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Inverse spectral problems for \(2m\)-dimensional canonical Dirac operators. (English) Zbl 1153.34006
The authors generalize the Ambarzumyan theorem to Dirac operators defined by \[ L(Y):=\left( B\frac{d}{dx}+V\right) Y=\lambda Y, \] where
\[ Y=\left( Y_{1},\;Y_{2}\right) ^{t},\quad V=\left( \begin{matrix} P & Q \\ Q & -P \end{matrix} \right)\text{ and }B=\left(\begin{matrix} 0 & I_{m} \\ -I_{m} & 0 \end{matrix} \right). \] The boundary conditions are either \(Y_{1}(0)=Y_{1}(\pi )=0,\) or \(Y_{1}(\pi )=\beta Y_{1}(0),\;\)and \(Y_{2}(0)=\beta Y_{2}(\pi )\), where \(\beta =0,\pm 1.\) They prove that if \(Q(0)=Q(\pi )\) and \(\lambda _{n}=n\in \mathbb{Z}\) with multiplicity \(m\), then \(V(x)=0\). A similar result holds for \( \beta =0\) and \(Q(0)=-Q(\pi )\) and also for \(\beta =\pm 1\).

MSC:
34A55 Inverse problems involving ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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