{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# THE SCHRODINGER KE RNEL, ALIAS THE THETA FUNCTION " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "# 17 April 2000" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 663 "# NOTATION: t is in the fourth quadrant of t he complex plane. \+ # Imaginary t => heat kernel (at I*t); real t => Schrodinger kern el. \n# A small imaginary part in t also serves as a necessary cutoff \+ in the Schrodinger calculation. \+ # H = Laplacian on the circ le of circumference 2*Pi.\n# x = coordinate on the circle; other coord inate fixed at y = 0.\n# eigen(t,x,k) = kth term of the eigenfunction \+ expansion.\n# image(t,x,n) = nth term of the image expansion.\n# kerne l(t,x,f,N) = Nth partial sum of the bidirectional sequence f.\n\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "eigen := (t,x,k) -> evalf( ( 1/(2*Pi))*exp(I*k*x)*exp(-I*k^2*t) ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "image := (t,x,n) -> evalf( (1/sqrt(4*Pi*I*t))*exp(I*( x-2*Pi*n)^2/(4*t)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ke rnel := (t,x,f,N) -> f(t,x,0) + sum( f(t,x,k)+f(t,x,-k), k=1..(N-1)); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# Fix the time and vary x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "# Keep increasing N until the two curves \+ agree closely." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot(\{ R e(kernel(1-0.1*I,x,eigen,10)), Re(kernel(1-0.1*I,x,image,10)) \}, x=0. .Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# Try a smaller ti me." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot(\{ Re(kernel(0. 1-0.1*I,x,eigen,10)), Re(kernel(0.1-0.1*I,x,image,10)) \}, x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# Try a smaller imaginary cutoff." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(\{ Re(kernel(1-0.01*I,x,eigen, 20)), Re(kernel(1-0.01*I,x,image,10)) \}, x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "\nplot(\{ Re(kernel(0.1-0.01*I,x,eigen,20 )), Re(kernel(0.1-0.01*I,x,image,10)) \}, x=0..Pi);\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(\{ Re(kernel(1-.001*I,x,eigen, 40)), Re(kernel(1-.001*I,x,image,10)) \}, x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(\{ Re(kernel(0.1-.001*I,x,eigen,40)) , Re(kernel(0.1-.001*I,x,image,10)) \}, x=0..Pi);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "# Take a closeup." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot(\{ Re(kernel(1-.001*I,x,eigen,40)), Re(kernel (1-.001*I,x,image,10)) \}, x=1..1.2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot(\{ Re(kernel(0.1-.001*I,x,eigen,40)), Re(kernel( 0.1-.001*I,x,image,10)) \}, x=1..1.2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Get even closer to the real axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(\{ Re(kernel(1-.0001*I,x,eigen,200)) , Re(kernel(1-.0001*I,x,image,50)) \}, x=1..1.2);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 92 "plot(\{ Re(kernel(0.1-.0001*I,x,eigen,200)), Re(kernel(0.1-.0001*I,x,image,20)) \}, x=1..1.2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot(\{ Re(kernel(1-.00001*I,x,eigen,300)), Re(kernel(1-.00001*I,x,image,100)) \}, x=1..1.2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plot(\{ Re(kernel(0.1-.00001*I,x,eigen,400) ), Re(kernel(0.1-.00001*I,x,image,40)) \}, x=1..1.2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# Now fix x (as 0) and vary t." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " plot(\{ Re(kernel(t-0.1*I,0,eigen,10)), Re(kernel(t-0.1*I,0,image,10)) \}, t=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# Try a s maller cutoff." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(\{ Re(kernel(t-0.01*I,0 ,eigen,10)), Re(kernel(t-0.01*I,0,image,10)) \}, t=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# Take a close-up." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(\{ Re(kernel(t-0.01*I,0,eigen,50)), Re(kernel(t- 0.01*I,0,image,10)) \}, t=1..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot(\{ Re(kernel(t-0.001*I,0,eigen,50)), Re(kernel(t -0.001*I,0,image,10)) \}, t=1..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# Look even closer up.\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot(\{ Re(kernel(t-0.001*I,0,eigen,50)), Re(kerne l(t-0.001*I,0,image,50)) \}, t=1.5..1.6);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "plot(\{ Re(kernel(t-0.0001*I,0,eigen,100)), Re(kern el(t-0.0001*I,0,image,50)) \}, t=1.5..1.6);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "47" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }