“Self-Adjoint Operator”的概念、定义、习题解题方法及翻译-数学辞典

单词

Self-Adjoint Operator

释义

Self-Adjoint Operator

Given a differential equation

(1)

where

and

, the Adjoint Operator

is defined by

(2)

In order for the operator to be self-adjoint, i.e.,

(3)

the second terms in (1) and (2) must be equal, so

(4)

This also guarantees that the third terms are equal, since

(5)

so (2) becomes

			(6)
			(7)

The Legendre Differential Equation and the equation of Simple Harmonic Motion are self-adjoint, but theLaguerre Differential Equation and Hermite Differential Equation are not.

A nonself-adjoint second-order linear differential operator can always be transformed into a self-adjoint one usingSturm-Liouville Theory. In the special case , (7) gives

(8)

(9)

(10)

(11)

where

is a constant of integration.

A self-adjoint operator which satisfies the Boundary Conditions

(12)

is automatically a Hermitian Operator.

See also Adjoint Operator, Hermitian Operator, Sturm-Liouville Theory

References

Arfken, G. ``Self-Adjoint Differential Equations.'' §9.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 497-509, 1985.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。