Probability and Mathematical Physics Seminar

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.