We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary $\xi$-progressive paths. This approach can provide higher flexibility in practical applications of the method.

We make some observations relating the theory of finite-dimensional differential algebraic groups (the ∂₀-groups of [2]) to the Galois theory of linear differential equations. Given a differential field (K,∂), we exhibit a surjective functor from (absolutely) split (in the sense of Buium) ∂₀-groups G over K to Picard-Vessiot extensions L of K, such that G is K-split iff L = K. In fact we give a generalization to "K-good" ∂₀-groups. We also point out that the "Katz group" (a certain linear algebraic...

We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.

We consider a singularity perturbed nonlinear differential equation $\epsilon u^{\text{'}}=f\left(x\right)u++\epsilon P(x,u,\epsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\left|u\right|$, $\left|\epsilon \right|$. We furthermore suppose that 0 is a turning point, namely that $xf\left(x\right)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $ϵ\to 0$) local (at $x=0$) or global, real analytic or $C^{\infty }$ solutions is equivalent to the existence of a formal series solution $\sum u_n\left(x\right){\epsilon }^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions....

On étudie les systèmes différentiels singulièrement perturbés de dimension 3 du type$$\{\begin{array}{ccc}\dot{x}\hfill & =& f(x,y,z,\epsilon ),\hfill \\ \dot{y}\hfill & =& g(x,y,z,\epsilon ),\hfill \\ \epsilon \dot{z}\hfill & =& h(x,y,z,\epsilon ),\hfill \end{array}$$où $f$, $g$, $h$ sont analytiques quelconques. Les travaux antérieurs étudiaient les points réguliers où la surface lente $h=0$ est transverse au champ rapide vertical. C’est le domaine d’application du théorème de Tikhonov. Dans d’autres travaux antérieurs, on étudiait les singularités de certains types : plis et fronces de la surface lente, ainsi que certaines singularités plus compliquées, analogues aux points tournants...

In this paper I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under $𝒞𝒫𝒯$, but not symmetric under $𝒫$ and $𝒯$ separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are $-{\phi }^4$ and $i{\phi }^3$ theories. These theories all have unexpected and remarkable properties. I discuss...