Матеріал з Вікіпедії — вільної енциклопедії.
Оператор
L
=
a
(
x
,
y
)
∂
2
∂
x
2
+
2
b
(
x
,
y
)
∂
2
∂
x
∂
y
+
c
(
x
,
y
)
∂
2
∂
y
2
+
d
(
x
,
y
)
∂
∂
x
+
e
(
x
,
y
)
∂
∂
y
+
f
(
x
,
y
)
{\displaystyle L=a(x,y){\frac {\partial ^{2}}{\partial x^{2}}}+2b(x,y){\frac {\partial ^{2}}{\partial x\partial y}}+c(x,y){\frac {\partial ^{2}}{\partial y^{2}}}+d(x,y){\frac {\partial }{\partial x}}+e(x,y){\frac {\partial }{\partial y}}+f(x,y)}
з поліноміальними коефіцієнтами
a
(
x
,
y
)
{\displaystyle a(x,y)}
b
(
x
,
y
)
{\displaystyle b(x,y)}
c
(
x
,
y
)
{\displaystyle c(x,y)}
d
(
x
,
y
)
{\displaystyle d(x,y)}
e
(
x
,
y
)
{\displaystyle e(x,y)}
f
(
x
,
y
)
{\displaystyle f(x,y)}
V
n
{\displaystyle V_{n}}
P
(
x
,
y
)
{\displaystyle P(x,y)}
deg
P
≤
n
{\displaystyle \deg P\leq n}
n
≥
2
{\displaystyle n\geq 2}
L
{\displaystyle L}
a
=
q
1
x
4
+
q
2
x
3
y
+
q
3
x
2
y
2
+
k
1
x
3
+
k
2
x
2
y
+
k
3
x
y
2
+
a
1
x
2
+
a
2
x
y
+
a
3
y
2
+
a
4
x
+
a
5
y
+
a
6
,
{\displaystyle a=q_{1}x^{4}+q_{2}x^{3}y+q_{3}x^{2}y^{2}+k_{1}x^{3}+k_{2}x^{2}y+k_{3}xy^{2}+a_{1}x^{2}+a_{2}xy+a_{3}y^{2}+a_{4}x+a_{5}y+a_{6},}
b
=
q
1
x
3
y
+
q
2
x
2
y
2
+
q
3
x
y
3
+
1
2
[
k
4
x
3
+
(
k
1
+
k
5
)
x
2
y
+
(
k
2
+
k
6
)
x
y
2
+
k
3
y
3
]
+
b
1
x
2
+
b
2
x
y
+
b
3
y
2
+
b
4
x
+
b
5
y
+
b
6
,
{\displaystyle b=q_{1}x^{3}y+q_{2}x^{2}y^{2}+q_{3}xy^{3}+{\frac {1}{2}}[k_{4}x^{3}+(k_{1}+k_{5})x^{2}y+(k_{2}+k_{6})xy^{2}+k_{3}y^{3}]+b_{1}x^{2}+b_{2}xy+b_{3}y^{2}+b_{4}x+b_{5}y+b_{6},}
c
=
q
1
x
2
y
2
+
q
2
x
y
3
+
q
3
y
4
+
k
4
x
2
y
+
k
5
x
y
2
+
k
6
y
3
+
c
1
x
2
+
c
2
x
y
+
c
3
y
2
+
c
4
x
+
c
5
y
+
c
6
,
{\displaystyle c=q_{1}x^{2}y^{2}+q_{2}xy^{3}+q_{3}y^{4}+k_{4}x^{2}y+k_{5}xy^{2}+k_{6}y^{3}+c_{1}x^{2}+c_{2}xy+c_{3}y^{2}+c_{4}x+c_{5}y+c_{6},}
d
=
(
1
−
n
)
[
2
(
q
1
x
3
+
q
2
x
2
y
+
q
3
x
y
2
)
+
k
7
x
2
+
(
k
2
+
k
8
−
k
6
)
x
y
+
k
3
y
2
]
+
d
1
x
+
d
2
y
+
d
3
,
{\displaystyle d=(1-n)[2(q_{1}x^{3}+q_{2}x^{2}y+q_{3}xy^{2})+k_{7}x^{2}+(k_{2}+k_{8}-k_{6})xy+k_{3}y^{2}]+d_{1}x+d_{2}y+d_{3},}
e
=
(
1
−
n
)
[
2
(
q
1
x
2
y
+
q
2
x
y
2
+
q
3
y
3
)
+
k
4
x
2
+
(
k
5
+
k
7
−
k
1
)
x
y
+
k
8
y
2
]
+
e
1
x
+
e
2
y
+
e
3
,
{\displaystyle e=(1-n)[2(q_{1}x^{2}y+q_{2}xy^{2}+q_{3}y^{3})+k_{4}x^{2}+(k_{5}+k_{7}-k_{1})xy+k_{8}y^{2}]+e_{1}x+e_{2}y+e_{3},}
f
=
n
(
n
−
1
)
[
q
1
x
2
+
q
2
x
y
+
q
3
y
2
+
(
k
7
−
k
1
)
x
+
(
k
8
−
k
6
)
y
]
+
f
1
.
{\displaystyle f=n(n-1)[q_{1}x^{2}+q_{2}xy+q_{3}y^{2}+(k_{7}-k_{1})x+(k_{8}-k_{6})y]+f_{1}.}
Соколов В.В., Алгебраические квантовые гамильтонианы на плоскости. Теоретическая и математическая физика , т.184.- 2015.- С.57--70.
![{\displaystyle b=q_{1}x^{3}y+q_{2}x^{2}y^{2}+q_{3}xy^{3}+{\frac {1}{2}}[k_{4}x^{3}+(k_{1}+k_{5})x^{2}y+(k_{2}+k_{6})xy^{2}+k_{3}y^{3}]+b_{1}x^{2}+b_{2}xy+b_{3}y^{2}+b_{4}x+b_{5}y+b_{6},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a48a6b86e03a42fcf79e20b7db25d7a56b813e5)
![{\displaystyle d=(1-n)[2(q_{1}x^{3}+q_{2}x^{2}y+q_{3}xy^{2})+k_{7}x^{2}+(k_{2}+k_{8}-k_{6})xy+k_{3}y^{2}]+d_{1}x+d_{2}y+d_{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7bd07028dcccc0735cea28b40b0ce589cd4fe2b)
![{\displaystyle e=(1-n)[2(q_{1}x^{2}y+q_{2}xy^{2}+q_{3}y^{3})+k_{4}x^{2}+(k_{5}+k_{7}-k_{1})xy+k_{8}y^{2}]+e_{1}x+e_{2}y+e_{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/713bc5fc54a0e91c8943ae62380fd99ac5a880db)
![{\displaystyle f=n(n-1)[q_{1}x^{2}+q_{2}xy+q_{3}y^{2}+(k_{7}-k_{1})x+(k_{8}-k_{6})y]+f_{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31d907b3fa54b1025609b289515e38487ae30163)
