On Borg’s method for non-selfadjoint Sturm–Liouville operators

Abstract

We prove local solvability and stability for the inverse problem of recovering a complex-valued square integrable potential in the Sturm–Liouville equation on a finite interval from spectra of two boundary value problems with one common boundary condition. For this purpose we generalize classical Borg’s method to the case of multiple spectra.

Appendix

Here we prove Riesz basisness of the functional sequence ${\tilde{A}}(\cos 2\rho x)$ appeared in Sect. 3. For this purpose, we use the following ”multiple” version of Proposition 1.8.6 in [3].

Proposition 1

Let complex numbers $\{\rho _n\}_{n\ge 0}$ of the form

$$\begin{aligned} \rho _n=n+\varkappa _n, \quad \{\varkappa _n\}\in l_2, \end{aligned}$$

(50)

be given. Denote by $m_n$ the multiplicity of $\rho _n$ and, without loss of generality, assume that

$$\begin{aligned} \rho _n=\rho _{n+1}=\ldots =\rho _{n+m_n-1}, \quad n\in \mathcal{S}:=\{n:\rho _n\ne \rho _{n-1},n\in {{\mathbb {N}}}\}\cup \{0\}. \end{aligned}$$

Then $\{c_n(x)\}_{n\ge 0}$ is a Riesz basis in $L_2(0,\pi ),$ where

$$\begin{aligned} c_{k+\nu }(x)=\frac{\partial ^\nu }{\partial \lambda ^\nu }\cos \rho x\Big |_{\rho =\rho _k}, \quad k\in \mathcal{S}, \quad \nu =\overline{0,m_k-1}, \quad \lambda =\rho ^2. \end{aligned}$$

Proof

Since, by virtue of (50), $m_n=1$ for sufficiently large n,  it is easy to show that the sequence $\{c_n(x)\}_{n\ge 0}$ is quadratically close to the orthogonal basis $\{\cos nx\}_{n\ge 0}.$ Thus, it remains to prove that $\{c_n(x)\}_{n\ge 0}$ is complete in $L_2(0,\pi ).$ In other words, one should show that if the numbers $\lambda _n:=\rho _n^2,\,n\ge 0,$ are zeros of the function

$$\begin{aligned} F(\lambda )=\int _0^{\pi } f(x)\cos \rho x\,dx, \quad f(x)\in L_2(0,\pi ), \end{aligned}$$

with account of multiplicity, then $F(\lambda )\equiv 0.$ Consider the function

$$\begin{aligned} \Delta (\lambda ):=\pi (\lambda -\lambda _0)\prod _{n=1}^{\infty } \frac{\lambda _n-\lambda }{n^2}. \end{aligned}$$

It follows from Lemma 3.3 in [21] that $\Delta (\lambda )=\rho \sin \rho \pi +o(\rho \exp (|Im\rho |\pi )),\,\lambda \rightarrow \infty ,$ and, hence, for any $\delta >0$ there exist $C_\delta >0$ and $\lambda _\delta >0$ such that $|\Delta (\lambda )|\ge C_\delta |\rho |\exp (|Im\rho |\pi )$ for $\lambda \in G_\delta :=\{\lambda =\rho ^2: |\rho -n|\ge \delta ,\,n\in {{\mathbb {Z}}}\}$ and $|\lambda |\ge \lambda _\delta .$ Thus, we arrive at the estimate

$$\begin{aligned} \frac{F(\lambda )}{\Delta (\lambda )}=o\Big (\frac{1}{\rho }\Big ), \quad \lambda \rightarrow \infty , \quad \lambda \in G_\delta , \end{aligned}$$

which along with the maximum modulus principle and Liouville’s theorem yield $F(\lambda )\equiv 0.$$\square$

Let us return to proving Riesz basisness of the sequence ${\tilde{A}}(\cos 2\rho x).$ Put ${\tilde{\rho }}_{2k-j}=2{\tilde{\rho }}_{kj},$$k\ge 1,$$j=0,1,$ and ${\tilde{\rho }}_0=0.$ Obviously, the sequence $\{{\tilde{\rho }}_n\}_{n\ge 0}$ has the form (50). Rearranging a finite number of its elements, we get the sequence $\{\rho _n\}_{n\ge 0},$ in which multiple elements are neighboring. Construct the functional sequence $\{c_n(x)\}_{n\ge 0}$ as in Proposition 1. It remains to note that, according to (9), (10) and (15), the sequence ${\tilde{A}}(\cos 2\rho x)$ differs from $\{c_n(x)\}_{n\ge 0}$ only by order of a finite number of elements.