16 problems

Braak's eigenvalue-count conjecture. For a given parity, determined by whether $N$ is even or odd, the interval $I_N$ contains at most two shifted eigenvalues. Moreover, two interv…

Positive-root distribution conjecture. There are exactly $N-k$ positive roots in $x$ when

Density conjecture. The set of parameters $(g,\Delta)$ for which $H_{g,\Delta}$ has a Juddian eigenvalue is dense in the $(g,\Delta)$-plane.

Let $\{E_k\}$ be the ordered eigenvalues of the quantum Rabi Hamiltonian, including both parity classes, with $E_1\leq E_2\leq\cdots$, and define the nearest-neighbor spacings…

Polynomial-curve approximation conjecture. give an excellent approximation of the shape of the first $2\eta$ eigenvalues of the asymmetric quantum Rabi model. These curves are inte…

Let $H_{\text{Rabi}}^{\eta}$ be the asymmetric quantum Rabi Hamiltonian with bias parameter $\eta$, and suppose $0\neq 2\eta\in\mathbb{Z}_{\geq 0}$. Let $J_{2\eta}$ be its commutin…

For a fixed integer bias parameter $\ell$, define … The inclusion … expresses that the parameter values producing degeneracies do not exhaust the parameter plane. Density conjectur…

Polynomial identity conjecture. For all natural numbers $N,\ell\geq 0$,

Analyticity conjecture. All non-trivial formal solutions of this commutation relation are not analytic in $a$ and $a^{\dag}$ if $g$ and $\Delta$ do not satisfy an additional relati…

Consider the two-photon Rabi model and its numerically obtained energy spectrum. Emary and Bishop identified a class of level crossings associated with exact solutions expressible…

Finiteness conjecture for exceptional eigenvalues. The product

Factorization and positivity conjecture. For $\ell,N\in\mathbb{Z}_{\geq 0}$, there exists a polynomial $A^\ell_N(x,y)\in\mathbb{Z}[x,y]$ such that

The original quantum Rabi model describes a two-level system coupled to a quantum oscillator. An excited-state quantum phase transition is a transition at a critical energy, charac…

The quantum Rabi–Stark model is the quantum Rabi model augmented by a nonlinear coupling between the two-level system and the oscillator; its Stark coupling strength can be tuned i…

For the generalized quantum Rabi model with the $A^{2}$-term, let the bare and physical ground states be the ground states associated with the corresponding bare and renormalized d…

Zero-distribution conjecture. The number of zeros of $G_\pm(x)$ in each interval $[n,n+1]$, for $n\in\mathbb{N}_0$, is restricted to be $0$, $1$, or $2$. Moreover, an interval…