Abstract
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.
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Acknowledgements
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.
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Appendix A: Proof of Lemma 2.3
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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Bai, Y., Wei, G. The Local Borg–Marchenko Uniqueness Theorem for Potentials Locally Smooth at the Right Endpoint. Integr. Equ. Oper. Theory 91, 31 (2019). https://doi.org/10.1007/s00020-019-2529-z
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DOI: https://doi.org/10.1007/s00020-019-2529-z