Crack Paths 2012

(a) if sin(()\,, — l)7r) : 0, Eq.(2l) requires:

{[9211+ h 2 n+ ,u/f(aln+

+ [ — g 2+nh 2 n+ ,u/f(aln+

I O

This part of the solution is characterized by integer eigenvalues.

(b) if cos(()\n — l)7r) : 0, Eqs (l9),(20) and (21) require:

(23)

(a2"+ b2” + g2n + hZn): O

(24)

— 1)(cl/ln + gin) + +

+

I O

{[9171+ h l n+ ,u/f(a2n+

+ [ — g l+n h l n+ ,u/f(a2n+

I O

This part of the solution is characterized by fractional eigenvalues.

(a) Integer eigenvalues

Since g m and h mcan be written as functions of a1” and bln [6] through Eq. (13)

and (l5),the same expression used in the homogeneoucsase [1] can hold (to : 6 : 0).

F01‘ A 0 = 1 ,OIlC o b t a i n0s'0 : 0 1 0+ 010, IJ/fUQ : 2 0 2 0a n dO'IIpZQ: 2 0 1 0— 2 0 1 0 .

(b) Fractional eigenvalues

3 A”: -,

=0,1,2,... n + 2 n

Eqs(l4) and (16) allow one to express 9;” and hzn as functions of a2” and b2” [6].

Therefore, Eq.(23) gives:

_(H1k2>\n + #1 + Mk1 + M2>\n)a2 ( M M+ #2 + M1k2>m + Mk2) n

(26)

b2n :

Eqs(l3) and (15) allow one to express 91,, and h mas functions of ab, and b1n[6].

Therefore, Eq.(24) gives:

b1 : (—M1k2>\n + #1 + H2161 — M2>\n)a1 n (MM-Hi2+/J/lk2>\n+/J/lk2) n

(27)

and Eqs(26) and (27) in Eq.(25) give:

_al

(28)

a2n :

W

Substituting Eqs (26),(27) and (28) in Eqs (2) and (3) gives:

a y l fl z i fi: 2T2n2_+1I:2'n/3+ ( # 2 0 0 1+ — # 1 0 0 2+ l))a2n:I Sin 2 7 2 +3

it

(29)

2

#2 ‘till/t2

n 2 0

Txywziw : 27.2"?“ Izn‘l‘ 3(H2(k1—1)— ,ui(l — h2))alnI Sin 2n+3

it

(30)

H20

2

#2 + M k 2

673