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Register a new userStability for the multi-dimensional Borg--Levinson Theorem of the biharmonic operator
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We prove for the first time a conditional H{o}lder stability related to the multi-dimensional Borg--Levinson theorem, which is concerned with determining a potential from spectral data for the biharmonic operator. The proof depends on the theory of scattering resonances to obtain the resolvent estimate and a Weyl-type law for the biharmonic operator.
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This text deals with multidimensional Borg-Levinson inverse theory. Its main purpose is to establish that the Dirichlet eigenvalues and Neumann boundary data of the Dirichlet Laplacian acting in a bounded domain of dimension 2 or greater, uniquely determine the real-valued bounded potential. We first address the case of incomplete spectral data, where finitely many boundary spectral eigen-pairs remain unknown. Under suitable summability condition on the Neumann data, we also consider the case where only the asymptotic behavior of the eigenvalues is known. Finally, we use the multidimensional Borg-Levinson theory for solving parabolic inverse coefficient problems.
In this article, stability estimates are given for the determination of the zeroth-order bounded perturbations of the biharmonic operator when the boundary Neumann measurements are made on the whole boundary and on slightly more than half the boundary, respectively. For the case of measurements on the whole boundary, the stability estimates are of ln-type and for the case of measurements on slightly more than half of the boundary, we derive estimates that are of ln ln-type.
A result of Borg--Hochstadt in the theory of periodic Jacobi matrices states that such a matrix has constant diagonals as long as all gaps in its spectrum are closed (have zero length). We suggest a quantitative version of this result by proving the two-sided bounds between oscillations of the matrix entries along the diagonals and the length of the maximal gap in the spectrum.
We prove that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the perturbed biharmonic operator $Delta_g^2+q$ on a conformally transversally anisotropic Riemannian manifold of dimension $ge 3$ with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [51]. In particular, our result is applicable and new in the case of smooth bounded domains in the $3$-dimensional Euclidean space as well as in the case of $3$-dimensional admissible manifolds.
In this paper we prove stable determination of an inverse boundary value problem associated to a magnetic Schrodinger operator assuming that the magnetic and electric potentials are essentially bounded and the magnetic potentials admit a Holder-type modulus of continuity in the sense of $L^2$.
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