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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2402.03434 |
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| _version_ | 1866929234685984768 |
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| author | Sword, Lewis Vegh, David |
| author_facet | Sword, Lewis Vegh, David |
| contents | We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian $H=SZ(1-Z)S$, where $Z$ and $S$ satisfy canonical commutation relations. While this model may appear unusual, using an appropriate coordinate transformation, the Schrödinger equation can be cast into a standard form with a Pöschl-Teller-type potential. Since the system is defined on an interval, it is well-known that $S$ is not self-adjoint. Nevertheless, the bootstrap method can still be implemented, producing an infinite set of positivity constraints. Using a certain operator ordering, the energy eigenvalues are only constrained into bands. With an alternative ordering, however, we find that a finite number of constraints is sufficient to fix the low-lying energy levels exactly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03434 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum mechanical bootstrap on the interval: obtaining the exact spectrum Sword, Lewis Vegh, David High Energy Physics - Theory Quantum Physics We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian $H=SZ(1-Z)S$, where $Z$ and $S$ satisfy canonical commutation relations. While this model may appear unusual, using an appropriate coordinate transformation, the Schrödinger equation can be cast into a standard form with a Pöschl-Teller-type potential. Since the system is defined on an interval, it is well-known that $S$ is not self-adjoint. Nevertheless, the bootstrap method can still be implemented, producing an infinite set of positivity constraints. Using a certain operator ordering, the energy eigenvalues are only constrained into bands. With an alternative ordering, however, we find that a finite number of constraints is sufficient to fix the low-lying energy levels exactly. |
| title | Quantum mechanical bootstrap on the interval: obtaining the exact spectrum |
| topic | High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2402.03434 |