On the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent

Resolvent estimates for non-selfadjoint operators with double characteristics

We present recent progress in the understanding of the spectral and subelliptic properties of non-elliptic quadratic operators with application to the study of return to equilibrium for some systems of chains of oscillators. We then explain how these results allow to describe the spectral properties and to give sharp resolvent estimates for some classes of non-selfadjoint pseudodi erential oper...

Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities

Consider the Schrödinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h−1) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case wh...

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

The Horn Conjecture for Sums of Compact Selfadjoint Operators

We determine the possible nonzero eigenvalues of compact selfadjoint operators A, B(1), B(2), . . ., B(m) with the property that A = B(1) +B(2) +· · ·+B(m). When all these operators are positive, the eigenvalues were known to be subject to certain inequalities which extend Horn’s inequalities from the finite-dimensional case when m = 2. We find the proper extension of the Horn inequalities and ...

Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation is ≫ h (or sometimes only ≫ h2) and bounded from above by hδ for some δ > 0. We get a complete asymptotic descri...