‘Ž,(2.34) and from (2.29) and (2.32), we have 𝑒𝑅(π‘₯,πœ‰,πœ†)=β—‹βˆ’|Res||π‘₯+πœ‰βˆ’2π‘Ž|ξ‚Ά[]|𝑠|,π‘₯,πœ‰βˆˆπ‘Ž,πœ‹.(2.35) From (2.30) and (2.33) together with (2.34) and (2.35), the lemma is proved. In the following lemma, we prove an integral formula which is satisfied by 𝑅(π‘₯,πœ‰,πœ†) and help in proving the eigenfunction expansion formula

Lemma 2.5. If the function 𝑓(π‘₯) on [0,πœ‹] has a second-order integrable derivatives and satisfies the Dirichlet condition 𝑓(0)=𝑓(πœ‹)=0, then the following integral formula is true: ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=βˆ’π‘“(π‘₯)πœ†+ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)πœ†ξ€Ίβˆ’π‘“ξ…žξ…žξ€»(π‘₯)+π‘ž(π‘₯)𝑓(πœ‰)π‘‘πœ‰,(2.36) where 𝑅(π‘₯,πœ‰,πœ†) is the kernel of the resolvent of the nonhomogenous Dirichlet problem (2.2).

Proof. By the aid of (2.1), we have ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=βˆ’1ξ‚»ξ€œΞ¨(πœ†)πœ“(π‘₯,πœ†)π‘₯0ξ€œπœ‘(πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰+πœ‘(π‘₯,πœ†)πœ‹π‘₯ξ‚Ό,πœ“(πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰(2.37) where the functions πœ‘(π‘₯,πœ†) and πœ“(π‘₯,πœ†) are the solutions of the homogenous Dirichlet problem (1.2)-(1.3), so that ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=βˆ’1ξ‚»Ξ¨(πœ†)πœ“(π‘₯,πœ†)πœ†ξ€œπ‘₯0ξ€Ίπœ‘ξ…žξ…žξ€»+(πœ‰,πœ†)+π‘ž(πœ‰)πœ‘(πœ‰,πœ†)𝑓(πœ‰)π‘‘πœ‰πœ‘(π‘₯,πœ†)πœ†ξ€œπœ‹π‘₯ξ€Ίπœ“ξ…žξ…žξ€»ξ‚Ό(πœ‰,πœ†)+π‘ž(πœ‰)πœ“(πœ‰,πœ†)𝑓(πœ‰)π‘‘πœ‰(2.38) from which we have ξ€œπœ‹01𝑅(π‘₯,πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=ξ‚»Ξ¨(πœ†)πœ“(π‘₯,πœ†)πœ†ξ€œπ‘₯0πœ‘ξ…žξ…ž(πœ‰,πœ†)𝑓(πœ‰)π‘‘πœ‰+πœ‘(π‘₯,πœ†)πœ†ξ€œπœ‹π‘₯πœ“ξ…žξ…žξ‚Ό+1(πœ‰,πœ†)𝑓(πœ‰)π‘‘πœ‰πœ†ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)π‘ž(πœ‰)𝑓(πœ‰)π‘‘πœ‰.(2.39) Integrating by parts twice the terms ∫π‘₯0 and βˆ«πœ‹π‘₯, in (2.39), and then using the boundary conditions 𝑓(0)=𝑓(πœ‹)=πœ‘(0,πœ†)=0 and 𝑓(0)=𝑓(πœ‹)=πœ“(πœ‹,πœ†)=0, respectively, and keeping in mind (2.1), we deduce that ξ€œπœ“(π‘₯,πœ†)π‘₯0πœ‘ξ…žξ…žξ€œ(πœ‰,πœ†)𝑓(πœ‰)π‘‘πœ‰+πœ‘(π‘₯,πœ†)πœ‹π‘₯πœ“ξ…žξ…žξ€œ(πœ‰,πœ†)𝑓(πœ‰)π‘‘πœ‰=βˆ’Ξ¨(πœ†)𝑓(π‘₯)βˆ’Ξ¨(πœ†)πœ‹0𝑅(π‘₯,πœ‰,πœ†)π‘“ξ…žξ…ž(πœ‰)π‘‘πœ‰.(2.40) Substituting from (2.40) into (2.39), we get the required result.

3. The Eigenfunctions Expansion Formula

We now construct and prove the eigenfunction expansion formula for the Dirichlet problem (1.2)-(1.3). Let πœ†+𝑛,𝑛=0,1,2,… and πœ†βˆ’π‘›,𝑛=0,1,2,… be the nonnegative and the negative eigenvalues of the problem (1.2)-(1.3), and let also π‘Ž+𝑛=ξ€œπœ‹0𝜌(π‘₯)πœ‘2ξ€·π‘₯,πœ†+𝑛𝑑π‘₯,π‘Žβˆ’π‘›=ξ€œπœ‹0𝜌(π‘₯)πœ‘2ξ€·π‘₯,πœ†βˆ’π‘›ξ€Έπ‘‘π‘₯(3.1) be the normalization numbers of the corresponding eigenfunctions πœ‘(π‘₯,πœ†Β±π‘›). We put π‘£Β±π‘˜πœ‘ξ€·(π‘₯)=π‘₯,πœ†Β±π‘˜ξ€Έξ”π‘ŽΒ±π‘˜,π‘˜=0,1,2,….(3.2) The set {π‘£Β±π‘˜(π‘₯)}βˆžπ‘˜=0 is an orthonormal system of eigenfunctions of the Dirichlet problem (1.2)-(1.3).

Theorem 3.1. Let f(x) be a second-order integrable derivatives on [0,πœ‹] and satisfy the conditions 𝑓(0)=𝑓(πœ‹)=0; then the following formula of eigenfunction expansion is true: 𝑓(π‘₯)=βˆžξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+βˆžξ“π‘˜=0π‘βˆ’π‘˜π‘£βˆ’π‘˜(π‘₯),(3.3) where π‘Β±π‘˜=βˆ«πœ‹0π‘£Β±π‘˜(πœ‰)𝑓(πœ‰)𝜌(πœ‰)π‘‘πœ‰ and the series uniformly converges to 𝑓(π‘₯),π‘₯∈[0,πœ‹].

Notice that, the expansion (3.3) can be written, more explicitly, in terms of πœ‘(π‘₯,πœ†Β±π‘˜) as 𝑓(π‘₯)=βˆžξ“π‘˜=01π‘Ž+π‘˜πœ“ξ€·π‘₯,πœ†+π‘˜ξ€Έξ€œπœ‹0πœ“ξ€·πœ‰,πœ†+π‘˜ξ€Έπ‘“(πœ‰)𝜌(πœ‰)π‘‘πœ‰+βˆžξ“π‘˜=01π‘Žβˆ’π‘˜πœ“ξ€·π‘₯,πœ†βˆ’π‘˜ξ€Έξ€œπœ‹0πœ“ξ€·πœ‰,πœ†βˆ’π‘˜ξ€Έπ‘“(πœ‰)𝜌(πœ‰)π‘‘πœ‰(3.4) or in terms of πœ“(π‘₯,πœ†Β±π‘˜)𝑓(π‘₯)=βˆžξ“π‘˜=01ξ€·π‘Ÿ+π‘˜ξ€Έ2π‘Ž+π‘˜πœ‘ξ€·π‘₯,πœ†+π‘˜ξ€Έξ€œπœ‹0πœ‘ξ€·πœ‰,πœ†+π‘˜ξ€Έ+𝑓(πœ‰)𝜌(πœ‰)π‘‘πœ‰βˆžξ“π‘˜=01ξ€·π‘Ÿβˆ’π‘˜ξ€Έ2π‘Žβˆ’π‘˜πœ‘ξ€·π‘₯,πœ†βˆ’π‘˜ξ€Έξ€œπœ‹0πœ‘ξ€·πœ‰,πœ†βˆ’π‘˜ξ€Έπ‘“(πœ‰)𝜌(πœ‰)π‘‘πœ‰,(3.5) where π‘ŸΒ±π‘˜ are defined by πœ“(π‘₯,πœ†Β±π‘˜)=π‘ŸΒ±π‘˜πœ‘(π‘₯,πœ†π‘˜Β±,),  0≀π‘₯β‰€πœ‹.

Proof. We write (2.36) in the form ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)𝜌(πœ‰)𝑓(πœ‰)π‘‘πœ‰=βˆ’π‘“(π‘₯)πœ†+π‘Ÿ(π‘₯,πœ†),(3.6) where ξ€œπ‘Ÿ(π‘₯,πœ†)=πœ‹0𝑅(π‘₯,πœ‰,πœ†)πœ†ξ€Ίβˆ’π‘“ξ…žξ…žξ€»(π‘₯)+π‘ž(π‘₯)𝑓(πœ‰)π‘‘πœ‰.(3.7) By the aid of Lemma 2.3 and the condition of the theorem imposed on π‘ž(π‘₯), it can be easily seen that ||||β‰€π‘€π‘Ÿ(π‘₯,πœ†)π‘œ||πœ†||3/2,π‘ βˆˆΞ“π‘›,(3.8) where π‘€π‘œ is constant which is independent of π‘₯,πœ‰,πœ† and the contour Γ𝑛, as defined in [4], is given by (2.27). Let πœ†=𝑠2; we denote by Ξ“+𝑛 the upper half of the Γ𝑛; let also Ł𝑛 denote the image of the contour Ξ“+𝑛 under the transformation πœ†=𝑠2. We multiply both sides of (3.6) by 1/2πœ‹π‘– and integrating with respect to πœ† on the contour 𝐿𝑛: 1ξ€Ÿ2πœ‹π‘–πΏπ‘›ξ‚»ξ€œπœ‹0ξ‚Ό=𝑅(π‘₯,πœ‰,πœ†)𝑓(πœ‰)𝜌(πœ‰)π‘‘πœ‰βˆ’π‘“(π‘₯)ξ€Ÿ2πœ‹π‘–πΏπ‘›π‘‘πœ†πœ†+1ξ€Ÿ2πœ‹π‘–πΏπ‘›π‘Ÿ(π‘₯,πœ†)π‘‘πœ†.(3.9) Among the poles of the function 𝑅(π‘₯,πœ‰,πœ†), as a function of πœ†, lie only πœ†Β±π‘œ,πœ†Β±1,…,πœ†Β±π‘› inside 𝐿𝑛. By using the residues formula and (2.10), we have 1ξ€Ÿ2πœ‹π‘–πΏπ‘›ξ‚»ξ€œπœ‹0ξ‚Ό=𝑅(π‘₯,πœ‰,πœ†)𝑓(πœ‰)𝜌(πœ‰)π‘‘πœ‰π‘›ξ“π‘˜=0Resπœ†=πœ†Β±π‘˜ξ‚»ξ€œπœ‹0𝑅(π‘₯,πœ‰,πœ†)𝑓(πœ‰)𝜌(πœ‰)π‘‘πœ‰=βˆ’π‘›ξ“π‘˜=0πœ‘ξ€·π‘₯,πœ†+π‘˜ξ€Έπ‘Ž+π‘˜ξ€œπœ‹0πœ‘ξ€·πœ‰,πœ†+π‘˜ξ€ΈπœŒ(πœ‰)𝑓(πœ‰)π‘‘πœ‰βˆ’π‘›ξ“π‘˜=0πœ‘ξ€·π‘₯,πœ†βˆ’π‘˜ξ€Έπ‘Žβˆ’π‘˜ξ€œπœ‹0πœ‘ξ€·πœ‰,πœ†βˆ’π‘˜ξ€ΈπœŒ(πœ‰)𝑓(πœ‰)π‘‘πœ‰.(3.10) Further βˆ’π‘“(π‘₯)ξ€Ÿ2πœ‹π‘–πΏπ‘›π‘‘πœ†πœ†=βˆ’π‘“(π‘₯).(3.11) By using (3.8), we have ||||1ξ€Ÿ2πœ‹π‘–πΏπ‘›π‘Ÿ||||≀𝑀(π‘₯,πœ†)π‘‘πœ†π‘œξ€Ÿ2πœ‹πΏπ‘›||||π‘‘πœ†||πœ†||3/2,(3.12) from which, by using the substitution πœ†=𝑠2, we have ||||1ξ€Ÿ2πœ‹π‘–πΏπ‘›π‘Ÿ||||≀𝑀(π‘₯,πœ†)π‘‘πœ†π‘œπœ‹ξ€œΞ“+𝑛||||𝑑𝑠𝑠2≀constant𝑛.(3.13) By substitution from (3.10), (3.11), and (3.13) into (3.9), we obtain |||||𝑓(π‘₯)βˆ’π‘›ξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜+π‘βˆ’π‘˜π‘£βˆ’π‘˜ξ€Έ|||||≀constant𝑛[],π‘₯∈0,πœ‹βˆ€π‘›,(3.14) where π‘Β±π‘˜=ξ€œπœ‹0πœ‘ξ€·πœ‰,πœ†Β±π‘˜ξ€Έξ”π‘ŽΒ±π‘˜π‘“(πœ‰)𝜌(πœ‰)π‘‘πœ‰,π‘£Β±π‘˜=πœ‘ξ€·πœ‰,πœ†Β±π‘˜ξ€Έξ”π‘ŽΒ±π‘˜,(3.15) which completes the uniform convergence of the series βˆ‘βˆžπ‘˜=0[𝑏+π‘˜π‘£+π‘˜+π‘βˆ’π‘˜π‘£βˆ’π‘˜] to 𝑓(π‘₯),π‘₯∈[0,πœ‹]. That is, 𝑓(π‘₯)=βˆžξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜(ξ€»π‘₯).(3.16) It can be proved that the series (3.16) is not only uniformly but also absolutely convergent, to show this we use the asymptotic relations of π‘£Β±π‘˜ and π‘Β±π‘˜ for π‘›β†’βˆž. Following [4], we have π‘Ž+𝑛=𝑑1𝑛2ξ‚€1+○𝑛3,π‘Žβˆ’π‘›=βˆ’π‘‘22𝑒2𝑛𝑑2π‘’βˆ’π‘‘21𝑛2ξ‚€1+○𝑛3,(3.17) where 𝑑1=π‘Ž3/2πœ‹2 and 𝑑2=(πœ‹βˆ’π‘Ž)/πœ‹, from which we can write ξ”π‘ŽΒ±π‘›ξ‚€1=○𝑛.(3.18) Using (3.18), (2.11) and (3.2), we deduce that ||π‘£Β±π‘˜||≀𝑀±[],βˆ€π‘₯∈0,πœ‹,andallπ‘˜,(3.19) where 𝑀± are some constants. Further, arguing as in Lemma 2.4 and noticing that 𝑓(0)=𝑓(πœ‹)=0, we have π‘Β±π‘˜=ξ€œπœ‹0π‘£Β±π‘˜1(π‘₯)𝑓(π‘₯)𝜌(π‘₯)𝑑π‘₯=πœ†Β±π‘˜ξ€œπœ‹0ξ€Ίπ‘“ξ…žξ…žξ€»π‘£(π‘₯)+π‘ž(π‘₯)𝑓(π‘₯)Β±π‘˜(π‘₯)𝑑π‘₯.(3.20) From [4], we have πœ†Β±π‘˜=Β±π‘˜2+β—‹(1), and using (3.20) we have ||π‘Β±π‘˜π‘£Β±π‘˜(||≀π‘₯)costantπ‘˜2,π‘˜βŸΆβˆž,(3.21) which complete the proof of absolute convergence of the series (3.16). It should be noted here that, in the proof of the absolute convergence of the series (3.16) we did not give the sum of the series as in the proof of uniform convergence (Theorem 3.1). In the following lemma, as a consequence of Theorem 3.1, we prove the Parsval’s identity which insures the convergence of the series (3.16) and helps in the proof of Theorem 3.3.

Lemma 3.2. Let 𝑓(π‘₯) satisfy the conditions of Theorem 3.1; then the following Parsval’s identity holds true ξ€œπœ‹0||||𝜌(π‘₯)𝑓(π‘₯)2𝑑π‘₯=βˆžξ“π‘˜=0ξ‚€||𝑏+π‘˜||2+||π‘βˆ’π‘˜||2,(3.22) where π‘Β±π‘˜=ξ€œπœ‹0π‘£Β±π‘˜(π‘₯)𝑓(π‘₯)𝜌(π‘₯)𝑑π‘₯.(3.23)

Proof. From Theorem 3.1, we have 𝑓(π‘₯)=βˆžξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜(ξ€Έπ‘₯),(3.24) where π‘Β±π‘˜ are given by (3.22). Multiplying both sides of (3.24) by 𝑓(π‘₯)𝜌(π‘₯) and integrating with respect to π‘₯∈[0,πœ‹], we have ξ€œπœ‹0||||𝜌(π‘₯)𝑓(π‘₯)2ξ€œπ‘‘π‘₯=πœ‹0βˆžξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜(ξ€Έπ‘₯)𝑓(π‘₯)𝜌(π‘₯)𝑑π‘₯.(3.25) By the aid of uniform convergence of the series (3.16), the integration and summation can be interchanged and we have ξ€œπœ‹0||||𝜌(π‘₯)𝑓(π‘₯)2𝑑π‘₯=βˆžξ“π‘˜=0𝑏+π‘˜ξ€œπœ‹0𝑣+π‘˜(π‘₯)𝑓(π‘₯)𝜌(π‘₯)𝑑π‘₯+βˆžξ“π‘˜=0π‘βˆ’π‘˜ξ€œπœ‹0π‘£βˆ’π‘˜(π‘₯)𝑓(π‘₯)𝜌(π‘₯)𝑑π‘₯,(3.26) where 𝜌(π‘₯) and π‘£Β±π‘˜(π‘₯) are real (see introduction) which complete the proof of the lemma. In the following theorem, the validity of eigenfunction expansion and the Parsval’s identity can be extended to any function of 𝐿2(0,πœ‹;𝜌) but the convergence of the expansion will be in some weak sense, that is, in the metric sense of 𝐿2(0,πœ‹;𝜌).

Theorem 3.3. Suppose that 𝑓(π‘₯) is any function from 𝐿2(0,πœ‹;𝜌); then the following Parsval’s identity (3.22) and the eigenfunction expansion (3.24) are true and the convergence of the series (3.16) to 𝑓(π‘₯) is in the metric sense of the space 𝐿2(0,πœ‹;𝜌).

Proof. Let 𝑓(π‘₯) be any function that belongs to 𝐿2(0,πœ‹;𝜌). It is known that the set of infinitely differential functions which vanish at the neighbourhood of the points π‘₯=0,  π‘₯=πœ‹ are dense in 𝐿2(0,πœ‹;𝜌), so that there exists a sequence {𝑓𝑛(π‘₯)} of finite smooth functions (and consequently, satisfy the conditions of the theorem) which converges to 𝑓(π‘₯) in the metric of 𝐿2(0,πœ‹;𝜌); in equation notation this is can be written as ‖‖𝑓𝑛‖‖(π‘₯)βˆ’π‘“(π‘₯)𝐿2=ξ‚΅ξ€œπœ‹0||π‘“πœŒ(π‘₯)𝑛||(π‘₯)βˆ’π‘“(π‘₯)2𝑑π‘₯1/2⟢0,asπ‘›βŸΆβˆž.(3.27) By the last lemma, every function 𝑓𝑛(π‘₯) satisfies the parseval’s identity ξ€œπœ‹0||π‘“πœŒ(π‘₯)𝑛(||π‘₯)2𝑑π‘₯=βˆžξ“π‘˜=0ξ‚€||π‘π‘˜(𝑛)+||2+||π‘π‘˜(𝑛)βˆ’||2,(3.28) where π‘π‘˜(𝑛)Β±=βˆ«πœ‹0𝜌(π‘₯)𝑓𝑛(π‘₯)π‘£Β±π‘˜(π‘₯)𝑑π‘₯,β€‰β€‰π‘˜=0,1,2,…
The identity (3.28) can be written as ‖‖𝑓𝑛(β€–β€–π‘₯)2𝐿2=β€–β€–π‘π‘˜(𝑛)+β€–β€–2𝑙2+β€–β€–π‘π‘˜(𝑛)βˆ’β€–β€–2𝑙2.(3.29) Consider the difference ‖‖𝑓𝑛(π‘₯)βˆ’π‘“π‘š(β€–β€–π‘₯)2𝐿2=β€–β€–π‘π‘˜(𝑛)+βˆ’π‘π‘˜(π‘š)+β€–β€–2𝑙2+β€–β€–π‘π‘˜(𝑛)βˆ’βˆ’π‘π‘˜(π‘š)βˆ’β€–β€–2𝑙2.(3.30) By the aid of (3.27), it follows that {𝑓𝑛(π‘₯)} is a fundamental sequence and hence by the completeness of 𝑙2 the sequences {π‘π‘˜π‘›Β±} are fundamentals, so, that there exists a limiting 𝑏+π‘˜ and π‘βˆ’π‘˜ such that β€–π‘π‘˜(𝑛)+βˆ’π‘+π‘˜β€–2𝑙2β†’0 and β€–π‘π‘˜(𝑛)βˆ’βˆ’π‘βˆ’π‘˜β€–2𝑙2β†’0; by using the continuity of the norm and passing to the limit as π‘›β†’βˆž in (3.29), we obtain ‖𝑓(π‘₯)β€–2𝐿2=‖‖𝑏+π‘˜β€–β€–2𝑙2+β€–β€–π‘βˆ’π‘˜β€–β€–2𝑙2,(3.31) which is the Parsval’s identity. Now we prove the eigenfunction expansion formula by the help of Parsval’s identity. For any 𝑛, we have ξ€œπœ‹0|||||𝜌(π‘₯)𝑓(π‘₯)βˆ’π‘›ξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜ξ€Έ|||||(π‘₯)2=ξ€œπ‘‘π‘₯πœ‹0𝜌(π‘₯)𝑓(π‘₯)βˆ’π‘›ξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜ξ€Έξƒ­Γ—ξƒ¬(π‘₯)𝑓(π‘₯)βˆ’π‘›ξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜ξ‚(π‘₯)𝑑π‘₯;(3.32) after calculation, we have ξ€œπœ‹0|||||𝜌(π‘₯)𝑓(π‘₯)βˆ’π‘›ξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜ξ€Έ|||||ξ€œ(π‘₯)𝑑π‘₯=πœ‹0||||𝜌(π‘₯)𝑓(π‘₯)2𝑑π‘₯βˆ’π‘›ξ“π‘˜=0ξ‚€||𝑏+π‘˜||2+||π‘βˆ’π‘˜||2(3.33) from which, and by using Parseval identity (3.22), we have limπ‘›β†’βˆžξ€œπœ‹0|||||𝜌(π‘₯)𝑓(π‘₯)βˆ’π‘›ξ“π‘˜=0𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜ξ€Έ|||||(π‘₯)2𝑑π‘₯⟢0.(3.34) So that, βˆ‘βˆžπ‘˜=0(𝑏+π‘˜π‘£+π‘˜(π‘₯)+π‘βˆ’π‘˜π‘£βˆ’π‘˜(π‘₯))→𝑓(π‘₯) in the metric of 𝐿2(0,πœ‹;𝜌), which completes the proof.


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