A criterion for Hill operators to be spectral operators of scalar type

By Fritz Gesztesy, V. Trachenko. 2006

69 citations recorded by the source corpus.

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger operator) H = −d2/dx2 + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion.The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3536f9fdbaec0fccba181e5d79b447fdf9f5b789

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In Atomic and Molecular Physics, and Optics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators (2011, background): “, in the 2000’s, many authors [18, 26, 29, 34, 33, 35, 37, 38, 39, 40, 52] focused on the problem of convergence of eigenfunction (or more generally root function) decompositions in the case of regular but not strictly regular bc.”
A Schauder and Riesz Basis Criterion for Non-Self-Adjoint Schr\ (2011, background): “As discussed in [25], [26], this implies that no smoothness or analyticity conditions imposed on a periodic potential V on R can guarantee that a Hill operator H in L(2)(R; dx) as in (1.”