Calculus of Variations and Geometric Measure Theory

V. Felli - B. Noris - R. Ognibene - G. Siclari

Quantitative spectral stability for Aharonov-Bohm operators with many coalescing poles

created by ognibene on 09 Jun 2023



Inserted: 9 jun 2023

Year: 2023

ArXiv: 2306.05008 PDF


The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, we obtain an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed.