From Conformal Group to Symmetries of Hypergeometric Type Equations

We show that properties of hypergeometric type equations become transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, we deduce the symmetries of the hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and 3-dimensional Laplace equation. We also derive the symmetries of the confluent and Hermite equation from the so-called Schr\"odinger symmetries of the heat equation in 2 and 1 dimension. Finally, we also describe how properties of the ${}_0F_1$ equation follow from the Helmholtz equation in 2 dimensions.Comment: comment by PM: the editing done by SIGMA deteriorates the quality of the paper which is very sad, please visit http://www.fuw.edu.pl/~derezins/sym-hyp-submitted.pdf for the submitted and reviewed version which looks as the authors intended; part of PM's doctoral thesi

Group-theoretical origin of symmetries of hypergeometric class equations and functions

We show that properties of hypergeometric class equations and functions become transparent if we derive them from appropriate 2nd order differential equations with constant coefficients. More precisely, properties of the hypergeometric and Gegenbauer equation can be derived from generalized symmetries of the Laplace equation in 4, resp. 3 dimension. Properties of the confluent, resp. Hermite equation can be derived from generalized symmetries of the heat equation in 2, resp. 1 dimension. Finally, the theory of the ${}_1F_1$ equation (equivalent to the Bessel equation) follows from the symmetries of the Helmholtz equation in 2 dimensions. All these symmetries become very simple when viewed on the level of the 6- or 5-dimensional ambient space. Crucial role is played by the Lie algebra of generalized symmetries of these 2nd order PDE's, its Cartan algebra, the set of roots and the Weyl group. Standard hypergeometric class functions are special solutions of these PDE's diagonalizing the Cartan algebra. Recurrence relations of these functions correspond to the roots. Their discrete symmetries correspond to the elements of the Weyl group.Comment: Prepared for the Summer School "Complex Differential and Difference Equations" 02.09.2018 - 15.09.2018 in B\k{e}dlew

Extended Weak Coupling Limit for Friedrichs Hamiltonians

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a ``small subsystem'' and an infinite dimensional one -- a ``reservoir''. The operator, which we call a ``Friedrichs Hamiltonian'', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before \cite{AFL} in more complicated models and they are usually referred to as "stochastic Limit".Comment: changes in notation and title, minor correction

An Evolution Equation Approach to Linear Quantum Field Theory

In the first part of our paper we analyze bisolutions and inverses of (non-autonomous) evolution equations. We are mostly interested in pseudo-unitary evolutions on Krein spaces, which naturally arise in linear Quantum Field Theory. We prove that with boundary conditions given by a maximal positive and maximal negative space we can associate an inverse, which can be viewed as a generalization of the usual Feynman propagator. In the context of globally hyperbolic manifolds, the Feynman propagator turns out to be a distinguished inverse of the Klein-Gordon operator. Within the formalism of Quantum Field Theory on curved spacetimes, the Feynman propagator yields the expectation values of time-ordered products of fields between the in and out vacuum --the basic ingredient for Feynman diagrams.Comment: 61 page

Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_{\omega,\lambda}:=\begin{bmatrix} -\frac{\lambda+\omega}{x} & - \partial_x \\ \partial_x & -\frac{\lambda-\omega}{x} \end{bmatrix}$. We describe their closed realizations in the sense of the Hilbert space $L^2(\mathbb R_+,\mathbb C^2)$, allowing for complex values of the parameters $\lambda,\omega$. In physical situations, $\lambda$ is proportional to the electric charge and $\omega$ is related to the angular momentum. We focus on realizations of $D_{\omega,\lambda}$ homogeneous of degree $-1$. They can be organized in a single holomorphic family of closed operators parametrized by a certain 2-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator. It is well-known that $D_{\omega,\lambda}$ arise after separation of variables of the Dirac-Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint realizations of Dirac-Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac-Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions