### Abstract

We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same order of differentiability as the ground state energy. We prove that discontinuities of the inference map correspond to C^{1}-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at C^{1}-smooth points of the boundary of the numerical range. Discontinuities exist at all C^{2}-smooth non-Analytic boundary points and are essentially stronger than at analytic points or at points which are merely C^{1}-smooth (non-exposed points).

Original language | English (US) |
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Article number | 121901 |

Journal | Journal of Mathematical Physics |

Volume | 59 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2018 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*59*(12), [121901]. https://doi.org/10.1063/1.5017904