Using the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with two explicit levels
Abstract
The construction of Quasi-Exactly-Solvable quantum Hamiltonians, where only the first two eigenstates Φ 0 ( x ) and Φ 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 with the conservation of probability and with the steady state P ∗ ( x ) , while E 1 > 0 is the rate that governs the exponential relaxation towards the steady-state, and is associated with the slowest observable L 1 ( x ) that corresponds to the ratio Φ 1 ( x ) Φ 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 when 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.
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BibTeX
@article{monthus2026using,
title = {{Using the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with two explicit levels}},
author = {Cécile Monthus},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
year = {2026},
month = oct,
doi = {10.1088/1742-5468/aeab79},
eprint = {2604.15908},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05788911v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Using the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with two explicit levels
AU - Cécile Monthus
PY - 2026
DA - 2026/10/09
JO - Journal of Statistical Mechanics: Theory and Experiment
DO - 10.1088/1742-5468/aeab79
UR - https://hal.science/hal-05788911v1
AB - The construction of Quasi-Exactly-Solvable quantum Hamiltonians, where only the first two eigenstates Φ 0 ( x ) and Φ 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 with the conservation of probability and with the steady state P ∗ ( x ) , while E 1 > 0 is the rate that governs the exponential relaxation towards the steady-state, and is associated with the slowest observable L 1 ( x ) that corresponds to the ratio Φ 1 ( x ) Φ 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 when 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.
SN - 1742-5468
ER -