4 problems

Let $M$ be a compact manifold and let $\varphi_t:M\to M$ be a $\mathcal{C}^\infty$ Anosov flow. Write $P=-iV$, where $V$ is the generator of $\varphi_t$. For each periodic orbit…

Let $\mathcal H$ be a Hilbert space, let $U\in B(\mathcal H)$ be a unitary operator, and let $Z\in \mathcal S^1(\mathcal H)$ be self-adjoint. For a function $f$ on the unit circle…

Let $\alpha$ be the order of the perturbation, let $\sigma_\alpha$ be the scaling factor appearing in the perturbative action shift, and let $\tilde{L}=2L/(\omega R_0^2)$, where…

Let $\rho(mu)$ denote the spectral density associated with the trace formulae, and let $w$ be the parameter controlling the contribution of back-scattered walks. For $w\ne 1$, the…