[seqfan] Re: Time-Indepenent Schrodinger Equation discrete version

[Ron Hardin]

Is the situation correctly described?  (I'm remembering some mathematical facts 
at a distance of decades):

/tmp/czd
T(n,k)=Number of -k..k arrays of length n whose end-around centered second 
difference is a constant times that array

Some solutions for n=6 k=5, alternating array and its second difference  
..2...-2....3...-3....1...-1....0....0....3...-9....3..-12....1...-3....2...-6..
..2...-2...-1....1....0....0....4...-4....2...-6...-3...12...-3....9...-4...12..
..0....0...-4....4...-1....1....4...-4...-5...15....3..-12....2...-6....2...-6..
.-2....2...-3....3...-1....1....0....0....3...-9...-3...12....1...-3....2...-6..
.-2....2....1...-1....0....0...-4....4....2...-6....3..-12...-3....9...-4...12..
..0....0....4...-4....1...-1...-4....4...-5...15...-3...12....2...-6....2...-6..
Some solutions for n=12 k=5, alternating array and its second difference  
.-3....3...-3....9...-5...10...-4...12....4...-4...-3....3....2...-8....1...-3..
..0....0....0....0...-2....4....2...-6...-1....1...-2....2...-2....8....2...-6..
..3...-3....3...-9....5..-10....2...-6...-5....5....1...-1....2...-8...-3....9..
..3...-3...-3....9....2...-4...-4...12...-4....4....3...-3...-2....8....1...-3..
..0....0....0....0...-5...10....2...-6....1...-1....2...-2....2...-8....2...-6..
.-3....3....3...-9...-2....4....2...-6....5...-5...-1....1...-2....8...-3....9..
.-3....3...-3....9....5..-10...-4...12....4...-4...-3....3....2...-8....1...-3..
..0....0....0....0....2...-4....2...-6...-1....1...-2....2...-2....8....2...-6..
..3...-3....3...-9...-5...10....2...-6...-5....5....1...-1....2...-8...-3....9..
..3...-3...-3....9...-2....4...-4...12...-4....4....3...-3...-2....8....1...-3..
..0....0....0....0....5..-10....2...-6....1...-1....2...-2....2...-8....2...-6..
.-3....3....3...-9....2...-4....2...-6....5...-5...-1....1...-2....8...-3....9..

.Empirical formula: t(n,k) { #pseudocode
.        if (n modulo 12 = 0) return 10*k^2+14*k+1;
.        if (n modulo 12 = 1 || n modulo 12 = 5 || n modulo 12 = 7 || n modulo 
12 = 11) return 2*k+1;
.        if (n modulo 12 = 2 || n modulo 12 = 10) return 4*k+1;
.        if (n modulo 12 = 3 || n modulo 12 = 9) return 3*k^2+5*k+1;
.        if (n modulo 12 = 4 || n modulo 12 = 8) return 4*k^2+8*k+1;
.        if (n modulo 12 = 6) return 6*k^2+10*k+1;
.}

Table starts
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73
..9.23..43..69.101.139.183.233.289..351..419..493..573..659..751..849..953.1063
.13.33..61..97.141.193.253.321.397..481..573..673..781..897.1021.1153.1293.1441
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
.17.45..85.137.201.277.365.465.577..701..837..985.1145.1317.1501.1697.1905.2125
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
.13.33..61..97.141.193.253.321.397..481..573..673..781..897.1021.1153.1293.1441
..9.23..43..69.101.139.183.233.289..351..419..493..573..659..751..849..953.1063
..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
.25.69.133.217.321.445.589.753.937.1141.1365.1609.1873.2157.2461.2785.3129.3493
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35.....

The empirical formula apparently depends on eigenvectors of the system existing 
only with periods of 1, 2, 3, 4 and 6.
The eigenvectors are sampled sines and cosines, and so the formula depends on no 
sampling x(j) of sin(j*2Pi/n+offset) existing with exclusively rational ratios 
of the x(j) except for n=1,2,3,4 or 6.
The empirical formula dependency on k then describes how those existing cycles 
are populated with new values.


 rhhardin at mindspring.com
rhhardin at att.net (either)