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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.)
##### Keywords:
Dirac operators; inverse problem
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