Reviews in Mathematical Physics

We construct in a rigorous mathematical way interacting quantum field theories on a $p$-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The advantage of the approach presented here is that all the perturbation calculations can be carried out in the standard way using functional derivatives, but in a mathematically rigorous way.

In this paper we are interested in generalizing Keller-type eigenvalue estimates for the non-selfadjoint Schrödinger operator to the Dirac operator, imposing some suitable rigidity conditions on the matricial structure of the potential, without necessarily requiring the smallness of its norm.

First, we want to give a complete classification of Yamabe solitons and gradient Yamabe solitons for real hypersurfaces in the complex hyperbolic two-plane Grassmannians $G_2^{*}({ℂ}^{m+2})$. Next, as an application we also give a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on real hypersurfaces in the complex hyperbolic two-plane Grassmannians $G_2^{*}({ℂ}^{m+2})$.

The operator that intertwines between the ${\mathbb{Z}}_2$-Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl intertwining operator maps the Hermite polynomials on the generalized Hermite polynomials.

Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.