特征算子谱表示与特征展开的研究

doi:10.11959/j.issn.1000-436x.2020096

Abstract

Abstract:

The spectral representation and expansion based on eigen-operators were deeply studied.The relations between Green’s function of characteristic differential equation and Hermitian differential and integral operators were given.The inverse relations between Hermitian differential operator and Hermitian integral operator were also studied.The spectral representation of Hermitian differential operators was given.It was also shown that the S-L eigen-equation cannot be used to realize the spectral representations of infinite dimensions in a finite interval.The method is much simpler and clearer than that of Neumann,and has advantages.The eigen-expansion (eigen-decomposition) of the Hermitian integral operator was given,which has the advantages of theoretical generality and comprehensiveness.The incorrect discussion in Wang et al<sup>[2]</sup>was correct that it is used to study the representation of characteristic spectrum.The physical and geometric meanings of naming for the long spherical wave function in optimal eigen-expansion were given.

Key words: Green’s function, S-L differential equation, eigen-operator, spectral representation, eigen-expansion, long spherical wave function

CLC Number:

TN911.7

Hongyu WANG,Tianshuang QIU. Study on spectral representation and eigenexpansion based on eigen-operators[J]. Journal on Communications, 2020, 41(5): 1-8.

Figures/Tables 4

微分方程	区间	边界条件	特征值	标准正交函数系
调和微分方程
$p (x) = 1, q (x) = 0, ρ (x) = 1$	(0,l)	y (0)=0	${(\frac{n π}{l})}^{2}$	$\sqrt{\frac{2}{l}} \sin \frac{n π}{l} x$
$\frac{d^{2} y}{d x^{2}} + λ y = 0$		y(l)=0	n=1,2,3,"
勒让德微分方程
$p (x) = (1 - x^{2}), q (x) = 0, ρ (x) = 1$	(-1,1)	y(±1)有界	n (n+1)	$\sqrt{n + \frac{1}{2}} p_{n} (x)$
$\frac{d}{d x} [(1 - x^{2}) \frac{d y}{d x}] + λ y = 0$			n=0,1,2,"
奇点x=±1
连带勒让德微分方程
$p (x) = (1 - x^{2}), q (x) = \frac{m^{2}}{1 - x^{2}}, ρ (x) = 1$	(-1,1)	y(±1)有界	n (n+1)	$\sqrt{n + \frac{1}{2}} p_{n} (x)$
$\frac{d}{d x} [(1 - x^{2}) \frac{d y}{d x}] + (λ - \frac{m^{2}}{1 - x^{2}}) y = 0$			n=0,1,2,"
奇点x=±1
贝塞尔微分方程
$p (r) = r, q (r) = \frac{n^{2}}{r}, ρ (r) = r$	(0,l)	y (0)有界	$k_{i}^{n}$	$\frac{\sqrt{2} r}{l J_{n + 1} (k_{i}^{n} l)} J_{n} (k_{i}^{n} r)$
$\frac{d}{d r} (r \frac{d y}{d r}) + (k^{2} r - \frac{n^{2}}{r}) y = 0$		y (kl)=0	i= 1,2,3,"
奇点r=0
埃尔米特微分方程
$p (x) = e^{- x^{2}}, q (x) = e^{- x^{2}}, ρ (x) = e^{- x^{2}},$	(-∞,∞)	y (±∞)与∞的有限次密同阶	2n+1,	$\frac{1}{\sqrt{2^{n} n! \sqrt{π}}} e^{- \frac{x^{2}}{2}} H_{n} (x)$
$\frac{d}{d x} (e^{- x^{2}} \frac{d}{d y}) + (\frac{λ}{α} e^{- x^{2}} - e^{- x^{2}}) y = 0$			n=0,1,2,"

References 10

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