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Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

Aug 13, 2020

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Page 1: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 2: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 3: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 4: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 5: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

12

Page 6: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 7: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

x xx = xc

E(x)

x = xc

T > 0

x x < xc

x > xc x = xc xx

x = xc x

Page 8: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

x

x

x

!

|!1,2" = |!1" ! |!2" = |!1"|!2"

|!1,2" = |!1" ! |!2" =!

i1

!

i2

ci1ci2 |"i1" ! |"i2",

|!!" =!

i!

ci! |"i!",

{|"i!"} #{|"i1" ! |"i2"}

H1 H2 H1 !H2

H1 !H2

|!1,2" =!

i1

!

i2

ci1i2 |"i1" ! |"i2"

H1!H2 {|"i1"} H1 {|"i2"}H2

n

|!" =!

i1

. . .!

i2

ci1...in |"i1" ! . . .! |"in"

ci1i2 = ci1ci2

|!S" = |0"A|1"B + |1"A|1"B|!E" = |0"A|1"B + |1"A|0"B

{|0"A, |1"A} {|0"B , |1"B}

|!A,B" = c1Ac1B |0"A|0"B + c1Ac2B |0"A|1"+ c2Ac1B |1"A|0"B + c2Ac2B |1"A|1"B

Page 9: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

|!S"c1A = c2A = c!1

2Bc1B = 0 c1A = c2A = c!1

2B= 1

c1B = 0 c1Ac1B = c2Ac2B = 0c1Ac2B = c2Ac1B = 1 ci1i2 = ci1ci2

c1A c2A c1B c2B

ci

ci

Page 10: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

HA ! HB HA HB

{|!Ai "} {|!B

i "}

|!" ="!

i

$i|!Ai "|!B

i "

$i

"!

i=1

$2i = 1

|!" {|!n"}

|!" =!

n

cn|!n"

|!" !

n

|cn|2 = 1,

#!|!" = 1

{|!Ai "} {|!B

i "} {|!Ai "|!B

i "}$i %

HA HB

$i %HA HB %

$i

%

Page 11: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

NN 1

|"" ="!

!1

$[1]!1|![1]

!1"|![2...N ]

!1"

N

{|i1"}i #1

{|![1]!1"} {|i1"} #1

{|i1"} #[1]i1!1

|"" ="!

i1,!1

#[1]i1!1

$[1]!1|i1"|![2...N ]

!1"

|![2...N ]!1 "

|![2...N ]!1

" ="!

!2

$[2]!2|![2]

!1!2"|![3...N ]

!2"

="!

i2,!2

#[2]i2!1!2

$[2]!2|i2"|![3...N ]

!2"

{|i2"}

|"" ="!

i1,i2,!1,!2

#[1]i1!1

$[1]!1

#[2]i2!1!2

$[2]!2|i1i2"|![3...N ]

!2"

#[2] #1 #2 |![2]!1!2"

#[n] #[1] #[N ] #[n]in!n!1!n

#[n]in!n!1!n

N $ 1N N

{|in"}

|"" =!

i1,i2...iN

ci1i2...iN |i1, i2 . . . iN!1, iN "

ci1i2...iN ="!

!1...!N!1

#[1]i1!1

$[1]!1

#[2]i2!1!2

$[2]!2

. . .#[N!1]iN!1!N!2

$[N!1]!N!1

#[N ]iN!N!1

{|in"}ci1i2...iN

#[n]in!n!1!n $[n]

!n

Page 12: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

m% n A

A = UDV †

U V U†U = I V †V = I IU V D

DA D

D U V

m% n A U m%m V n% n D m% n

#[n] #n #n!1 #[1] #1

i1 #[N ] #N!1 iN #n

%n in q in#[n] {|in"} {|in"}

{|in"} qn {|in"} q

{|i1, i2 . . . iN!1, iN "} qN qN

# q%n%n!1 #[1] #[N ]

q%n q%n!1 $n %n

$[n!1] %n!1 %n!1

$[n!1]1 $[n!1]

2 $[n!1]3 $[n]

x = $[n]!n |!n=x #[n]

%n!1 % q % %n

#n!1 (v, 1, 1) = #[n]in!n!1!n |!n!1=v,in=1,!n=1

#n (1, 1, v) = #[n]in!n!1!n |!n!1=1,in=1,!n=v

in

!

!2

$[1]!1

#[2]i2!1!2

$[2]!2

#[3]i3!2!3

$[3]!3

#

Page 13: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

& #1 i2 #[2]

#3 i3 #[3] q%1 q%3 #[2]

$[2] #[3] q%1 q%1 q%1 q%3 q%3 q%3 ##[2] #[3] #[2] #[3]

$[1] $[3]

#2 q%1 q%3

%2 $[2]

%2 & $[2]

#2 %2

#

e!iHt/!

qN

qN qN !H e!iHt

e!iHt

#

H =N!1!

l=1

H [l,l!1]

= H [1,2] + H [2,3] + H [3,4] + H [4,5] + H [5,6] + . . .

H = H [odd] + H [even]

H [odd] = H [1,2] + H [3,4] + H [5,6] + . . .

H [even] = H [2,3] + H [4,5] + H [6,7] + . . .

e!iHt

Page 14: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

H [odd] H [even]

e!iHt H

[A, B] &= 0 ' eA+B &= eAeB

e(A+B) ="e(A+B)/n

#n= lim

n"#

"e

An e

Bn

#n

n n = 1/'

lim#"0

"e(A+B)#

#1/#= lim

#"0

$e#Ae#B + O('2)

%1/#

' ( 1 ' ) 0

e#(A+B) * e#Ae#B + O('2)

O('2)'2 ' |'|( 1

O('2)' = it A + B = $H

T = 1n

e!iHT =$e!iH#

%T/# *$e#F e#G

%T"

iH [odd] iH [even] F G

e!iHT *$e#F e#G

%T" =

"e!#H[odd]

e!#H[even]#$

( = T#

H [odd] H [even]

G[n,n+1]

q %

Page 15: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

H

e!#H[n,n+1]G[n,n+1]

#[1]

qN qN

#

Page 16: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

{|in"} n

{|in"}

$[n!1] #[n] $[n] #[n+1] $[n+1]

& G[n,n+1]

& & G[n,n+1] &$

Page 17: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

& #[n] $[n] #[n+1]

$[n!1] #[n] $[n] #[n+1] $[n+1]

#$[n] $$[n] #$[n+1] #[n]

$[n!1] #n!1 #[n+1] $[n+1] #n+1 $

#n %n q%n!1 q%n+1

#

O(q!t) %n * %n+1 q

N $ 1$[n] qn q(N n) %n q n = 1

qN/2 n = N/2 %n = q n = N

%n

%n % # % q %$ % % $ %

# %

%n

$[n] %

%%

%

%q$ (

%%

O(qN )

ci1i2...iN qN N q %N % q %

N $ 1 $ % O(Nq%2)+ Nq%2 + N%$ %

NN N

$

Page 18: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

O(q%3) %

%

% $

#

%

$

S()) = $Tr() log ))

= $!

i

$2i log $2

i

$2i )

$i

$%

%# $

$

# $#

|1|# # $

# $

%

Page 19: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

O('2)'

$t$t $t

$t'

''

$t

k k + NN

NO(qN )

k kk

q2k%k k NNk q2k%k

N/k & k # (k + 1) $N/k &

& k # (k + 1) $ %3&k

i=2 qi + 2q%

$ %&k

i=2 qi!1 + %3&k

i=2 qi k = 2 %&k

i=2 qi!1 = q%

%3&k

i=2 qi = q2%3 k = 2N/k

kk i = 2 k N/k

kk

%&k

i=2 qi!1 %3&k

i=2 qi i = 2 kk = 1 #$# k = 1

H N

k = 2

Page 20: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

NN

'

tf $ ti = i,

tsE(ts)

ts i,ts

ts

tits

ti tsts

''

''

' '

' ( 1

'b

Page 21: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

' = 'b

' = 'b

' 'a 'b 'c

'a 'c

' = 'c ' = 'a

'a 'b ' = 'b 'b 'c

' = 'c

' t!#

t!# '

'

Page 22: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

#"|H|"" - E0

E0

|""

O(Nq%2)

Page 23: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

#[1]i1!1 $[1]

!1#[2]i2!1!2$

[2]!2 . . .#[N!1]iN!1

!N!2 $[N!1]!N!1 #[N ]iN

!N!1

#[1]i1!1 $[1]

!1#[2]i2!1!2$

[2]!2 . . .#[N!1]iN!1

!N!2 $[N!1]!N!1 #[N ]iN

!N!1

|"" #"|

#

Page 24: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

$$% $

ci1i2...iN ci1i2...iN

qN q N {|in"}in q in|in"

in #

#A[2]" = #"|A[2]|""

A

A|in"

A n

n

Page 25: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

#$ n # 5(n$1)$2

5(N $ n)$ 2

A

An in n

#n n#n %

n

A{|in"} A A

A

#A[n]" =#"|A[n]|""#"|""

Page 26: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

A

B

%n b 4(N $ 1) + N + b

b in in+1 # $B

B

B

Page 27: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

O(qn%3)O(q%3)

O(q2%3) NO(qN%3) N

Mx(h)

Mx(h) =1N

N!

i=1

#"|*[i]x |""

#"|""

$%

$ #

# $

$

$#

$

$

Page 28: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

S($[n]) = $!

i

$2i log $2

i

$i i $[n] $[n]

S()))

$[n] S()) - 0 S()) = 0

E1($[n]) = $ log $2max

= $2 log $max

$max $[n] eln(x) . loge(x) log(x) . loge(x)

log 1 = 0

- 0

0 + $i < 1 /i log $i < 0 /i| log $i| -| log $max| /i

1 + | log $i|| log $max|

=log $i

log $max

log $i < 0 /i

S($[n])E1($[n])

=$

&i $2

i log $2i

$ log $2max

=$2

&i $2

i log $i

$2 log $max

Page 29: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

=&

i $2i log $i

log $max

=&

i $2i log $i&

i $2i log $max

$i

| log $i| -| log $max|

S($[n])E1($[n])

- 1

S($[n]) - E1($[n]) S($[n]) = E1($[n]) $i = $max

/i $i $i

$i = 1&"/i

S($[n]) = E1($[n])0 + $i < 1 /i $i = 1

0$i = 1 0$i = 1$max = 1 $i = 0 i

S($[n]) = $!

$2i log $2

i

= $2 log $max

= E1($[n])

S($[n]) = E1($[n]) = 0 S($[n]) > E1($[n])S($[n]) = E1($[n]) = 0 S($[n]) = E1($[n]) &= 0

S())N N N * ,N ),

S - E1

SE1

SE1

= 2

Page 30: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

S = $k!

i

Pi log Pi

Pi i kPi k = kB

)

S()) = $Tr() log ))

)

) =!

i

pi|!i"#!i|

pi |!i")

) =!

k

Pk|k"#k|

S()) = $!

i

#i|) log())|i"

= $!

i

#i|!

k

Pk|k"#k| log

'!

k

Pk|k"#k|(|i"

{|k"}

S()) = $!

i

Pi log Pi

Page 31: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

|!" {|n"}

|!" =!

n

cn|n"

) =!

n,m

c%ncm|n"#m|

#!|A|!" =!

m

!

n

c%mcn#m|A|n"

#m|)|n" = cmc%n

#!|A|!" =!

m

!

n

#n|)|m"#m|A|n"

=!

n

#n|)A|n"

= Tr()A)

$2i )

S()) = $!

i

$2i log $2

i

$ log Pmax

$

Page 32: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% qN h

h

12

12

H =!

'i,j(

J [i,j]*[i]x *[j]

x + h!

i

K [i]*[i]z

*z *x

*x =)

0 11 0

**y =

)0 $ii 0

**z =

)1 00 $1

*

q = 2

{|in"} =+)

10

*,

)01

*,

*z

#i, j"i j j

j = i+1

#i, j"

J [i,j]

J [i,j] = J J J

Page 33: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

J

J < 0 J > 0

J = $1

J [i,j] K [i] J [i,j]

K [i] = K*x *z K = $1 K = J = $1

H = $!

i

*[i]x *[i+1]

x $ h

2

!

i

"*[i]

z + *[i+1]z

#

H [i,i+1] = $*[i]x *[i+1]

x $ h

2

"*[i]

z + *[i+1]z

#

h&

i *[i]z = h

2

&i

"*[i]

z + *[i+1]z

#

!

i

*[i]z =

"*[1]

z + *[2]z + *[3]

z + . . .#

=12

"*[1]

z +"*[1]

z + *[2]z

#+

"*[2]

z + *[3]z

#+ *[3]

z + . . .#

=12

N!1!

i=1

"*[i]

z + *[i+1]z

#+

12*[1]

z +12*[N ]

z

*[1]z *[N ]

z

N ),

H [1,2] =-$*[i]

x *[i+1]x $ h

2

"*[i]

z + *[i+1]z

#.

i=1

+h

2*[1]

z

= $*[1]x *[2]

x $ h

2

"*[1]

z + *[2]z

#+

h

2*[1]

z

= $*[1]x *[2]

x $ h

)*[1]

z +12*[2]

z

*

H [N!1,N ] = $*[N!1]x *[N ]

x $ h

)12*[N!1]

z + *[N ]z

*

*x

*z hh h = 0 h = 2

h = 1 h = 1h = 1

J < 0 h = 0

h = 0

Page 34: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h < hc h > hc

% = 10 % = 20 % = 30 % = 4% - 10

N %

N # % q % N $ 1 %#

i

% % $# #

$

h = 1

h = 1 % = 30

Page 35: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 30 %

% = 30h = 1

#$

%% = 20 % = 10

h = 1 % = 10

Page 36: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

%

'b

'c 'a 'b 'a

'b 'c

Page 37: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 0.7 % = 30

Page 38: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 0.7 % = 30

Page 39: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 0 h = 1

h = 0.7 h = 1.4

Page 40: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 1.4 % = 30

% = 30 h = 0.7 h = 1.5h = 0.9

h = 0.7

h = 1h = 1

h = 1h = 1 h = 0

Page 41: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 0 |#*x"| = 1h = 0

|#*x"|

|#*x"| % = 20

% % = 10

h = 1%

Page 42: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

hh = 1

h h = 1 h = 0.9 h = 0.8 |#*x"||#*x"| = 1

h = 1h = 1 h = 1

#*x" = 0

% = 30 h = 0

h = 0 h = 0|#*x"| = 1 h = 0.5

h = 0.7

Mx(h) =1N

N!

i=1

#"|*[i]x |""

#"|""

h ) 1

Page 43: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

n = 50 h > 1#*x" h > 1

#*x"

h < hc h > hc

h = 1 h#*x"

h < 1 #*x" h > 1#*x"

h > 1h < 1

h < 1

G G|!0" =ei%|!0" [H,G] = 0G|!0" &= ei%|!0"G|!0(h)" = |!0(h)" h > 1 G =

&i *[i]

z

#*x" &= 0G G

G

hh h > hc

h hh h > hc h < hc

h h

Page 44: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 20

h = 1 h = 1.1% %

#*x"h < 1

h = 1h > 1

h = 0J J J

Page 45: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 10 h = 1

hh

h = 1 h = 1h = 1 h = 0.9 h = 1

Page 46: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h % = 30

h = 0.9 h = 1

h

% = 30 h h = 0 h = 2

Page 47: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h h = 1h = 0.9

h = 1.9 h = 2

h = 0 h = 2 h = 1 h = 0.7

ht($'1) t($'2)

t($'1) hh = 1.1 h = 2 t($'1) h = 1

t($'1)t($'2)

t($'2) h ) 1h ) 1

h ) 1

h h = 1

% = 30h = 0 h = 0.9 h = 1.1 h = 2

h ) 1 h = 1

h = 1

Page 48: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h < 1 h ) 1

h ) 1n = 1 n = N

n = 1 n = Nh

h - 1h > 1

h > 1h

% = 30 h h = 0h = 2

h

Page 49: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h % = 30

% = 30h = 0 h = 0.9 h = 1.1 h = 2

Page 50: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h

h h

SE1

h E1S

h

% = 30

h = 1SE1

(h)

E1S (h)

Page 51: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 30

hh = 0

E1S h

S E1

h = 1

Page 52: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 30

h = 0 h = 0.9 h = 1.1 h = 2

E1S h = 1

SE1* 2

SE1

= 2 N = 100SE1

E1S

%

E1S h

h = 1E1S

Page 53: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

|1" =

/

0100

1

2 |2" =

/

0010

1

2 |3" =

/

0001

1

2

H =!

'i,j(

J [i,j]"M [i]

x M [j]2x + M [i]2

x M [j]x

#+ h

!

i

K [i]M [i]z

Mx Mz

Mz =

/

02 0 00 $1 00 0 $1

1

2 Mx =

/

00 1 00 0 11 0 0

1

2 M2x = (Mx)2 =

/

00 0 11 0 00 1 0

1

2

#*x"

Mx + M2x =

/

00 1 11 0 11 1 0

1

2

Mz *z

*x12

12 #Mx+M2

x"

J = $1J = $1

H = $!

i

"M [i]

x M [i+1]2x + M [i]2

x M [i+1]x

#$ h

2

!

i

"M [i]

z + M [i+1]z

#

H [i,i+1] = $"M [i]

x M [i+1]2x + M [i]2

x M [i+1]x

#$ h

2

"M [i]

z + M [i+1]z

#

H [1,2] = $"M [1]

x M [2]2x + M [1]2

x M [2]x

#$ h

2

"2M [1]

z + M [2]z

#

H [N!1,N ] = $"M [N!1]

x M [N ]2x + M [N!1]2

x M [N ]x

#$ h

2

"M [N!1]

z + 2M [N ]z

#

J = $1 h = 1

h h = 1

Page 54: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 0.5 h = 0.7 % = 25

h = 0.9 h = 1.1 % = 25

h = 0.5 h = 0.7h = 1 h = 1.1

h = 0.5h = 0.5 h = 0.7

h = 0.7h = 0.7

Page 55: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 0.5 h = 0.9h = 1.1

h ) 1h = 1 h = 1

h = 0.5 h = 0.9h = 0.5 h = 0.9

h ) 1,h > 1

12 #Mx + M2

x"

12 #Mx + M2

x" % = 20

h h = 0 h = 2h = 1 h = 2

h = 1

Page 56: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h h = 1h < 1

|1|h h = 1

% = 25

h = 1 h = 1.1

h > 1 h ) 1

Page 57: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 25

h = 1h = 0

h = 1h > 1

h

Page 58: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 25

h = 1 h = 1.1

h = 0.9 h > 1h > 1

h = 1h ) 2

% = 25 h = 0 h = 0.9 h = 1 h = 2

Page 59: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 25

% = 25

Page 60: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

% = 25 h = 0 h = 0.9 h = 1 h = 2

S E1

SE1

E1S % = 25

Page 61: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

E1S = 2

%

% = 30

h = 1 SE1

= 2

h = 1

% = 25 h = 0 h = 0.9 h = 1 h = 2

Page 62: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h < 1h = 1

h > 1h > 1

i

' = 0.01

h = 2h = 1.8

h > 1 |#*x"|

h = 1.8h = 1.9 % = 20

Page 63: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 2

10!8

h = 2

h = 1.8 h = 1.9 % = 20

h = 2 h = 1.8 h = 1.9h = hg h > hg

h < hg

h ( hg

%%

h = 2 h = 1.8 h = 1.9 % = 20

Page 64: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h = 1.8h = 1.9 % = 25

h = 1.8 h = 1.9 % = 25

Page 65: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

%%

Page 66: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

h

Page 67: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 68: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 69: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 70: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 71: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 72: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 73: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 74: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 75: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 76: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 77: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 78: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 79: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 80: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 81: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 82: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 83: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]
Page 84: Pages 1 to 5 of Thesis Contents - Roman Orus · θ α 1 i 2 Γ[2] α 3 i 3 Γ[3] qχ 1 qχ 3 ˜Γ[2] λ[2] Γ˜[3] qχ 1 qχ 1 qχ 1 qχ 3 qχ 3 qχ 3 Γ Γ [2]Γ[3] ˜Γ Γ˜[3]

A1

A†AXX = Y X =

1Y

#[1] #1 i1 #[N ] #N!1 iN #[n]

in #n #n!1,#[1] #[N ]

# #[1] #[N ]

#$# &q%n!1 $[n] #

O(x) x x

' = ite!#H ' = t

e!#H

O('2)' O('2)

'

J J