Borg-Levinson theorem for perturbations of the bi-harmonic operator (Q2806025)

In this paper, the author studies the operator \(H_{4}\) defined as follows: NEWLINE\[NEWLINEH_4 \equiv H_4(x,\delta) := \Delta^2 - \nabla\cdot (F(x)\nabla) -2i \vec{G}(x)\cdot \nabla -i\nabla\cdot \vec{G}(x)+V(x),NEWLINE\]NEWLINE where \(\nabla\) denotes the gradient in \(\mathbb R^n\), \(\Delta\) denotes the Laplacian in \(\mathbb R^n\) and the coefficients \(F(x)\), \(\vec{G}(x)\) and \(V(x)\) are assumed to be real-valued. He also assumes that NEWLINE\[NEWLINEF(x) \in W_p^2(\Omega),\quad \vec{G}(x) \in (W_p^1(\Omega))^n,\quad V(x)\in L^p(\Omega),\quad p>\frac{n}{2},\quad n\geq 3,NEWLINE\]NEWLINE where it is supposed without loss of generality that the value of \(p\) is the same for all these spaces. This operator is the second order perturbation of the biharmonic operator. The author shows that the coefficients of this operator, in the smooth bounded domain \(\Omega\subset \mathbb R^{n}\), \(n\geq3\), are determined by the knowledge of the discrete Dirichlet spectrum and of some special derivatives up to the third order of the normalized eigenfunctions at the boundary. Then, the author studies the classical inverse spectral problem. He does not assume knowledge of the Dirichlet-Neumann map but proves that the Dirichlet-to-Neumann map is determined uniquely by the Borg-Levinson data, that is, by the spectral data. To obtain the main results, the author uses the method of Carleman estimate and by thus constructs complex geometric optics (CGO) solutions for the equation \(H_{4} =0\) in \(\Omega\). The result proved in this paper states an analog of the Borg-Levinson theorem for the biharmonic operators with second order perturbation.