In this section, we try to obtain some asymptotic results and a reconstruction formula for the potential q, which has been obtained as a solution of an inverse nodal problem.
Lemma 2.1 The solution of problem (1.1)-(1.3) has the following form:
(2.1)
where .
Proof Because satisfies equation (1.1), we get
By integrating the first term twice on the right-hand side by parts and taking the conditions into account (1.2), we find that
where . □
Lemma 2.2 The eigenvalues of problem (1.1)-1.3) are the roots of (1.3). This spectral characteristic satisfies the following asymptotic expression [23]:
(2.2)
where
Lemma 2.3 Assume that . Then, as ,
Proof By using some iterations and trigonometric calculations in (2.1), we obtain
If is equal to zero and is not close to zero, then
Now, we take and . Because Taylor’s expansion for the arctangent function is given by
for some integer i, then
Therefore
The nodal length is
This completes the proof of Lemma 2.3. □
Lemma 2.4 Suppose . Then, for almost every with ,
Proof Since , almost everywhere. Thus, given any , when n is sufficiently large and for almost every ,
This proves Lemma 2.4. □
Theorem 2.1
The potential function
satisfies
for almost every with . We note that the asymptotic expression for in Theorem 2.1 implies that .
Proof When we consider (2.4) in the form
so that
By Lemma 2.4
for almost every .
It remains to show that for almost every ,
tends to zero as . Take a sequence of continuous functions which converges to q in . Then has a subsequence converging to q almost everywhere in . We call this subsequence . Take any x such that converges to . Then for a given , we can fix a large k such that . Hence
By Lemma 2.3,
and so it tends to zero as . By Lemma 2.4, the first term satisfies, when n is sufficiently large,
On the other hand,
Because is continuous, this term is arbitrarily every . Hence we conclude that . This proves Theorem 2.1. □
Lemma 2.5 We take a sequence converges to , then, for any large enough n, with as
Proof By (2.4) and observation that the integral is constant on any nodal interval , we obtain
and for this term converges to zero. □
Lemma 2.6 Suppose that , then as with ,
Proof Firstly, let us show that if q is continuous on , the result is satisfied. Let . By using the intermediate value theorem, there exists such that
If x is close enough to a, the difference can be arbitrarily small. Then, for all , when n is large enough, with we get
In the above process, we assume that . The estimate also holds if . Hence if , converges to uniformly on . Thus can be arbitrarily small. Because is dense in , for any , there exists a sequence convergent to q in . Hence, fix n sufficiently large,
From the above process and Lemma 2.5, when k is large enough, the first two terms are arbitrarily small. Hence, as ,
□
Theorem 2.2 converges to q in .
Proof When we consider the value of , we obtain that
It suffices to show that as
By using (2.4) we have
Hence, we only need to prove that for
and
(2.5)
From Lemma 2.6, the first limit holds and the second limit also holds. On the other hand, the sequence of functions
converges to 0 for almost every . Furthermore,
and
Then, we may apply the Lebesque dominated convergence theorem to show that (2.5) is valid. The proof of Theorem 2.2 is completed. □