Self-adjoint differential equation

A linear ordinary differential equation $ l ( y) = 0 $ that coincides with the adjoint differential equation $ l ^ {*} ( y) = 0 $. Here

$$ l ( y) \equiv a _ {0} ( t) y ^ {(} n) + \dots + a _ {n} ( t) y, $$

$$ l ^ {*} ( y) \equiv (- 1) ^ {n} ( \overline{a}\; _ {0} ( t) y) ^ {(} n) + \dots + (- 1) ^ {0} \overline{a}\; _ {n} ( t) y, $$

where

$$ y ^ {( \nu ) } = \ \frac{d ^ \nu y }{dt ^ \nu } ,\ \ y ( \cdot ) \in C ^ {n} ( I),\ \ a _ {k} ( \cdot ) \in C ^ {n - k } ( I), $$

$$ a _ {0} ( t) \neq 0,\ t \in I, $$

$ C ^ {m} ( I) $ is the space of $ m $- times continuously-differentiable complex-valued functions on $ I = ( \alpha , \beta ) $, and the bar denotes complex conjugation.

The left-hand side of every self-adjoint differential equation $ l ( y) = 0 $ is a sum of expressions of the form

$$ l _ {2m} ( y) = ( p _ {m} y ^ {(} m) ) ^ {(} m) , $$

$$ l _ {2m - 1 } ( y) = { \frac{1}{2} } [( iq _ {m} y ^ {( m - 1) } ) ^ {(} m) + ( iq _ {m} y ^ {(} m) ) ^ {( m - 1) } ], $$

where $ p _ {m} ( t) $ and $ q _ {m} ( t) $ are sufficiently-smooth real-valued functions and $ i ^ {2} = - 1 $. A self-adjoint differential equation with real coefficients is necessarily of even order, and has the form

$$ ( p _ {0} y ^ {(} m) ) ^ {(} m) + ( p _ {1} y ^ {( m - 1) } ) ^ {( m - 1) } + \dots + p _ {m} y = 0 $$

(see [1]–[3]).

A linear system of differential equations

$$ L ( x) = 0,\ \ L ( x) \equiv \dot{x} + A ( t) x,\ \ t \in I, $$

with a continuous complex-valued $ ( n \times n) $- matrix $ A ( t) $, is called self-adjoint if $ A ( t) = - A ^ {*} ( t) $, where $ A ^ {*} ( t) $ is the Hermitian conjugate of $ A ( t) $( see [1], [4], and Hermitian operator). This definition is not consistent with the definition of a self-adjoint differential equation. For example, the system

$$ \dot{x} _ {1} - x _ {2} = 0,\ \ \dot{x} _ {2} + p ( t) x _ {1} = 0, $$

which is equivalent to the self-adjoint differential equation

$$ \dot{y} dot + p ( t) y = 0, $$

is self-adjoint as a linear system if and only if $ p ( t) \equiv 1 $.

The boundary value problem

$$ \tag{1 } l ( y) = 0,\ \ t \in \Delta = \ [ t _ {0} , t _ {1} ], $$

$$ \tag{2 } U _ {k} ( y) = 0 ,\ k = 1 \dots n, $$

where the $ U _ {k} : C ^ {(} n) ( \Delta ) \rightarrow \mathbf R ^ {1} $ are linear and linearly independent functionals describing the boundary conditions, is called self-adjoint if it coincides with the adjoint boundary value problem, that is, (1) is a self-adjoint differential equation and $ U _ {k} ( y) = U _ {k} ^ {*} ( y) $ for all $ y ( \cdot ) \in C ^ {n} ( \Delta ) $ and for all $ k = 1 \dots n $( see [1]–[3], [5]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see Green formulas)

$$ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline \xi \; l ( y) dt = \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline{l}\; ( \xi ) y dt $$

holds for any pair $ y ( \cdot ), \xi ( \cdot ) \in C ^ {(} n) ( \Delta ) $ that satisfy the boundary conditions (2).

All the eigenvalues of the self-adjoint problem

$$ l ( y) = \lambda y,\ \ U _ {k} ( y) = 0,\ \ k = 1 \dots n, $$

are real, and the eigenfunctions $ \phi _ {1} , \phi _ {2} $ corresponding to distinct eigenvalues $ \lambda _ {1} , \lambda _ {2} $ are orthogonal:

$$ \int\limits _ { t _ {0} } ^ { {t _ 1 } } \overline \phi \; _ {1} \phi _ {2} ( t) dt = 0. $$

The linear boundary value problem

$$ \tag{3 } L ( x) \equiv \dot{x} + A ( t) x = 0,\ \ U ( x) = 0,\ \ t \in \Delta , $$

where $ A ( t) $ is a continuous complex-valued $ ( n \times n) $- matrix and $ U $ is an $ n $- vector functional on the space $ C _ {n} ^ {1} ( \Delta ) $ of continuous complex-valued functions $ x: \Delta \rightarrow \mathbf R ^ {n} $, is called self-adjoint if it coincides with its adjoint boundary value problem

$$ L ^ {*} ( x) = 0,\ \ U ^ {*} ( x) = 0,\ \ t \in \Delta , $$

that is,

$$ L ( x) = - L ^ {*} ( x),\ \ U ( x) = U ^ {*} ( x) $$

for all $ x ( \cdot ) \in C _ {n} ^ {1} ( \Delta ) $. A self-adjoint boundary value problem has properties analogous to those of the problem (1), (2) (see [4]).

The concepts of a self-adjoint differential equation and of a self-adjoint boundary value problem are closely connected with that of a self-adjoint operator [6] (cf. also Spectral theory of differential operators). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [5], [7]).

References

[1]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[2]	M.A. Naimark, "Linear differential operators" , 1–2 , Harrap (1968) (Translated from Russian)
[3]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[4]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[5]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[6]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[7]	V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)

Comments

In general, the system of eigenfunctions is complete.

References

[a1]	S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965)