1549301742-The_Theory_of_Difference_Schemes__Samarskii

Shrodinger time-dependent equation 353

2. Three-layer schemes. Of special interest is the three-layer scheme with
   weights

(3)

Yo= YN = 0, y O ( X) = u 0 ( X) ,

where f; = yj+i, fJ = yj-l, y = yj and O" = 0" 0 is a real number. For any

O" the scheme is of second-approximation order with 1/; = 0( h^2 + r^2 ). For

later use, we seek a particular solution in the form yf (xs) =ck qf Xk(x 8 ).
Substituting this expression into equation (3), recalling the definition of
eigenfunctions AXk = ->.k Xk and omitting the subscript k of qk and >.k>
we eventually get the quadratic equation for q:

( i + 2 μ ()") q^2 - 2 μ (2 ()" - 1) q + 2 μ q - i = 0 ,

with discriminant

-=μ D 2(2 ()"- 1)2 - 1 4 2 2 - μ ()" = ( 1-4 ) 2 ()" μ -1.

4

Plain calculations show that

1 1

D < 0 for ()" >

4 4 T^2 6.^2

and give the roots of the quadratic equation

q (I '^2 ) - ~~~~~-(2 ()" - 1) μ ± i Jl + 4μ2 ()"2 - (2 ()" - 1)2 μ2

• i+2μ0" ,

Under the constraint

1

()" > -

4

1 4

6. = h2 ,

I q( I, 2) I = 1.

the particular solutions Yi do not increase wi tb increasing j:

II yf +

1

II < II Yi II·

Being concerned with q~^1 '^2 > = e±i'Pk, we look for the general solution of the
problem in view as a sum

N-l

yj = L ( o:k cos J'Pk + /]k sin J'Pk) Xk(x)

k=l