Difference between revisions of "Self-adjoint differential equation"
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− | + | A linear ordinary differential equation $ l ( y) = 0 $ | |
+ | that coincides with the [[Adjoint differential equation|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 | 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 [[#References|[1]]]–[[#References|[3]]]). | (see [[#References|[1]]]–[[#References|[3]]]). | ||
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A linear system of differential equations | 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 | + | 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 [[#References|[1]]], [[#References|[4]]], and [[Hermitian operator|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 | 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 | + | is self-adjoint as a linear system if and only if $ p ( t) \equiv 1 $. |
The boundary value problem | 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 | + | 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 [[#References|[1]]]–[[#References|[3]]], [[#References|[5]]]). If (1), (2) is a self-adjoint boundary value problem, then the equality (see [[Green formulas|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 | + | 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 | 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 | + | 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 | The linear boundary value problem | ||
− | + | $$ \tag{3 } | |
+ | L ( x) \equiv \dot{x} + A ( t) x = 0,\ \ | ||
+ | U ( x) = 0,\ \ | ||
+ | t \in \Delta , | ||
+ | $$ | ||
− | where | + | 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, | that is, | ||
− | + | $$ | |
+ | L ( x) = - L ^ {*} ( x),\ \ | ||
+ | U ( x) = U ^ {*} ( x) | ||
+ | $$ | ||
− | for all | + | 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 [[#References|[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|self-adjoint operator]] [[#References|[6]]] (cf. also [[Spectral theory of differential operators|Spectral theory of differential operators]]). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [[#References|[5]]], [[#References|[7]]]). | 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|self-adjoint operator]] [[#References|[6]]] (cf. also [[Spectral theory of differential operators|Spectral theory of differential operators]]). Self-adjointness and a self-adjoint boundary value problem are also defined for a linear partial differential equation (see [[#References|[5]]], [[#References|[7]]]). | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Naimark, "Linear differential operators" , '''1–2''' , Harrap (1968) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , '''2''' , Interscience (1963)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Naimark, "Linear differential operators" , '''1–2''' , Harrap (1968) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , '''2''' , Interscience (1963)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== |
Latest revision as of 08:13, 6 June 2020
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
$$ ( p _ {0} y ^ {(} m) ) ^ {(} m) + ( p _ {1} y ^ {( m - 1) } ) ^ {( m - 1) } + \dots + p _ {m} y = 0 $$
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) |
Self-adjoint differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-adjoint_differential_equation&oldid=19159