Research Article | Open Access
The Eigenfunction Expansion for a Dirichlet Problem with Explosive Factor
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 .
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 . He gave a new approach in the spectral theory of singular differential operator of the second order by using contour integration. Also Levitan  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 . 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 (1.4) as ± signs causes many analytical difficulties, see , because the problem is to be treated as two separated problems and so the formula of eigenfunctions expansion is written as two sums. In  the author considered the weight function of the form 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: where the nonnegative real function has a second piecewise integrable derivatives on , is a spectral parameter, and the weight function or the explosive factor is of the form In  the author proved that the eigenvalues , of problem (1.2)-(1.3) are real and the corresponding eigenfunctions are orthogonal with weight function . We prove, here, the reality of these eigenfunctions under the condition that is real. Indeed, let be the solution of the differential equation (1.2), which satisfies the conditions so that taking the complex conjugate we have By the aid of the uniqueness theorem, we have . In a similar way, we can see that where is the solution of (1.2), , , , that is, the eigenfunctions of the problem (1.2)-(1.3) are real. As we know from , the eigenvalues of problem (1.2)-(1.3) coincide with the roots of the function , where is the Wronskian of the two solutions , of (1.2)-(1.3) In the following lemma, under the reality of , we prove the simplicity of the eigenvalues, that is, we prove that the roots of (1.8) are simple, in other cases for is complex the roots of (1.8) may not be simple.
Proof. We prove that where the dot means differentiation with respect to . Let be the solution of the problem and let be the solution of the problem From (1.9), we have Differentiating (1.11) with respect to , we have Multiplying (1.11) by and (1.12) by and then subtracting the two results we have from which, by integrating with respect to , from 0 to and using the conditions in (1.10), we obtain In a similar way, from (1.10), we can write From (1.8), we have From which we have Substituting from (1.16) into (1.13) and (1.14), we can see by adding that where and the numbers are the normalization numbers of the eigenfunctions of problem (1.2)-(1.3). Following , we have and which complete the proof of lemma.
2. The Function
We introduce the function by which is called the Green’s function of the nonhomogenous Dirichlet problem where and is defined by (1.4). The function is, also, called the kernel of the resolvent , where . In the following lemmas, we prove some essential properties of which are useful in the forthcoming study of the eigenfunction expansion of the problem (1.2)-(1.3)
Lemma 2.1. Let be any function belonging to , then the function 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 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 where are given together with their asymptotic formulas in . By using the standard calculation, we find 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 satisfies the following formula: where is regular in the neighborhood of and .
Proof. So long as, from Lemma 1.1, the roots of the function are simple; it follows that the poles of the function are simple. So that can be represented in the form from (2.1); for , we have From (1.17) and (1.18), the relation (2.9) takes the form 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 , 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 and this, in turn, needs to extend the asymptotic formulas of over all the interval . In , the asymptotic formulas for , were deduced for and , respectively. In the following lemma, we write this asymptotic formulas over all for both and .
Proof. Following , the solutions and of the Dirichlet problem (1.2)-(1.3) have the representation We can see, from , that the solution of the equation , has the representation and the solution of the equation has the representation where and are given by (2.13), so that, can be extended to in terms of the two linearly independent solutions , as From the continuity of at and by using the asymptotic relations, (2.14), of and (2.17) of , the constants , are calculated in the form Substituting from (2.19) into (2.18), we have, for 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 is a linear combination of the two linearly independent solutions , as , where and by using the asymptotic formulas (2.16) of and (2.15) of , we have, for , 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 .
Lemma 2.4. Under the conditions of Lemma 2.3, the resolvent satisfies the following inequality:
Proof. From (2.11) and (2.12), we have
It can be easily seen that, for , we have
where is the quadratic contour, as defined in 
From (2.1) we have six possibilities, three of which for and the other three for . Now for we have the following situation: (i) , (ii) , and (iii) . 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
In the case of , we discuss (i*) , (ii*) , and (iii*) .
Again by substituting (2.24), (2.25), and (2.26) into the second branch of (2.1), we get From (2.28) and (2.31), we have and from (2.29) and (2.32), we have 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 and help in proving the eigenfunction expansion formula
Lemma 2.5. If the function on has a second-order integrable derivatives and satisfies the Dirichlet condition , then the following integral formula is true: where is the kernel of the resolvent of the nonhomogenous Dirichlet problem (2.2).
Proof. By the aid of (2.1), we have where the functions and are the solutions of the homogenous Dirichlet problem (1.2)-(1.3), so that from which we have Integrating by parts twice the terms and , in (2.39), and then using the boundary conditions and , respectively, and keeping in mind (2.1), we deduce that 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 and be the nonnegative and the negative eigenvalues of the problem (1.2)-(1.3), and let also be the normalization numbers of the corresponding eigenfunctions . We put The set 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 and satisfy the conditions ; then the following formula of eigenfunction expansion is true: where and the series uniformly converges to .
Notice that, the expansion (3.3) can be written, more explicitly, in terms of as or in terms of where are defined by , .
Proof. We write (2.36) in the form where By the aid of Lemma 2.3 and the condition of the theorem imposed on , it can be easily seen that where is constant which is independent of and the contour , as defined in , is given by (2.27). Let ; we denote by the upper half of the ; let also Ł denote the image of the contour under the transformation . We multiply both sides of (3.6) by and integrating with respect to on the contour : Among the poles of the function , as a function of , lie only inside . By using the residues formula and (2.10), we have Further By using (3.8), we have from which, by using the substitution , we have By substitution from (3.10), (3.11), and (3.13) into (3.9), we obtain where which completes the uniform convergence of the series to . That is, 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 and for . Following , we have where and , from which we can write Using (3.18), (2.11) and (3.2), we deduce that where are some constants. Further, arguing as in Lemma 2.4 and noticing that , we have From , we have , and using (3.20) we have 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 satisfy the conditions of Theorem 3.1; then the following Parsval’s identity holds true where
Proof. From Theorem 3.1, we have where are given by (3.22). Multiplying both sides of (3.24) by and integrating with respect to , we have By the aid of uniform convergence of the series (3.16), the integration and summation can be interchanged and we have where and 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 but the convergence of the expansion will be in some weak sense, that is, in the metric sense of .
Theorem 3.3. Suppose that is any function from ; 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 is in the metric sense of the space .
Proof. Let be any function that belongs to . It is known that the set of infinitely differential functions which vanish at the neighbourhood of the points , are dense in , so that there exists a sequence of finite smooth functions (and consequently, satisfy the conditions of the theorem) which converges to in the metric of ; in equation notation this is can be written as
By the last lemma, every function satisfies the parseval’s identity
The identity (3.28) can be written as Consider the difference By the aid of (3.27), it follows that is a fundamental sequence and hence by the completeness of the sequences are fundamentals, so, that there exists a limiting and such that and ; by using the continuity of the norm and passing to the limit as in (3.29), we obtain which is the Parsval’s identity. Now we prove the eigenfunction expansion formula by the help of Parsval’s identity. For any , we have after calculation, we have from which, and by using Parseval identity (3.22), we have So that, in the metric of , which completes the proof.
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Copyright © 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.