Multisummability and Stokes multipliers of linear meromorphic differential equations. (English) Zbl 0729.34005

Braaksma, B. L. J.

J. Differ. Equations 92, No. 1, 45-75 (1991).

Consider the differential equation \(xy'(x)=A(x)y(x)\) where \(y(x)\in {\mathbb{C}}^ n\) and \(A(x)\) is an analytic \((n\times n)\)-matrix having a pole at \(x=0\). It is well known that there exists a formal fundamental matrix \(\hat Y(x)=\hat F(t)t^{\Lambda}\exp Q(t^{-1}),\quad t=x^{1/p},\) where \(p\in {\mathbb{N}}\), Q(t) is a diagonal matrix whose entries are polynomials, \(\Lambda\) is a constant matrix satisfying \(\Lambda Q(t)=Q(t)\Lambda\), and \(\hat F(t)\) is an \((n\times n)\)-matrix whose entries are formal power series in t. In general, the series \(\hat F(t)\) does not converge. It is shown that \(\hat F(t)\) is multisummable in certain sectors such that the multisums \(F(t)\) satisfy \(F(t)\sim \hat F(t)\) as \(t\to 0\) and such that \(Y(x)=F(t)t^{\Lambda}\exp Q(t^{- 1}),\quad t=x^{1/p},\) is a fundamental matrix of the differential equation.

Reviewer: W.Bergweiler (Aachen)

MSC:

34M99

Ordinary differential equations in the complex domain

Keywords:

Stokes multiplier; meromorphic differential equation; asymptotic integration; multisummability

References:

[2]	Balser, W.; Jurkat, W. B.; Lutz, D. A., Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., 71, 48-94 (1979) · Zbl 0415.34008
[3]	Balser, W.; Jurkat, W. B.; Lutz, D. A., On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities, II, SIAM J. Math. Anal., 19, 396-443 (1988) · Zbl 0639.34034
[4]	Braaksma, B. L.J, Laplace integrals, factorial series and singular differential systems, (Proceedings, Bicentennial Congress Wiskundig Genootschap (1978)), 101-121, MCTract 100 · Zbl 0417.34013
[5]	Braaksma, B. L.J, Laplace integrals in singular differential and difference equations, (Proceedings, Ordinary and Partial Differential Equations. Proceedings, Ordinary and Partial Differential Equations, Dundee 1978. Proceedings, Ordinary and Partial Differential Equations. Proceedings, Ordinary and Partial Differential Equations, Dundee 1978, Lecture Notes in Mathematics, Vol. 827 (1980), Springer-Verlag: Springer-Verlag New York/Berlin), 25-53 · Zbl 0449.34005
[6]	Braaksma, B. L.J; Harris, W. A., Laplace integrals and factorial series in singular functional differential systems, Appl. Anal., 8, 23-45 (1978) · Zbl 0406.34058
[7]	Ecalle, J., Les fonctions résurgentes, III, Publ. Math. Orsay (1985) · Zbl 0602.30029
[8]	Ecalle, J., L’accélération des fonctions résurgentes (1987), manuscript
[10]	Harris, W. A.; Sibuya, J. Y.; Weinberg, L., Holomorphic solutions of linear differential systems at singular points, Arch. Rational Mech. Anal., 35, 245-248 (1969) · Zbl 0227.34003
[11]	Horn, J., Integration linearer Differentialgleichungen durch Laplacesche Integrale und Fakultätenreihen, Jahresber. Deutsch. Math. Ver., 25, 74-83 (1917) · JFM 45.0487.01
[12]	Immink, G. K., A note on the relation between Stokes multipliers and formal solutions of analytic differential equations, SIAM J. Math. Anal., 21, 782-792 (1990) · Zbl 0715.34011
[13]	Jurkat, W. B., Meromorphe Differentialgleichungen, (Lecture Notes in Mathematics, Vol. 637 (1978), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0408.34004
[15]	Ramis, J.-P, Les séries \(k\)-sommables et leurs applications, (Proceedings, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory. Proceedings, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Les Houches 1979. Proceedings, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory. Proceedings, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Les Houches 1979, Lecture Notes in Physics, Vol. 126 (1980), Springer-Verlag: Springer-Verlag New York/Berlin), 178-199 · Zbl 1251.32008
[16]	Schäfke, R., Über das globale Verhalten der Normallösungen von \(X\)′\((t) = (B + t^{−1}A)X(t)\) und zweier Arten von assozierten Funktionen, Math. Nachr., 121, 123-145 (1985) · Zbl 0563.34003
[17]	Sibuya, Y., Linear differential equations in the complex domain: Problems of analytic continuation, (Transl. Math. Monographs, 82 (1990), AMS) · Zbl 0151.12503
[18]	Turrittin, H. L., Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math., 93, 27-66 (1955) · Zbl 0064.33603