The Local Borg–Marchenko Uniqueness Theorem for Potentials Locally Smooth at the Right Endpoint

In this note we are concerned with the local Borg–Marchenko uniqueness theorem for potentials locally smooth at the right endpoint. We establish the so-called high-energy asymptotics of the difference \((m_{1}-m_{2})(z)\) of two Weyl–Titchmarsh functions \(m_{j}(z)\) corresponding to two Schrödinger operators \(H_j=-d^2/dx^2+q_j\), for \(j=1,2\) and \(q_1=q_2\) a.e. on [0, a], in \(L_{2}(0,b)\) with \(0<a<b\le \infty \), where the potentials \( q_j\) are sufficiently smooth in a right neighbourhood of the point a and their right derivatives at a coincide up to a certain order. Moreover, we also provide a new proof of the local Borg–Marchenko theorem.

The authors would like to express their great appreciation to Professor Christiane Tretter and the anonymous referees for careful reading of the manuscript and giving us very helpful suggestions for the improvement of the original manuscript. The research was supported in part by the NNSF (11571212) of China.

Authors and Affiliations

1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

   Yu Bai & Guangsheng Wei

Corresponding author

Correspondence to Guangsheng Wei.

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Appendix A: Proof of Lemma 2.3

In this appendix we give a new proof of Lemma 2.3. In the following we always let \(z=\lambda ^{2}\) with \(\mathrm {Im}\lambda \ge 0\) for symbolic simplification. Note that \(|z|\rightarrow \infty \) in the sector \(\Lambda (\delta )\) is equivalent to \(|\lambda |\rightarrow \infty \) in the sector \( \Lambda (\delta /2)\).

Before proving Lemma 2.3, for \(q\in C^{2k}[a,c]\) \((0\le a<c<\infty ),\) we use [11, Lemma 1.4.1] to give a solution, denoted by \(\hat{y} (x,\lambda ),\) of (1.4) defined on [a, c],  which has the following form:

where \(v_{0}(x)=1,\) \(v_{i}(x)=\int _{a}^{x}H(v_{i-1})(t)dt\) for \( i=1,2,\ldots ,2k+1\) and

in which both \(K(x,\xi )\) and \(v_{2k+1}^{\prime }(x+a-\xi )\) are continuous on [a, c] in the variable \(\xi \) for every \(x\in [a,c]\). The proof of (A.1) is similar to that of [11, Lemma 1.4.1].

Lemma A.1

For \(q\in C^{2k}[a,c]\), let \(\hat{y}(x,\lambda )\) be the solution of (1.4) defined by (A.1). Then

as \(\left| \lambda \right| \rightarrow \infty \) in the sector \( \Lambda (\delta )\).

Proof

Note that, for any given \(c_{0},c_{1},c_{2}>0,\) the inequality \(e^{c_{0} \mathrm {Im}\lambda }\ge c_{1}|\lambda |^{c_{2}}\) remains true for sufficiently large \(|\lambda |\) with \(\lambda \in \Lambda (\delta )\). From (A.1) and (A.2) we have

as \(\left| \lambda \right| \rightarrow \infty \) in \(\Lambda (\delta )\). In order to prove (A.3) we only need to prove

as \(\left| \lambda \right| \rightarrow \infty \) in \(\Lambda (\delta )\).

Let \(\epsilon =1/\sqrt{\mathrm {Im}\lambda }\) and \(\lambda \in \Lambda (\delta )\). Consider the function

Taking \(\int _{c-\epsilon }^{c}2i\lambda e^{2i\lambda (c-\xi )}d\xi =-1+e^{2i\lambda \epsilon }\) into account, we get

By the fact that \(\mathrm {Im}\lambda \ge \left| \lambda \right| \sin \delta \) for all \(\lambda \in \Lambda (\delta )\), it follows that

which implies, for \(\lambda \rightarrow \infty \) in \(\Lambda (\delta ),\) \( g_{1}(\lambda )=o(1)\).

On the other hand, we have

where \(C_{0}=\max _{\xi \in [a,c-\epsilon ]}\left| v_{2k+1}^{\prime }(c+a-\xi )\right| \), which implies \(g_{2}(\lambda )=o(1)\) as \(\lambda \rightarrow \infty \) in \(\Lambda (\delta )\).

The above discussion shows that \(g(\lambda )=g_{1}(\lambda )+g_{2}(\lambda )=o(1)\) as \(\lambda \rightarrow \infty \) in \(\Lambda (\delta )\).

It follows form (A.1) and (A.2) that

as \(\left| \lambda \right| \rightarrow \infty \) in the sector \( \Lambda (\delta )\).

We are now in a position to prove Lemma 2.3.

Proof of Lemma 2.3

Let \(c=a+\varepsilon \) in Lemma 2.3. For \(m_{+}(a,z)\) defined by (2.2), by the same proof of Lemma 2.1 and (2.7), we have

where \(y_{N}(x,\lambda ),\) \(y_{D}(x,\lambda )\) are the solutions of (1.4) under the initial conditions \(y_{N}(a)=1,y_{N}^{\prime }(a)=0;\) \( y_{D}(a)=0,y_{D}^{\prime }(a)=1\) and \(m_{-,a}(c,z)=-y_{N}^{\prime }(c,\lambda )/y_{N}(c,\lambda )\).

By [11, p.58] we arrive at

where

(see [11, p.55-56]) with \(\sigma _{1}(a)=q(a)\), \(\sigma _{2}(a)=-q^{\prime }(a)\) and

From [11, p.56], we have \(v_{2k+1}^{\prime }(a)=\sigma _{2k+1}(a)\). From (A.3) and (A.7), we easily see that

which implies

and

as \(|\lambda |\rightarrow \infty \) in the sector \(\Lambda (\delta /2)\). From (A.9)–(A.10) and (A.12)–(A.13) we have

as \(\left| \lambda \right| \rightarrow \infty \) in \(\Lambda (\delta /2)\). Let \(C_{j}(a)=\sigma _{j}(a)/(2i)^{j}\). From (A.11) and (A.14) it is easy to see that (2.9) holds and

as \(\left| \lambda \right| \rightarrow \infty \) in \(\Lambda (\delta /2)\).

On the other hand, it is known [9] that the solutions \( y_{N}(x,\lambda )\) and \(y_{D}(x,\lambda )\) on [a, c] have the following asymptotic expansions:

and it is also known [1] that both \(m_{+}(c,z)\) and \(m_{-,a}(c,z)\) have the same asymptotics: \(i\lambda +o(1)\) as \(\left| \lambda \right| \rightarrow \infty \) in the sector \(\Lambda (\delta /2)\). Consequently, together with (A.16) and (A.17) we have

as \(\left| \lambda \right| \rightarrow \infty \) in the same sector. From (A.8), (A.15) and (A.18) we infer

as \(\left| \lambda \right| \rightarrow \infty \) in \(\Lambda (\delta /2),\) which yields (2.8) since \(z=\lambda ^{2}\).

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