Uniqueness and stability results for an inverse spectral problem in a periodic waveguide

作者：Kavian Otared^*; Kian Yavar; Soccorsi Eric

来源：Journal de Mathematiques Pures et Appliquees, 2015, 104(6): 1160-1189.

DOI：10.1016/j.matpur.2015.09.002

• Inverse spectral problem
• Stability
• Uniqueness
• Periodic waveguide
• Schrodinger operator
• Periodic potential

Let Omega := omega x R where omega subset of R-2 is a bounded domain, and let V : Omega -> R be a bounded potential which is 2 pi-periodic in the variable x(3) is an element of R. We study the inverse problem consisting in the determination of V, through the boundary spectral data of the operator u bar right arrow Au := -del u + Vu, acting on L-2(omega x (0, 2 pi)), with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for j = 1,2 two potentials V-j are given so that parallel to V-j parallel to(infinity) <= R and if we denote by (lambda(j,k))(k) the eigenvalues of the operators A(j) (that is the operator A with V := V-j), then for a constant c > 0, depending on omega and R > 0, we have parallel to F((V-1 - V-2)1(omega x(0,2 pi)))parallel to infinity <= c lim supk ->infinity vertical bar lambda(1,k) - lambda(2,k)vertical bar, provided that Sigma(k >= 1)parallel to psi(1,k) - psi(1,k)parallel to(2)(L2)((partial derivative omega) (x [0,2 pi])) < infinity, where psi(j,k) := partial derivative phi(j,k)/partial derivative n (here F denotes the Fourier transform). The arguments developed here may be applied to other spectral inverse problems, and similar results can be obtained.

• 出版日期2015-12