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PureMathematics
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,2022,12(8),1381-1391
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128151
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UniformStationaryMeasureof
Space-InhomogeneousOne-Dimensional
Three-StateQuantumWalks
PengYe,LixiaZhang,CaishiWang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Jul.19
th
,2022;accepted:Aug.19
th
,2022;published:Aug.31
st
,2022
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[J].
n
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,2022,12(8):
1381-1391.DOI:10.12677/pm.2022.128151
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Abstract
Inthispaper,weconsidertheuniformstationarymeasureofspace-inhomogeneous
threestatequantumwalksonthelineandcycles.Firstly,theeigenvalueproblemis
solvedbytransfermatrixandthecorrespondinguniformstationarymeasureisgiven
ontheline.Inaddition,wegivetheperiodicrepresentationoftheevolutionmatrix
underthemodelontheline.Then,weshowtheuniformstationarymeasureofthe
cyclesbyrestrictingthepositionspacetothecycles.
Keywords
Quantum,Inhomogeneous,StationaryMeasure,UniformMeasure
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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O
(1)
Ψ
R
(1)
,
Ψ
L
(2)
Ψ
O
(2)
Ψ
R
(2)
,
···
T
=
···
,
0
0
0
,
0
0
0
,
α
β
γ
,
0
0
0
,
0
0
0
,
···
T
Ù
¥
|
α
|
2
+
|
β
|
2
+
|
γ
|
2
= 1.
Ú
\
A
Š
¯
K
U
(
S
)
Ψ =
λ
Ψ
,
(
λ
∈
C,
|
λ
|
= 1).
Ï
•
U
(
S
)
´
j
Ý
,
¤
±
k
φ
(Ψ)
∈
Θ
s
.
Ú
n
2.6[11]
U
y
´
¹
y
ë
ê
š
à
g
n
þ
f
i
j
ü
z
Ý
¤
8
Ü
, Ψ
n
(
x
)=
(Ψ
L
n
(
x
)
,
Ψ
O
n
(
x
)
,
Ψ
R
n
(
x
))
T
´
V
Ç
Å
Ì
.
e
‰
½
Ð
©
Ψ(0),
ƒ
A
A
Š
¯
K
U
(
S
)
Ψ=
λ
Ψ
,
K
k
A
•
þ
Ψ(
x
) =
Π
x
y
=1
D
+
y
Ψ(0)(
x
≥
1)
,
Ψ(0)(
x
= 0)
,
Π
x
y
=
−
1
D
−
y
Ψ(0)(
x
≤−
1)
,
DOI:10.12677/pm.2022.1281511385
n
Ø
ê
Æ
“
+
=
£
Ý
D
±
y
©
OL
«
•
D
+
y
=
d
+
11
d
+
12
d
+
13
d
+
21
d
+
22
d
+
23
d
+
31
d
+
32
d
+
33
Ú
D
−
y
=
d
−
11
d
−
12
d
−
13
d
−
21
d
−
22
d
−
23
d
−
31
d
−
32
d
−
33
.
Ù
¥
d
+
11
=
(
λ
−
e
y
)(
λ
2
−
g
y
−
1
c
y
)
−
g
y
−
1
b
y
f
y
λ
{
a
y
(
λ
−
e
y
)+
b
y
d
y
}
,d
+
12
=
−
h
y
−
1
{
b
y
f
y
+
c
y
(
λ
−
e
y
)
}
λ
{
a
y
(
λ
−
e
y
)+
b
y
d
y
}
d
+
13
=
−
l
y
−
1
{
b
y
f
y
+
c
y
(
λ
−
e
y
)
}
λ
{
a
y
(
λ
−
e
y
)+
b
y
d
y
}
,d
+
21
=
λ
2
d
y
+
g
y
−
1
(
a
y
f
y
−
c
y
d
y
)
λ
{
a
y
(
λ
−
e
y
)+
b
y
d
y
}
d
+
22
=
h
y
−
1
(
a
y
f
y
−
c
y
d
y
)
λ
{
a
y
(
λ
−
e
y
)+
b
y
d
y
}
,d
+
23
=
l
y
−
1
(
a
y
f
y
−
c
y
d
y
)
λ
{
a
y
(
λ
−
e
y
)+
b
y
d
y
}
d
+
31
=
g
y
−
1
λ
,d
+
32
=
h
y
−
1
λ
,d
+
33
=
l
y
−
1
λ
,
d
−
11
=
a
y
+1
λ
,d
−
12
=
b
y
+1
λ
,d
−
13
=
c
y
+1
λ
d
−
21
=
−
a
y
+1
(
f
y
g
y
−
l
y
d
y
)
λ
{
l
y
(
λ
−
e
y
)+
h
y
f
y
}
,d
−
22
=
−
b
y
+1
(
f
y
g
y
−
l
y
d
y
)
λ
{
l
y
(
λ
−
e
y
)+
h
y
f
y
}
d
−
23
=
λ
2
f
y
−
c
y
+1
(
g
y
f
y
−
l
y
d
y
)
λ
{
l
y
(
λ
−
e
y
)+
h
y
f
y
}
,d
−
31
=
−
a
y
+1
{
h
y
d
y
+
g
y
(
λ
−
e
y
)
}
λ
{
l
y
(
λ
−
e
y
)+
h
y
f
y
}
d
−
32
=
−
b
y
+1
{
h
y
d
y
+
g
y
(
λ
−
e
y
)
}
λ
{
l
y
(
λ
−
e
y
)+
h
y
f
y
}
,d
−
33
=
−
(
λ
−
e
y
)(
λ
2
−
g
y
c
y
+1
)
−
h
y
c
y
+1
d
y
λ
{
l
y
(
λ
−
e
y
)+
h
y
f
y
}
.
3.
Ì
‡
(
J
9
Ù
y
²
!
Ì
‡
ï
Ä
˜
m
š
à
g
˜
‘
n
þ
f
i
.
,
Ù
ü
z
Ý
X
e
U
x
=
cos
θ
0
e
ω
x
i
sin
θ
0
e
ω
x
i
0
e
−
ω
x
i
sin
θ
0
−
cos
θ
(
ω
x
∈
[0
,
2
π
)
,θ
∈
(0
,
2
π
))
.
Ù
¥
x
L
«
˜
,
A
O
•
Ä
é
Ý
¥
ë
ê
ω
x
,
•
3
φ
∈
[0
,
2
π
),
÷
v
^
‡
ω
x
−
ω
x
−
1
= 2
φ,x
∈
Z
.
DOI:10.12677/pm.2022.1281511386
n
Ø
ê
Æ
“
+
3.1.
†
‚
þ
i
˜
—
²
-
ÿ
Ý
9
ü
z
Ý
±
Ï
5
L
«
¼
T
.
A
Š
9
ƒ
A
A
•
þ
,
½
n
3.1.1
e
ü
z
Ý
÷
v
U
(
s
)
Ψ=
λ
Ψ,
-
λ
=
e
φi
,
l
A
Š
¯
K
C
†
•
U
(
s
)
Ψ=
e
φi
Ψ,
Ù
¥
Ψ(
x
) =
Π
x
y
=1
D
+
y
Ψ(0)(
x
≥
1)
,
Ψ(0)(
x
= 0)
,
Π
x
y
=
−
1
D
−
y
Ψ(0)(
x
≤−
1)
,
=
£
Ý
D
±
y
•
D
+
x
=
e
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
i
sin
θ
0
−
e
−
φi
cos
θ
Ú
D
−
x
=
e
−
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
+1
i
sin
θ
0
−
e
φi
cos
θ
,
Ù
¥
α
x
=
ω
x
−
1
+
φ
=
ω
x
−
φ
,
ω
x
∈
[0
,
2
π
)
,θ
∈
(0
,
2
π
)
.
q
φ
(Φ)
∈
Θ
u
∩
Θ,
K3
†
‚
þ
k
˜
—
²
-
ÿ
Ý
.
y
²
:
O
Ž
Ù
=
£
Ý
•
D
+
x
=
λ
2
−
e
−
ω
x
−
1
i
sin
2
θe
ω
x
i
λ
cos
θ
0
e
ω
x
i
sin
θ
λ
000
e
−
ω
x
−
1
i
sin
θ
λ
0
−
cosθ
λ
=
e
2
φi
−
e
(
ω
x
−
ω
x
−
1
)
i
sin
2
θ
e
φi
cos
θ
0
e
ω
x
i
sin
θ
e
φi
000
e
−
ω
x
−
1
i
sin
θ
e
φi
0
−
cosθ
e
φi
=
e
2
φi
−
e
2
φi
sin
2
θ
e
φi
cos
θ
0
e
(
ω
x
−
φ
)
i
sin
θ
000
e
(
φ
−
ω
x
−
1
)
i
sin
θ
0
−
e
−
φi
cos
θ
=
e
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
i
sin
θ
0
−
e
−
φi
cos
θ
.
DOI:10.12677/pm.2022.1281511387
n
Ø
ê
Æ
“
+
a
q
,
D
−
x
=
cos
θ
λ
0
e
ω
x
i
sin
θ
λ
000
e
−
ω
x
i
sin
θ
λ
0
λ
2
−
e
(
ω
x
+1
−
ω
x
)
i
sin
2
θ
−
λ
cos
θ
=
cos
θ
e
φi
0
e
ω
x
i
sin
θ
e
φi
000
e
−
ω
x
i
sin
θ
e
φi
0
e
2
φi
−
e
(
ω
x
+1
−
ω
x
)
i
sin
2
θ
−
e
φi
cos
θ
=
cos
θ
e
φi
0
e
(
ω
x
−
φ
)
i
sin
θ
000
e
−
(
ω
x
+
φ
)
i
sin
θ
0
e
2
φi
cos
2
θ
−
e
φi
cos
θ
=
e
−
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
+1
i
sin
θ
0
−
e
φi
cos
θ
.
N
´
y
D
+
x
,
D
−
x
´
j
Ý
,
d
Ú
n
2.6,Ψ(
x
)
‰
ê
†
˜
x
Ã
'
.
¤
±
φ
(
x
)
∈
Θ
u
.
Ï
d
,
†
‚
þ
T
þ
f
i
ä
k
˜
—
²
-
ÿ
Ý
.
e
¡
•
Ä
†
‚
þ
ü
z
Ý
S
{
U
x
,x
∈
Z
}
±
Ï
5
.
½
Â
3.1.2
é
u
þ
f
i
.
ü
z
Ý
S
{
U
x
,x
∈
Z
}
,
e
U
x
+
n
=U
x
(
n
∈
N
)
,
K
¡
S
{
U
x
}
ä
k
N
±
Ï
,
Ä
K
v
k
±
Ï
.
í
Ø
3.1.3
U
x
=
cos
θ
0
e
ω
x
i
0
e
ω
x
i
0
e
−
ω
x
i
sin
θ
0
−
cos
θ
(
ω
x
∈
[0
,
2
π
)
,θ
∈
(0
,
2
π
))
,
Ù
¥
ω
x
−
ω
x
−
1
=
2
π
N
.
l
k
ω
x
+
N
= 2
π
+
ω
x
.
¤
±
U
x
+
N
=
cos
θ
0
e
ω
x
+
N
i
0
e
ω
x
+
N
i
0
e
−
ω
x
+
N
i
sin
θ
0
−
cos
θ
=
cos
θ
0
e
ω
x
i
0
e
ω
x
i
0
e
−
ω
x
i
sin
θ
0
−
cos
θ
= U
x
.
K
ü
z
Ý
S
{
U
x
}
±
N
•
±
Ï
.
3.2.
‚
þ
i
˜
—
²
-
ÿ
Ý
‚
C
2
N
þ
þ
f
i
Ï
L
ü
z
Ý
U
x
?
1ü
z
.
3
†
‚
þ
þ
f
i
ü
z
d
(2.1)
ª
¤
û
½
.
‚
þ
ž
m
ü
z
Ž
f
a
q
u
U
(
s
)
•
U
(
s
)
c
.
Ù
ü
z
•
§
•
Ψ
n
+1
=U
(
s
)
c
Ψ
n
(
n
≥
0)
,
Ψ
n
=
(Ψ
n
(1)
,
···
,
Ψ
n
(2
N
))
,
DOI:10.12677/pm.2022.1281511388
n
Ø
ê
Æ
“
+
Ù
¥
Ψ
n
(1) = (Ψ
L
n
(1)
,
Ψ
O
n
(1)
,
Ψ
Q
n
(1))
T
;Ψ
n
(2
N
) = (Ψ
L
n
(2
N
)
,
Ψ
O
n
(2
N
)
,
Ψ
Q
n
(2
N
))
T
.
•
Ä
‚
C
2
m
þ
˜
m
š
à
g
n
þ
f
i
,
3
½
n
3.1
¥
^
U
(
s
)
c
“
O
U
(
s
)
?
1ü
z
k
e
¡
(
J
,
Ù
¥
2
m
(
m
∈
N
)
“
L
º:
‡
ê
.
é
u
‚
C
2
m
=(
V,E
),
∀
m
∈
N
,
º:
8
Ú
>
8
©
O
•
V
=
{
x
∈
Z/mZ
}
Ú
E
=
{
(
x,x
+1)
,
(
x
+1
,x
):
x
∈
V
}
.
e
¡
ò
þ
f
i
ü
z
˜
m
˜
•
›
‚
C
2
m
þ
.
Ù
ü
z
•
§
÷
v
e
ª
Ψ
n
+1
(
x
) =
P
x
+1
Ψ
n
(
x
+1)+
O
x
Ψ
n
(
x
)+
Q
x
−
1
Ψ
n
(
x
−
1)
,
(
x
∈
Z/mZ
)
.
·
K
3.2.1
‚
C
2
m
(
m
∈
N
)
þ
˜
m
š
à
g
n
þ
f
i
,
3
?
¿
˜
x
?
=
£
Ý
X
e
D
+
x
=
e
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
i
sin
θ
0
−
e
−
φi
cos
θ
ω
x
∈
[0
,
2
π
)
,θ
∈
(0
,
2
π
)
,
Ù
¥
ω
x
−
ω
x
−
1
=
2
π
m
,
∀
x
∈{
1
,
2
,
3
,
···
,
2
m
}
.
d
.
3
‚
C
2
m
þ
k
˜
—
²
-
ÿ
Ý
.
é
u
Ð
©
Ψ(0) = (
α,β,γ
)
T
,
˜
—
²
-
ÿ
Ý
•
ν
(Ψ(
x
)) =
|
α
|
2
+
|
γ
|
2
.
Ù
¥
α,β,γ
∈
C
.
y
²
:
®
•
ω
x
−
ω
x
−
1
=
2
π
m
,
K
φ
=
2
π
m
,
¤
±
D
+
x
=
e
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
i
sin
θ
0
−
e
−
φi
cos
θ
=
e
π
m
i
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
i
sin
θ
0
−
e
−
π
m
i
cos
θ
q
Ï
•
α
x
=
π
m
+
ω
x
−
1
=
ω
x
−
π
m
,
?
˜
Ú
,
Œ
±
α
x
+1
−
α
x
= (
π
m
+
ω
x
)
−
(
π
m
+
ω
x
−
1
) =
2
π
m
.
¤
±
,
k
D
+
x
+1
D
+
x
=
e
φi
cos
θ
0
e
α
x
+1
i
sin
θ
000
e
−
α
x
+1
i
sin
θ
0
−
e
−
φi
cos
θ
e
φi
cos
θ
0
e
α
x
i
sin
θ
000
e
−
α
x
i
sin
θ
0
−
e
−
φi
cos
θ
=
e
2
π
m
i
00
000
00
e
−
2
π
m
i
.
d
d
2
m
Y
x
=1
D
+
y
=
e
2
π
m
i
00
000
00
e
−
2
π
m
i
m
=
100
000
001
.
é
u
Ð
©
Ψ(0) = (
α,β,γ
)
T
,
k
Ψ(
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