Imaginary Powers of the Dunkl Harmonic Oscillator

A Perturbation of the Dunkl Harmonic Oscillator on the Line

Let Jσ be the Dunkl harmonic oscillator on R (σ > −1/2). For 0 < u < 1 and ξ > 0, it is proved that, if σ > u − 1/2, then the operator U = Jσ + ξ|x|−2u, with appropriate domain, is essentially self-adjoint in L(R, |x|dx), the Schwartz space S is a core of U 1/2 , and U has a discrete spectrum, which is estimated in terms of the spectrum of Jσ. A generalization Jσ,τ of Jσ is also considered by t...

Embedding Theorems for the Dunkl Harmonic Oscillator on the Line

Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line.

Imaginary Powers of Laplace Operators

We show that if L is a second-order uniformly elliptic operator in divergence form on R, then C1(1+ |α|) ≤ ‖L‖L1→L1,∞ ≤ C2(1+ |α|). We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

Harmonic Oscillator

Imaginary Exceptions : On the Powers and Limits of Thought Experiment

Thought experiment is one of the most widely-used and least understood techniques in philosophy. A thought experiment is a process of reasoning carried out within the context of a well-articulated imaginary scenario in order to answer a specific question about a non-imaginary situation. The aim of my dissertation is to show that both the powers and the limits of this methodology can be traced t...