Probability and Mathematical Physics Seminar

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On the spectrum of the hierarchical Schrödinger – type operators

Speaker: Stanislav Molchanov, UNC Charlotte and HSE Moscow

Location: Warren Weaver Hall 512

Date: Friday, February 15, 2019, 11 a.m.

Synopsis:

The hierarchical Laplacian was initially introduced in the works of N. Bogolubov and his school (V. Vladimirov, I. Volovich, E. Zelinov) as an essential object in the \(p\)–adic analysis. Similar ideas were developed by F. Dyson in his famous paper on the phase transitions in \(1D\) Ising model with the long range potentials.

We define Dyson–Vladimirov hierarchical Laplacian \(\Delta\) as the non-local operator in \(L^2 (\mathbb{R}, dx)\) associated the Dyson metric on \(\mathbb{R}\). Such Laplacian has many features of the classical fractals (renorm group etc.).

The talk will present the elements of the spectral theory of the hierarchical Hamiltonian \(H = -\Delta + V(x)\). The theory includes the standard results (on the essential self-adjointness, negative spectrum etc.) for the deterministic operators and the results in the spirit of the Anderson localization for the class of the random Schrödinger operators.