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On Persson’s formula: an étale groupoid approach

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Abstract

Persson’s formula expresses the infimum of the essential spectrum of a suitable self-adjoint Schrödinger operator in ℝn in terms of the lower spectral points of a family of restrictions of the operator to complements of relatively compact subsets. It has been extended to other situations. We present an approach based on C*-algebras associated to étale groupoids. In such a setting there are intrinsic versions, referring to self-adjoint elements of the groupoid C*-algebras. When representations in Hilbert spaces are considered, the results not always involve only the essential spectrum. The range of applications is quite distinct from the traditional one. We indicate examples related to symbolic dynamics and band dominated operators on discrete metric spaces. The treatment needs only a small amout of symmetry. Even when group actions are involved, restrictions to non-invariant subsets are needed and have to be treated carefully.

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Acknowledgements

The author was supported by the Fondecyt Project 1200884. He is grateful to an anonymous referee for a careful reading of the manuscript and for valuable remarks. A useful discussion with Professor Jiawen Zhang is also acknowledged.

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  1. Departmento de Matemáticas, Universida de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile

    Marius Măntoiu

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  1. Marius Măntoiu
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Correspondence to Marius Măntoiu.

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Măntoiu, M. On Persson’s formula: an étale groupoid approach. Isr. J. Math. 249, 899–933 (2022). https://doi.org/10.1007/s11856-022-2329-z

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  • DOI: https://doi.org/10.1007/s11856-022-2329-z

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