Abstract
We calculate the regularized trace formula of the infinite sequence of eigenvalues for some version of a Dirichlet boundary value problem with turning points.
1. Introduction
The study of regularized traces of ordinary differential operators has a long history and there are a large number of papers and books studying this issue. The trace formulae for the scalar differential operators have been found by Gelfand and Levitan [1]. The formula obtained there gave rise to a large and very important theory, which started from the investigation of specific operators and further embraced the analysis of regularized traces of discrete operators in general form. In a short time, a number of authors turned their attention to trace theory and obtained interesting results. Dikiĭ [2] demonstrated a technique of using the trace of a resolvent for finding traces. Dikiĭ provided a proof of the Gelfand-Levitan formula in [2] on the basis of direct methods of perturbation theory, and in [3], he derived trace formulas of all orders for the Sturm-Liouville operator by constructing the fractional powers of the operator in closed form and by computing an analytic extension for its zeta function. Later, Levitan [4] suggested one more method for computing the traces of the Sturm-Liouville operator: by matching the expressions for the characteristic determinant via the solution of an appropriate Cauchy problem and via the corresponding infinite product, he found and compared the coefficients of the asymptotic expansions of these expressions thus obtaining trace formulas. The investigation carried out in 1957 by Faddeev [5] linked the trace theory to a substantially new class of problems, singular differential operators. Gasymov's paper [6] was the first paper in which a singular differential operator with discrete spectrum was considered. Afterwards these investigations were continued in many directions, such as Dirac operators, differential operators with abstract operator-valued coefficients, and the case of matrix-valued Sturm-Liouville operators (see, [7]).
Beyond their aesthetic appeal, trace formulas play an important role in the inverse spectral theory [8, 9]. Equation (1.1) is called differential equation with turning points if the weight function , which is given by (1.3), changes sign. The turning points appear in elasticity, optics, geophysics, and other branches of natural sciences. The inverse problems for equations with turning points and singularities help to study blow-up solutions for some nonlinear integrable evolution equations of mathematical physics. The turning points cause analytical difficulties, not only in calculating the eigenvalues asymptotes, but also in calculating the trace formula. In [10–12], the authors studied the spectral analysis of problem (1.1)-(1.2). They investigated the asymptotic relations of both eigenvalues and eigenfunctions also studied the eigenfunction expansion formula and proved the equiconvergence formula of that eigenfunctions. To complement the picture of spectral analysis of problem (1.1)-(1.2), we crown the series of papers [10–12] with the present work. In the present work, we evaluate the regularized trace formula for the problem (1.1)-(1.2) by using contour integration method as in [13]. It should be noted here that in [13] the author studied such formula for continuous spectrum in the whole line, while the present work contains point spectrum in a finite interval.
Consider the following Dirichlet problem where is a nonnegative real function, which has a second piecewise integrable derivatives of the second order in , is a spectral parameter and the weight function or the explosive factor is of the form
Following [10], we state the basic notations and results that are needed in the subsequent calculation. In [10], the author proved that the Dirichlet problem (1.1)-(1.2) has a countable number of eigenvalues where are the nonnegative eigenvalues and are the negative eigenvalues which admit the asymptotic formulas where and the constants are given by
We introduced in [10] the function as the Wronskian of the two solutions and of (1.1)-(1.2). We denote by the function , for , which has the following asymptotic formula where where , and are expressed, in (1.6), in terms of and . We also prove that the roots of coincide with the eigenvalues of the Dirichlet problem (1.1)-(1.2) and these eigenvalues are simple. Our aim is to calculate the summation of these eigenvalues which we call the trace formula or more precisely the regularized trace formula. During the calculation of the eigenvalues and the eigenfunctions, the condition (1.2) forced us to evaluate up to the term containing . We must also notice that, the formula obtained in the present work, due to the Dirichlet condition (1.2), contained together with its first derivatives on . We used the methodology as in [4, 14], by Levitan, but our problem contains more difficulties because of the presence of the as sign. In the following theorem we calculate the summation of these eigenvalues in a certain form called the regularized trace formula for the Dirichlet problem (1.1)-(1.2).
Theorem 1.1. Suppose that has a second-order piecewise integrable derivatives on then, in view of the introduced notations, (1.4) and (1.7), the following regularized trace formula takes place where the constants , and are given by
Proof. We use the well-known formula from the theory of functions of a complex variable
where is given by (1.7) and the contour is a quadratic contour on the s-domain as defined in [10]
From (1.7) we have
where, , and are given by (1.8). It is clear that, on the contour , the term satisfies the following inequality from below
which can be shown by expressing , and in terms of the exponential functions. So that
From (1.14) the Equation (1.12) can be put in the form
further,
where are the eigenvalues of (1.1)-(1.2) for . Integrating by part the last term of (1.17) we obtain
Substituting from (1.16) into (1.19) we have
From (1.18) and (1.20), (1.17) takes the form
We evaluate each term of (1.21), first of all, by direct calculation it can be easily shown that
To evaluate the integration , notice that the function is odd, so that from (1.8) we have
From which (1.23) can be written in the form
We notice that in the integration the function is bounded while
so that becomes of the form
By the help of the relations
(1.26) becomes
To evaluate we have is bounded and
and hence becomes of the form
As before, by applying the relations
to (1.30), we have
From (1.28) and (1.32) by substitution into (1.24) we have
To evaluate we notice that is odd function, then
From (1.34) we have
By using (1.25) and keeping in mind that is bounded we have
By the help of (1.27), after some elementary calculation, takes the form
Now we evaluate , by using (1.29) and the roundedness of we have from (1.35)
From (1.38) and (1.31) after calculation we have
By substitution from (1.39) and (1.36) into (1.35), we have
We evaluate the third integral of (1.21)
After simplification (1.41) becomes
For we have
by calculating the integrations in (1.43) we have
On the Other hand,
from which after calculation we have
Substituting from (1.44) and (1.46) into (1.42), we have
Substituting from (1.47), (1.40), (1.33), and (1.22) into (1.21) we have
Passing to the limit as and by using (1.9) and (1.6) we reach to the required formula (1.10).
2. Conclusion and Comment
It should be noted here that, we cannot expect that the trace formula must become the Gelfand-Levitan trace formula as the point approaches the point “”, because we do not put the condition at the point .
Due to the presence of the turning point (1.3), as it is well known for all unbounded operator, the convergent series is actually the sum of two infinite siriases, each of which cannot be summed separately.
Moreover the presence of the turning point helps in the solution of inverse problem for different densities medium [15]. The present work belongs to the school of Gasymov [6].
Conflict of Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors are very grateful to the referees for their fruitful comments and advice, also thanks for Professor Gussein G. Sh. for his advice.