On the role of the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with $N=2$ explicit levels
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913040924934144 |
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| author | Monthus, Cecile |
| author_facet | Monthus, Cecile |
| contents | The construction of Quasi-Exactly-Solvable quantum Hamiltonians where only the two first 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 the conservation of probability and to the steady state $P_*(x)$, 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{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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15908 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the role of the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with $N=2$ explicit levels Monthus, Cecile Statistical Mechanics Probability The construction of Quasi-Exactly-Solvable quantum Hamiltonians where only the two first 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 the conservation of probability and to the steady state $P_*(x)$, 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{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 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. |
| title | On the role of the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with $N=2$ explicit levels |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2604.15908 |