Erfan Shalchian | Searcharxiv

arXiv2019

A Convergent Continuum Strong Coupling Expansion For Quantum Mechanics & Quantum Field Theory

We generalize the notion of an asymptotic weak coupling expansion about an exactly solvable model in quantum mechanics and quantum field theory to an all positive value coupling convergent expansion. This is done by rescaling the variables available in the theory by free parameters, then adding and subtracting the exactly solvable model. The rest (initial rescaled theory by free parameters + the subtracted exactly solvable model) is expanded about the added exactly solvable model. Evaluating finite orders of this expansion at its extremum points with respect to the free parameter(s) gives a sequence that converges to the result of the previous asymptotic expansion, with a good convergence rate, at relative strong coupling. We solve for the eigenenergies of the anharmonic, pure anharmonic and double well potential problems using this method by expanding about the symmetrical point of these potentials. Accurate results for the eigenenergies can be obtained for all positive values of the coupling for the anharmonic and pure anharmonic oscillators and at strong coupling for the double well potential. To provide confirmation for the convergent formalism developed for $ϕ^4$ theory and QED we improve the electron g-factor calculation at the one loop level using the convergent formalism. Applications of this method are not limited to quantum mechanics or quantum field theory, for example it can also have applications in the context of differential equations.

hep-th↗

arXiv2018

A generalization of Picard-Lindelof theorem/ the method of characteristics to systems of PDE

We generalize Picard-Lindelof theorem/ the method of characteristics to the following system of PDE: $C_{il}(x,y) {\partial y_i / \partial x_l} + {\partial y_i / \partial x_m} = D_i(x,y)$. With a Lipschitz or $C^r$ $C_{il},D_i: [-a, a]^{m} \times [-b, b]^{n} \rightarrow \mathbb{R}$ and initial condition $I_i: [-\bar{a}, \bar{a}]^{m-1} \rightarrow (-b,b)$, $\bar{a} \leq a$, we obtain a local unique Lipschitz or $C^r$ solution $f$, respectively that satisfies the initial condition, $f_i (v, 0 ) = I_i(v)$, $v \in [-\bar{a}, \bar{a}]^{m-1}$. To construct the solution we set bounds on the value of the solution by discretizing the domain of the solution along the direction perpendicular to the initial condition hyperplane. As the number of discretization hyperplanes is taken to infinity the upper and lower bounds of the solution approach each other, hence this gives a unique function for the solution ($Ufs$). A locality condition is derived based on the constants of the problem. The dependence of $C_{il}$, $D_i$ and $I_i$ on parameters, the generalization to nonlinear systems of PDE and the application to hyperbolic quasilinear systems of first order PDE in two independent variables is discussed.

math.AP↗