Fuchsian equation

equation of Fuchsian class

A linear homogeneous ordinary differential equation in the complex domain,

$$ \tag{1 } w ^ {(} n) + p _ {1} ( z) w ^ {( n - 1) } + \dots + p _ {n} ( z) w = 0 , $$

with analytic coefficients, all singular points of which on the Riemann sphere are regular singular points (cf. Regular singular point). Equation (1) belongs to the Fuchsian class if and only if its coefficients have the form

$$ p _ {j} ( z) = \ \prod _ {m = 1 } ^ { k } ( z - z _ {m} ) ^ {-} j q _ {j} ( z), $$

where $ z _ {1} \dots z _ {k} $ are distinct points and $ q _ {j} ( z) $ is a polynomial of degree $ \leq j ( k - 1) $. A system $ w ^ \prime = A ( z) w $ of $ n $ equations belongs to the Fuchsian class if it has the form

$$ \tag{2 } \frac{dw }{dz } = \ \sum _ {m = 1 } ^ { k } \frac{A _ {m} }{z - z _ {m} } w, $$

where $ z _ {1} \dots z _ {k} $ are distinct points and the $ A _ {m} \neq 0 $ are constant $ ( n \times n) $- dimensional matrices. The points $ z _ {1} \dots z _ {k} , \infty $ are singular for the equation (1) and the system (2). Fuchs' identity holds for (1):

$$ \sum _ {j = 1 } ^ { n } \left ( \sum _ {m = 1 } ^ { k } \rho _ {j} ^ {m} + \rho _ {j} ^ \infty \right ) = ( k - 1) \frac{n ( n - 1) }{2} , $$

where $ \rho _ {1} ^ {m} \dots \rho _ {n} ^ {m} $ are the characteristic exponents at $ z _ {m} $, and $ \rho _ {1} ^ \infty \dots \rho _ {n} ^ \infty $ those at $ \infty $( cf. Characteristic exponent). Fuchsian equations (and systems) are also called regular equations (systems). This class of equations and systems was introduced by J.L. Fuchs .

Let $ D $ be the Riemann sphere with punctures at the points $ z _ {1} \dots z _ {k} , \infty $. Every non-trivial solution of (1) (respectively, every component of a solution of (2)) is an analytic function in $ D $. In general, this function is infinite-valued, and all the singular points of (1) (or (2)) are branch points of it of infinite order.

A second-order Fuchsian equation with singular points $ z _ {1} \dots z _ {k} , \infty $ has the form

$$ \tag{3 } w ^ {\prime\prime} + \sum _ {m = 1 } ^ { k } \frac{1 - ( \rho _ {1} ^ {m} + \rho _ {2} ^ {m} ) }{z - z _ {m} } w ^ \prime + $$

$$ + \sum _ {m = 1 } ^ { k } \left [ \frac{\rho _ {1} ^ {m} \rho _ {2} ^ {m} \prod _ {j = 1 } ^ { k } {} ^ \prime ( z _ {m} - z _ {j} ) }{z - z _ {m} } + Q _ {k - 2 } ( z) \right ] { \frac{w}{\prod _ {m = 1 } ^ { k } ( z - z _ {m} ) } } = 0, $$

where $ Q _ {k - 2 } ( z) $ is a polynomial of degree $ k - 2 $. The transformation $ w = ( z - z _ {m} ) ^ {l} w $ takes a Fuchsian equation to a Fuchsian equation, with

$$ ( \rho _ {1} ^ {m} , \rho _ {2} ^ {m} ) \rightarrow \ ( \rho _ {1} ^ {m} - l, \rho _ {2} ^ {m} - l), $$

$$ ( \rho _ {1} ^ \infty , \rho _ {2} ^ \infty ) \rightarrow \ ( \rho _ {1} ^ \infty + l, \rho _ {2} ^ \infty + l), $$

and the characteristic exponents at the other singular points are unchanged. By means of such transformations, equation (3) can be reduced to the form

$$ w ^ {\prime\prime} + \sum _ {m = 1 } ^ { k } \frac{1 - ( \rho _ {2} ^ {m} + \rho _ {1} ^ {m} ) }{z - z _ {m} } w ^ \prime + $$

$$ + ( \overline \rho \; {} _ {1} ^ \infty \overline \rho \; {} _ {2} ^ \infty z ^ {n - 2 } + d _ {1} z ^ {n - 3 } + \dots + d _ {n - 2 } ) { \frac{w}{\prod _ {m = 1 } ^ { k } ( z - z _ {m} ) } } = 0, $$

$$ \overline \rho \; {} _ {j} ^ \infty = \rho _ {j} ^ \infty + \sum _ {m = 1 } ^ { k } \rho _ {j} ^ {m} . $$

A second-order Fuchsian equation with $ N $ singular points is completely determined by specifying the values of the characteristic exponents at these points if and only if $ N < 4 $. Using a Möbius transformation the equation can be reduced to the form: a) $ N = 1 $, $ \widetilde{w} {} ^ {\prime\prime} = 0 $; b) $ N = 2 $, $ \zeta ^ {2} \widetilde{w} {} ^ {\prime\prime} + a \zeta \widetilde{w} {} ^ {\prime\prime} + b \widetilde{w} = 0 $( the Euler equation); c) $ N = 3 $— the Papperitz equation (or Riemann equation).

A matrix Fuchsian equation has the form

$$ \tag{4 } \frac{dW }{dz } = \ \sum _ {m = 1 } ^ { k } \frac{W U _ {m} }{z - z _ {m} } , $$

where $ z _ {1} \dots z _ {k} $ are distinct points, $ W $ is an $ ( n \times n) $- dimensional matrix function, and the $ U _ {m} \neq 0 $ are constant matrices. The matrix $ U _ {m} $ is called a differential substitution at $ z _ {m} $. Let $ \gamma $ be a closed curve that starts at a non-singular point $ b $, is positively oriented and contains only the singular point $ z _ {m} $ inside it. If $ W ( z) $ is a solution of (4) that is holomorphic at $ b $, then under analytic continuation along $ \gamma $,

$$ \tag{5 } W \rightarrow V _ {m} W, $$

where $ V _ {m} $ is a constant matrix, called an integral substitution at $ z _ {m} $. H. Poincaré (see [2]) posed the so-called first regular Poincaré problem for a system of the form (4). It consists of the following three problems:

A) to represent the solution $ W ( z) $ in its whole domain of existence;

B) to construct the integral substitutions at the points $ z _ {m} $;

C) to give an analytic characterization of the singularities of the solutions.

In particular, solving problem B) enables one to construct the monodromy group of (4). A solution of the Poincaré problem was obtained by I.A. Lappo-Danilevskii [3]. Let $ L _ {b} ( z _ {j _ {1} } \dots z _ {j _ \nu } \mid z) $, $ j _ {m} \in \{ 1 \dots k \} $, $ \nu = 1, 2 \dots $ be the hyperlogarithms:

$$ L _ {b} ( z _ {m} \mid z) = \ \int\limits _ { b } ^ { z } \frac{dz }{z - z _ {m} } ,\ \ $$

$$ L _ {b} ( z _ {j _ {1} } \dots z _ {j _ \nu } \mid z) = \int\limits _ { b } ^ { z } \frac{L _ {b} ( z _ {j _ {1} } \dots z _ {j _ {\nu - 1 } } \mid z) }{z - z _ {j _ \nu } } dz, $$

let $ W _ {0} ( z) $ be the element (germ) at $ b $ of a solution of (4), normalized by the condition $ W _ {0} ( b) = I $, and let $ W ( z) $ be the analytic function in $ D $ generated by this element. Then $ W ( z) $ is an entire function of the matrices $ U _ {1} \dots U _ {k} $ and has a series expansion

$$ W ( z) = I + \sum _ {\nu = 1 } ^ \infty \ \sum _ {j _ {1} \dots j _ \nu } ^ { {( } 1 \dots k) } U _ {j _ {1} } \dots U _ {j _ \nu } L _ {b} ( z _ {j _ {1} } \dots z _ {j _ \nu } \mid z), $$

which converges uniformly in $ z $ on every compact set $ K \subset D $. The integral substitution $ V _ {m} $ at $ z _ {m} $ corresponding to the solution $ W ( z) $ is an entire function of $ U _ {1} \dots U _ {k} $ and has a series expansion

$$ V _ {m} = I + \sum _ {\nu = 1 } ^ \infty \ \sum _ {j _ {1} \dots j _ \nu } ^ { {( } 1 \dots k) } U _ {j _ {1} } \dots U _ {j _ \nu } P _ {j} ( z _ {j _ {1} } \dots z _ {j _ \nu } \mid b), $$

where $ P _ {j} $ can be expressed in terms of hyperlogarithms (see [3], [6]).

Formulas that give a solution to problem C) have also been obtained (see [3]).

References

[1a]	J.L. Fuchs, "Zur Theorie der linearen Differentialgleichungen mit Veränderlichen Koeffizienten" J. Reine Angew. Math. , 66 (1866) pp. 121–160
[1b]	J.L. Fuchs, "Zur Theorie der linearen Differentialgleichungen mit Veränderlichen Koeffizienten. Ergänzung" J. Reine Angew. Math. , 68 (1868) pp. 354–385
[2]	H. Poincaré, "Papers on Fuchsian functions" , Springer (1985) (Translated from French) MR0809181 Zbl 0577.01048
[3]	I.A. Lappo-Danilevskii, "Applications des fonctions matrices dans la theorie des systèeme des équations différentielles ordinaires lineaires" , Moscow (1957) (In Russian; translated from French)
[4]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 MR0069338 Zbl 0064.33002
[5]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119
[6]	V.I. Smirnov, "A course of higher mathematics" , 3 , Addison-Wesley (1964) pp. Part 2 (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[7]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) MR1570308 MR0010757 MR1524980 MR0000325 MR1522581 Zbl 0612.34002 Zbl 0191.09801 Zbl 0063.02971 Zbl 0022.13601 Zbl 65.1253.02 Zbl 53.0399.07

Comments

The matrix $ V _ {m} $ of equation (5) is also called the local monodromy at $ z _ {m} $ or the monodromy matrix at $ z _ {m} $ of the Fuchsian system (4). Riemann posed the problem, the Riemann monodromy problem, of finding for given $ V _ {i} $ a Fuchsian system with these given monodromy matrices. This problem was essentially solved by J. Plemelj [a3], G. Birkhoff [a4], [a5] and I.A. Lappo-Danilevskii [a2]. By taking a contour $ \gamma $ through all the $ z _ {i} $ and $ \infty $ and a piecewise-constant matrix function on $ \gamma $( value $ V _ {1} $ between $ z _ {1} $ and $ z _ {2} $, value $ V _ {2} V _ {1} $ between $ z _ {2} $ and $ z _ {3} ,\dots $) the problem can be turned into a Riemann–Hilbert problem. The conditions on the points $ z _ {i} $ and the matrices $ U _ {m} $ which are necessary and sufficient for the systems to retain the same monodromy under smooth changes in these parameters take the form of differential equations known as the isomonodromy equations or Schlessinger equations. These equations have links to (completely) integrable systems (cf. Integrable system) and quantum fields, cf., e.g., [a6], [a7].

References

[a1]	E. Hille, "Ordinary differential equations in the complex domain" , Wiley (1976) MR0499382 Zbl 0343.34007
[a2]	I.A. Lappo-Danilevskii, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Dover, reprint (1953)
[a3]	J. Plemelj, "Problems in the sense of Riemann and Klein" , Wiley (1964) MR0174815 Zbl 0124.28203
[a4]	G.D. Birkhoff, "Singular points of ordinary linear differential equations" Trans. Amer. Math. Soc. , 10 (1909) pp. 434–470 MR1500848 Zbl 46.0695.02 Zbl 45.0484.01 Zbl 44.0373.01
[a5]	G.D. Birkhoff, "A simplified treatment of the regular singular point" Trans. Amer. Math. Soc. , 11 (1910) pp. 199–202 MR1500860 Zbl 41.0350.01
[a6]	D.V. Chudnovsky, "Riemann, monodromy problem, isomonodromy deformations and completely integrable systems" C. Bardos (ed.) D. Bessis (ed.) , Bifurcation phenomena in mathematical physics and related topics , Reidel (1980) pp. 385–447 MR0580306
[a7]	M. Jimbo, T. Miwa, M. Sato, "Holonomic quantum fields—the unanticipated link between deformation theory of differential equations and quantum fields" K. Osterwalder (ed.) , Mathematical problems in theoretical physics , Springer (1980) pp. 119–142 MR0582615