Exact instanton transseries for quantum mechanics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914756020928512 |
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| author | van Spaendonck, Alexander Vonk, Marcel |
| author_facet | van Spaendonck, Alexander Vonk, Marcel |
| contents | We calculate the instanton corrections to energy spectra of one-dimensional quantum mechanical oscillators to all orders and unify them in a closed form transseries description. Using alien calculus, we clarify the resurgent structure of these transseries and demonstrate two approaches in which the Stokes constants can be derived. As a result, we formulate a minimal one-parameter transseries for the natural nonperturbative extension to the perturbative energy, which captures the Stokes phenomenon in a single stroke. We derive these results in three models: quantum oscillators with cubic, symmetric double well and cosine potentials. In the latter two examples, we find that the resulting full transseries for the energy has a more convoluted structure that we can factorise in terms of a minimal and a median transseries. For the cosine potential we briefly discuss this more complicated transseries structure in conjunction with topology and the concept of the resurgence triangle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_05700 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exact instanton transseries for quantum mechanics van Spaendonck, Alexander Vonk, Marcel High Energy Physics - Theory Mathematical Physics We calculate the instanton corrections to energy spectra of one-dimensional quantum mechanical oscillators to all orders and unify them in a closed form transseries description. Using alien calculus, we clarify the resurgent structure of these transseries and demonstrate two approaches in which the Stokes constants can be derived. As a result, we formulate a minimal one-parameter transseries for the natural nonperturbative extension to the perturbative energy, which captures the Stokes phenomenon in a single stroke. We derive these results in three models: quantum oscillators with cubic, symmetric double well and cosine potentials. In the latter two examples, we find that the resulting full transseries for the energy has a more convoluted structure that we can factorise in terms of a minimal and a median transseries. For the cosine potential we briefly discuss this more complicated transseries structure in conjunction with topology and the concept of the resurgence triangle. |
| title | Exact instanton transseries for quantum mechanics |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2309.05700 |