id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ap-4589	Ohya, Satoshi	ALGEBRAIC DESCRIPTION OF SHAPE INVARIANCE REVISITED	2017	8	.pdf	application/pdf	4619	258	69	d dx − j x + g j . For fixed j ∈ { 1 2 , 3 2 , · · · , νmax}, the energy eigenvalues and eigenfunctions are given by En = −(j + n)2 − g2 (j + n)2 , n ∈ {0, 1, · · · , N}, (4.11) and ψEn,j(x) ∝ A−(j)A−(j + 1) · · ·A−(j + n− 1)ψEn,j+n(x), (4.12) where N = max{n ∈ Z≥0 : j + n < √ g} = νmax − j and ψEn,j+n(x) ∝ (sinh x)j+n exp(− g j+nx).	cache/ap-4589.pdf	txt/ap-4589.txt