On the role of the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with $N=2$ explicit levels

The construction of Quasi-Exactly-Solvable quantum Hamiltonians where only the two first eigenstates ${\mathrm{\Phi }}_0(x)$ and ${\mathrm{\Phi }}_1(x)$ of energies $E_0$ and $E_1$ are explicit is revisited from the point of view of one-dimensional Markov processes satisfying detailed-balance, whose generators are related to quantum Hamiltonians via similarity transformations. Here the lowest energy vanishes $E_0=0$ and is associated the conservation of probability and to the steady state $P_{\ast }(x)$, while $E_1>0$ is the rate that governs the exponential relaxation towards the steady-state, and is associated to the slowest observable $L_1(x)$ that corresponds to the ratio $\frac{{\mathrm{\Phi }}_1(x)}{{\mathrm{\Phi }}_0(x)}$ of the two quantum eigenstates. Our main conclusion is that the Markov perspective leads to interesting re-interpretations and that the construction of quasi-exactly-solvable models with $N=2$ explicit levels is more intuitive and technically simpler if one takes the slowest observable $L_1(x)$ as the central object from which all the other properties can be reconstructed. This general approach is then applied to Fokker-Planck generators in continuous space and to Markov jump generators on the lattice.