We present a new approach (distinct from Gel’fand-Levitan) to the theorem of Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schrödinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(−κ2) = −κ− ∫ b 0 A(α)e−2ακ dα...

We extend the well-known trace formula for Hill's equation to general one-dimensional Schrodinger operators. The new function <J , which we introduce, is used to study absolutely continuous spectrum and inverse problems. In this note we will consider one-dimensional Schrodinger operators d2 (IS) H = -j-1 + V(x) onL2(R;dx)

The results of this paper concern the “large spectra” of sets, by which we mean the set of points in Fp at which the Fourier transform of a characteristic function χA, A ⊆ Fp, can be large. We show that a recent result of Chang concerning the structure of the large spectrum is best possible. Chang’s result has already found a number of applications in combinatorial number theory. We also show t...

For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl’s function, the problem of reconstructing the matrix by two spectra of CMV operators with different “boundary conditions”, and the problem of reconstructing a CMV matrix by its spectrum and the spectrum of the CMV matrix obtained fro...

Let A be an unbounded from above self-adjoint operator in a separable Hilbert space H and EA(·) its spectral measure. We discuss the inverse spectral problem for singular perturbations à of A (à and A coincide on a dense set in H). We show that for any a ∈ R there exists a singular perturbation à of A such that à and A coincide in the subspace EA((−∞, a))H and simultaneously à has an additional...

We introduce a new mathematical method, based on the inverse spectral theory of Gel'fand and Levitan, of designing dispersive coatings for use in femtosecond lasers. We fabricated an example in AlGaAs by metal-organic chemical-vapor deposition. The mirror has a high value of group delay dispersion over a bandwidth of 10 nm, reaching an extreme of -1200 fs(2) at the center (805 nm) and falling t...

For any positive integer n and any given n distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of n numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.

We show that the family of spectral triples for quantum projective spaces introduced by D’Andrea and Da̧browski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K2ρ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connecti...

We link boundary control theory and inverse spectral theory for the Schrödinger operatorH = @2 x+q (x) on L2 (0;1) with Dirichlet boundary condition at x = 0: This provides a shortcut to some results on inverse spectral theory due to Simon, Gesztesy-Simon and Remling. The approach also has a clear physical interpritation in terms of boundary control theory for the wave equation. 1. Introduction...