Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators

Uniqueness Theorems in Inverse Spectral Theory for One-dimensional Schrödinger Operators

New unique characterization results for the potential V (x) in connection with Schrödinger operators on R and on the half-line [0,∞) are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary sp...

Inverse Uniqueness Results for One-dimensional Weighted Dirac Operators

Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl–Titchmarsh–Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one...

Inverse Spectral Theory for One-dimensional Schrödinger Operators: the a Function

We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simon’s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simon’s fundamental equation.

Inverse Spectral Theory for One

Symplectic inverse spectral theory for pseudodifferential operators

We prove, under some generic assumptions, that the semiclassical spectrum modulo O(~) of a one dimensional pseudodifferential operator completely determines the symplectic geometry of the underlying classical system. In particular, the spectrum determines the hamiltonian dynamics of the principal symbol.