The inverse problem for the impulsive differential pencil (Q6586710)

In this paper, the boundary value problem \(L:=L(q_{1}, q_{0}, h, H, \omega_{1}, \omega_{2})\) for the the impulsive differential pencil\N\[\N-y^{\prime \prime }+(2\varrho q_{1}(x)+q_{0}(x))y=\mu r(x)y,\ \ \ x\in(0,1),\N\tag{1}\N\]\Nwith the boundary conditions\N\[\NU(y):=y^{\prime }(0)-hy(0)=0, \ \ \ \\NV(y):=y^{\prime }(1)+Hy(1)=0,\tag{2}\N\]\Nwhere the real-valued functions \(q_{1}(x)\in W_{2}^{1}(0,1),\ q_{0}(x)\in L_{2}(0,1)\) has been considered. The parameters \(h, H\) and \(\mu=\varrho^{2} \) are real and spectral, respectively. Also \(r(x)=\omega_{1}^{2}\) for \(x<\frac{1}{2}\) and \(r(x)=\omega_{2}^{2}\) for \(x>\frac{1}{2}\), where \(\omega_{2}>\omega_{1}.\)\N\NThe authors investigate the inverse problem for \(L\) in the finite interval \((0,1)\). Taking Mochizuki-Trooshin's theorem, they prove that two potentials and the boundary conditions are uniquely given by one spectra together with a set of values of eigenfunctions in the point \(x=1/2\). Moreover, applying Gesztesy-Simon's theorem, the authors demonstrate that if the potentials are assumed on the interval \([1/2(1 - \theta), 1]\) where \(\theta \in (0, 1)\), a finite number of the spectrum is enough to give potentials on the interval \([0, 1]\) and other boundary condition.\N\NThe main results of the paper are Theorem 1 and Theorem 2 where the uniqueness of the inverse problem is proved by Mochizuki-Trooshin's and Gesztesy-Simon's methods.