Abstract
In this article, it is found that the asymptotic formulas for nodal points and nodal length for the differential operators having singularity type at the points 0 and π, it is shown that the potential function can be determined from the positions of the nodes for the eigenfunctions.
1. Introduction
Inverse problems of spectral analysis imply the reconstruction of a linear operator from some or other of its spectral characteristics. Such characteristics are spectra (for different boundary conditions), normalizing constants, spectral functions, scattering data etc. An early important result in this direction, which gave vital impetus for the further development of inverse problem theory, was obtained Citation1. At present, inverse problems are studied for certain special classes of ordinary differential operators. Inverse problems from spectra are the most simple in their formulation and well-studied in Citation18,Citation24. An effective method of constructing a regular and singular Sturm--Liouville operator from a spectral function or from two spectra are given Citation9,Citation13,Citation19,Citation22,Citation23.
We note that the detail of inverse problem for singular equations are given in the monographs and references therein Citation3,Citation5,Citation6,Citation8,Citation12,Citation17.
In some recent interesting works Citation2,Citation11, Browne and Sleeman, and Hald and Mclaughlin have taken a new approach to inverse spectral theory for the Sturm--Liouville problem. The novelty of this work lies in the use of nodal points as the given spectral data. In recent years, inverse nodal problems have been studied by several authors Citation2,Citation4,Citation7,Citation21,Citation25 etc.
In this article, we study inverse spectral theory for a singular Sturm--Liouville problem using a new kind of spectral data.
Consider the Sturm--Liouville problem:
(1)
(2)
(3)
where q is integrable and h and H are finite numbers.
Let λn be nth eigenvalue and be jth nodal point of the nth eigenfunction yn. It is well known that the problem (1.1–1.3) has only real and simple eigenvalues
. The corresponding eigenfunction yn has n − 1 nodal points in (0,1). So,
, j=1,2,…,n-1. Let
and
be the nodal length.
Before giving the main results of this article, we mention some properties of the classical Legendre equation. This equation is written as
we let
then y satisfies
where λ =n(n+1). For
and n sufficiently large, we conclude that Legendre functions are Citation14
(4)
We consider the problem
(5)
(6)
(7)
where h is finite number and ϵ positive number. We accept that the solution of the problem (1.5–1.7) is Citation16
(8)
In this case, the kernel K(x, t) satisfies the equation
and the boundary conditions
Let
be the eigenvalues of this problem (1.5–1.7) and
, j=1,2,…,n-1 be nodal points of the nth eigenfunction. It is shown that the set of all nodal points
is dense in (0,π) ; in fact, a judicious choice of one nodal point
for each yn, n>1 also forms a dense set in (0,π). The simplest method for choosing a dense subset of the nodes is to choose all the eigenfunctions. In addition, it will be found that the nodes for yn are roughly equally spaced. Hence, we can choose the first node in
and the second in
The third, fourth, fifth and sixth nodes should lie in
and
and so on. This method gives a dense set of nodes and it is not even necessary to choose the next node from the next eigenfunction and any finite amount of nodes can be deleted.
2. Main results
In this section, our purpose is to develop asymptotic expressions for the points and
at which yn, the eigenfunction corresponding to the eigenvalue λn of the problem (1.4–1.6) vanishes.
THEOREM 2.1
We consider the equation
(9)
with the boundary conditions
(10)
(11)
Then, the nodal points and nodal length of the problem (2.1–2.3) satisfy
(12)
(13)
Proof
Zeros of (2.3) for λ are eigenvalues of the problem (2.1–2.3). So, the asymptotic expression of eigenvalues is Citation15
(14)
for constant a, where
m
In fact, for positive λ the estimate (1.4) assumes the form
Furthermore, differentiating the relation (1.4) with respect to x and inserting the values of the ϕ(x,λ) and
into the boundary condition (1.7) after some straightforward computations, we get
The above asymptotic formula can be improved substantially. Indeed, differentiating (1.8) with respect to x and inserting in (1.7), we obtain after simple transformations
then we yield (2.6). We recall that the above method was used by Levitan and Sargsjan Citation20 for regular Sturm--Liouville problem.
The eigenfunctions of the problem (2.1–2.3) are in the asymptotic form Citation14
Hence, we use the classical estimate
where M is a constant. Thus ϕ(x,λ) will vanish in the intervals whose end points are solution to
This equation can also be written as
Using the Taylor expansion for arc cos(M1), then we get
or
Inserting (2.6) in the last equation, we get
The nodal length is
The proof is complete.
Now, we will prove a uniqueness theorem. It says that the potential function q(x) for a singular Sturm--Liouville problem is uniquely determined by a dense subset of the nodes. We mentioned that this theorem was given for regular Sturm--Liouville problems by McLaughlin Citation21, Hald and McLaughlin Citation11 and Browne and Sleeman Citation2.
THEOREM 2.2
Suppose that q is integrable. Then, h and are uniquely determined by any dense set of nodal points.
Proof
Assume that we have two problems of the type (2.1–2.3) with the h, and q,
. Let the nodal points
,
satisfying
form a dense set in
We take solutions of (2.1–2.3) wn for
and
for (
. It follows from (2.1) that
(15)
Recall that
are uniformly bounded in n and the
are uniformly bounded in n and
. We select a subsequence of nodes from the dense set. To show that
we integrate both sides of (2.7) from
to π and choose a subsequence that tends to π, and we see that
From this results we can say
Let
and integrate both sides of (2.7) from 0 to
From the asymptotic forms of
and λn, we have
We take a sequence
accumulating at an arbitrary
. Hence,
this holds for all x. We can therefore conclude that
is uniquely determined by a dense set of nodes. This completes the proof.
COROLLARY
For the problem (2.1–2.3), the potential q is uniquely determined by a dense set of nodes and the constant .
Proof
Suppose that . Since
it follows that
. Hence, we can conclude from Theorem 2.2 that
almost everywhere on
3. Conclusion
We mention that the above problem is solved by several authors for regular differential operator. In this work, a uniqueness theorem of potential function for singular differential operator is given. Morever, when h is finite number and q is integrable function, nodal points and nodal length can be estimated for this problem.
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