A differential equation for a class of correlation kernels
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915289617137664 |
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| author | Johnson, Clifford V. |
| author_facet | Johnson, Clifford V. |
| contents | A new differential equation is derived for an object ${\widehat S}(E,E^\prime,x)$, which when integrated over the appropriate range in $x$, yields the kernel $K(E,E^\prime)$ with which $n$-point correlation functions can be computed in a wide class of models. When $E{=}E^\prime$, the equation reduces to the equation for the diagonal resolvent ${\widehat R}(E,x)$ of the Schrödinger Hamiltonian ${H}{=}{-}\hbar^2\partial_x^2{+}u(x)$ that is familiar from the classic work of Gel'fand and Dikii, and which appears in many areas of physics. This more general equation may also prove to be useful in a wide range of applications. Some special cases relevant to random matrix theory are explored using analytical and numerical methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_10622 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A differential equation for a class of correlation kernels Johnson, Clifford V. High Energy Physics - Theory Statistical Mechanics Mathematical Physics A new differential equation is derived for an object ${\widehat S}(E,E^\prime,x)$, which when integrated over the appropriate range in $x$, yields the kernel $K(E,E^\prime)$ with which $n$-point correlation functions can be computed in a wide class of models. When $E{=}E^\prime$, the equation reduces to the equation for the diagonal resolvent ${\widehat R}(E,x)$ of the Schrödinger Hamiltonian ${H}{=}{-}\hbar^2\partial_x^2{+}u(x)$ that is familiar from the classic work of Gel'fand and Dikii, and which appears in many areas of physics. This more general equation may also prove to be useful in a wide range of applications. Some special cases relevant to random matrix theory are explored using analytical and numerical methods. |
| title | A differential equation for a class of correlation kernels |
| topic | High Energy Physics - Theory Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2505.10622 |