On a theorem of Ambarzumian

Abstract

We consider the eigenvalue problem for the one-dimensional (stationary) Dirac operator with some boundary conditions. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, then the potential is actually zero. The analogous statement for the Schr odinger operator is due to Ambarzumian. The proof is based on the fact that the (generalized) moments of a function cannot have alternating signs unless the moments are zero (see x 2).

Inverse spectral problem for Dirac operators by spectral data

“…Analogously in [8], it is shown that the function a(x) ∈ W 1 2 [0, 2π]. It is easily found by using (16) and (17) that…”

“…Let the real numbers {λ n , α n }, (n ∈ Z) of the form (9) and (11) be given. Using these numbers, we construct the functions F 0 (x, t) and F(x, t) by the formulas (16) and (17) and determine A(x, t) from the main equation (15). Let us construct the function ϕ(x, λ) by the formula (4), the function Ω(x) by the formula (5), ∆(λ) by the formula (14) and β n by the formula (8) respectively, i.e.,…”

“…Solution of the inverse quasiperiodic problem for Dirac system was given in [27]. For Dirac operator, Ambarzumian-type theorems were proved in [16,17,35]. On a positive half line, inverse problem for Dirac system was investigated by two spectra in [13], inverse scattering problem for a system of Dirac equations of order 2n was completely solved in [12] and when the boundary condition contained spectral parameter, for Dirac operator, inverse scattering problem was worked in [5,22].…”

Necessary and sufficient conditions for the solvability of inverse problem for a class of Dirac operators

“…Inverse problem for weighted Dirac equations was investigated in [28]. For Dirac operator Ambarzumian-type theorems were proved in [12,13,30]. On a positive half line, inverse scattering problem for a system of Dirac equations of order 2n was completely solved in [11] and when boundary condition contained spectral parameter, for Dirac operator, inverse scattering problem was worked in [2,17].…”

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

“…The development of the theory of the onedimensional Dirac equation and two-dimensional canonical system occured more slowly than that of the Sturm-Liouville equation [7]. For example Ambarzumyan-type theorems for Dirac operators appeared in [2,[9][10][11][12][13][14][15]. Despite the parallels between Sturm-Liouville equations and canonical systems, there are important differences:(i) The operators associated with canonical systems are not lower-semi-bounded, thus the simple variational arguments used in Sturm-Liouville theory cannot be applied directly.…”