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{XPPMATH 20 ",(*&I\"aG6\"\"\"\")I\" xGF%\"\"%F&F&*&I\"bGF%F&)F(\"\"#F&!\"\"I\"cGF%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a:=1/32; b:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Vp:=diff(V,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&#\"\"\"\"\")F%)I\"xG6\"\"\"$F%F%*&\"\"#F%F(F%!\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sol:=solve(Vp,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!\"\"%!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x0:=sol[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "c:=solve(eval(V,x=x0 ),c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(V,x=-2*x0..2*x0);" }}{PARA 13 "" 1 "" {TEXT 218 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$!#!)! \"\"$\"$?(!\"\"7$$!1MLL$Q6G\"z!#:$\"184V\"e!z*y'!#97$$!1nmmmFiDy!#:$\" 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expGF%6#,$*&$\"\"&!\"\"\"\"\"*&)I&alphaGF%\"\"#F2)I\"xGF%F6F2F2F1F2-I \"HGF%6$F$*&F5F2F8F2F2F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "NI:=n->1/sqrt(int(phi(n)^2,x=-infinity..infinity)); # we could g et this from a formula!" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F %6$I)operatorGF%I&arrowGF%F%*$-I%sqrtG6$%*protectedGI(_syslibGF%6#-I$i ntGF%6$*$)-I$phiGF%F#\"\"#\"\"\"/I\"xGF%;,$I)infinityGF-!\"\"F=F>F%F%F %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "NI(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"/seS$*)QA%!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N:=20; # choosing a matrix size" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "NI_tab le:=[seq(NI(j),j=0..N)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "77$\"/=U\"f \"[tf!#9$\"/seS$*)QA%F%$\"/PHqY%>6#F%$\"/n,7!y>i)!#:$\"/#p(f&H$[IF,$\" /%)onWmR'*!#;$\"/)ewVJFy#F1$\"/XP*firV(!#<$\"/O%)\\1Hf=F6$\"/Q+*3!R#Q% !#=$\"/O$H9A$*z*!#>$\"/Gn+DA*3#F>$\"/<\\rvgkU!#?$\"/\">![Nej$)!#@$\"/z 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}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "od: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "En:=convert(Eigenvalues(HM), list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "77$\"38_H\\(oL/')*!#=$\"3I!pC& z4ns)*F%$\"3^\"=$\\$fFI!H!#<$\"3jJ.c2(f+$HF*$\"3&\\73_PImx%F*$\"3ee()e sH*[![F*$\"3:%fhSVzla'F*$\"3G%pyp'>w[lF*$\"3(=7OaZ3\"GyF*$\"3Q<`_!Q=oS )F*$\"3&>;ThT+44*F*$\"38RoiDSxw5!#;$\"3-,Bbuycm6F=$\"3*>GU5Zr\"H8F=$\" 3\\1&RBQwpj\"F=$\"3J$>7w[5`w#F=$\"3UW)*o*zsYV$F=$\"3dL5eF^m5fF=$\"3k'4 %>zG7HqF=$\"3iWA.wSTW7!#:$\"3Yc#o+9o_U\"FN" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 62 "# some of the low-lying eigenvalue estimates come i n 'pairs' ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Evec:=Eigenv ectors(HM);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q%EvecF'/%'italicGQ%trueF'/%,m athvariantGQ'italicF'-I#moGF$6-Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/% *separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimit 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