Criteria for discrete spectrum of singular selfadjoint differential operators

The Discrete Spectrum of Selfadjoint Operators under Perturbations of Variable Sign

On the Discrete Spectrum of a Family of Differential Operators

A family Aα of differential operators depending on a real parameter α is considered. The problem can be formulated in the language of perturbation theory of quadratic forms. The perturbation is only relatively bounded but not relatively compact with respect to the unperturbed form. The spectral properties of the operator Aα strongly depend on α. In particular, for α < √ 2 the spectrum of Aα bel...

On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators

Imbedded Singular Continuous Spectrum for Schrödinger Operators

We construct examples of potentials V (x) satisfying |V (x)| ≤ h(x) 1+x , where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen “twenty-first century” problems for Schrödinger operators posed by Barry Simon in [22]. The construction also provides the first example of a Sch...

Operators with Singular Continuous Spectrum

The Baire category theorem implies that the family, F,of dense sets G6 in a fixed metric space, X , is a candidate for generic sets since it is closed under countable intersections; and if X is perfect (has no isolated point), then A E F has uncountable intersections with any open ball in X. There is a long tradition of soft arguments to prove that certain surprising sets are generic. For examp...