5 problems

Let $v\in{\mathfrak V}_{{\mathsf r}{\mathsf e}{\mathsf g}}\setminus{\mathfrak V}_{{\mathsf U}{\mathsf V}}$, so that the ultraviolet cutoff is removed, and let…

Large-coupling absence conjecture. There exists $\lambda_{\mathsf c}>0$ such that $H(\lambda,0)$ has no ground state for

Consider the spin-boson Hamiltonian $H$ with form factor $\hat h$ and dispersion relation $\omega$; write $\hat h/\omega\notin L^2(\mathbb R^d)$ for failure of the infrared square-…

Let $H_{\rm N}$ be the Nelson Hamiltonian and let $\varphi_{\rm g}^{\rm N}$ be a ground state of $H_{\rm N}$. Denote by $N$ the boson number operator. Finite boson-number conjectur…

Let $N:=\mathrm{d}\Gamma(\mathbf{1})$ be the photon (or phonon) number operator, and let $\psi_t$ denote the time-evolved state. In the spin-boson model, a uniform bound of the for…