6 problems

Let $H=-\triangle+|x|^\beta$ on $\mathbb R^n$, where $\beta>0$. Let $\Omega\subset\mathbb R^n$ be a measurable sensor set satisfying the equidistribution condition … for every…

Let $E_a^p$ denote the class of functions $f:\mathbb R\to\mathbb C$ such that $C_a^p(f)$ and $C_a^p(\widehat f)$ are finite. Let $\Phi(x,t)$ solve the time-dependent harmonic-oscil…

Let $N_{\sigma,\tau}(\beta,\lambda)$ denote the shifted lattice-point counting function and let $R^\gamma_{-1/2}(\beta,\lambda)$ denote the corresponding Riesz mean for the harmoni…

For each $s>0$, consider the harmonic oscillator on $\mathbb R^2$ given by … with $u\to0$ as $|(x,y)|\to\infty$, and let $s_n$ be an $s$-value minimizing the $n$-th eigenvalue…

Let $H_{m,n}$ be the classical Hamiltonian of the two-dimensional anisotropic harmonic oscillator, let $K_{m,n}$ and $F_i(m,n)$ be its classical first integrals, and denote their W…

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…