Bessel potensiali
Matematikada
Bessel potentsiali Riesz potentsialiga o'xshash potentsialdir (Fridrix Vilgelm Bessel nomi bilan atalgan), lekin cheksizlikda yaxshiroq parchalanish xususiyatlariga ega.
Agar s musbat haqiqiy qismga ega kompleks son bo'lsa, s tartibli Bessel potensiali operatori:
(
I
−
Δ
)
−
s
/
2
{\displaystyle (I-\Delta )^{-s/2}}
bu yerda Δ - Laplas operatori va kasr quvvati Furye transformlari yordamida aniqlanadi.
Yukava potensiali Bessel potentsiallarining
s
=
2
{\displaystyle s=2}
3 o'lchovli fazoda xususiy holatlaridir
Furye fazosida ko'rinishi
Bessel potentsiali Furye o'zgarishlariga ko'paytirish orqali ta'sir qiladi: har bir
ξ
∈
R
d
{\displaystyle \xi \in \mathbb {R} ^{d}}
uchun:
F
(
(
I
−
Δ
)
−
s
/
2
u
)
(
ξ
)
=
F
u
(
ξ
)
(
1
+
4
π
2
|
ξ
|
2
)
s
/
2
.
{\displaystyle {\mathcal {F}}((I-\Delta )^{-s/2}u)(\xi )={\frac {{\mathcal {F}}u(\xi )}{(1+4\pi ^{2}\vert \xi \vert ^{2})^{s/2}}}.}
Integral ko'rinishlari
s
>
0
{\displaystyle s>0}
bo'lganda, Bessel potensiali
R
d
{\displaystyle \mathbb {R} ^{d}}
da quyidagicha ifodalanishi mumkin:
(
I
−
Δ
)
−
s
/
2
u
=
G
s
∗
u
,
{\displaystyle (I-\Delta )^{-s/2}u=G_{s}\ast u,}
bu yerda Bessel yadrosi
G
s
{\displaystyle G_{s}}
x
∈
R
d
∖
{
0
}
{\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}}
uchun integral formula bo'yicha ifodalanadi
G
s
(
x
)
=
1
(
4
π
)
s
/
2
Γ
(
s
/
2
)
∫
0
∞
e
−
π
|
x
|
2
y
−
y
4
π
y
1
+
d
−
s
2
d
y
.
{\displaystyle G_{s}(x)={\frac {1}{(4\pi )^{s/2}\Gamma (s/2)}}\int _{0}^{\infty }{\frac {e^{-{\frac {\pi \vert x\vert ^{2}}{y}}-{\frac {y}{4\pi }}}}{y^{1+{\frac {d-s}{2}}}}}\,\mathrm {d} y.}
Bu yerda
Γ
{\displaystyle \Gamma }
Gamma funksiyasini bildiradi. Bessel yadrosi orqali
x
∈
R
d
∖
{
0
}
{\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}}
ham ifodalanishi mumkin:
G
s
(
x
)
=
e
−
|
x
|
(
2
π
)
d
−
1
2
2
s
2
Γ
(
s
2
)
Γ
(
d
−
s
+
1
2
)
∫
0
∞
e
−
|
x
|
t
(
t
+
t
2
2
)
d
−
s
−
1
2
d
t
.
{\displaystyle G_{s}(x)={\frac {e^{-\vert x\vert }}{(2\pi )^{\frac {d-1}{2}}2^{\frac {s}{2}}\Gamma ({\frac {s}{2}})\Gamma ({\frac {d-s+1}{2}})}}\int _{0}^{\infty }e^{-\vert x\vert t}{\Big (}t+{\frac {t^{2}}{2}}{\Big )}^{\frac {d-s-1}{2}}\,\mathrm {d} t.}
Ushbu oxirgi ifodani o'zgartirilgan Bessel funksiyasi nuqtai nazaridan qisqaroq yozish mumkin, shu sababli potensial o'z nomini oladi:
G
s
(
x
)
=
1
2
(
s
−
2
)
/
2
(
2
π
)
d
/
2
Γ
(
s
2
)
K
(
d
−
s
)
/
2
(
|
x
|
)
|
x
|
(
s
−
d
)
/
2
.
{\displaystyle G_{s}(x)={\frac {1}{2^{(s-2)/2}(2\pi )^{d/2}\Gamma ({\frac {s}{2}})}}K_{(d-s)/2}(\vert x\vert )\vert x\vert ^{(s-d)/2}.}
Asimptotiklar
Kelib chiqishida birida
|
x
|
→
0
{\displaystyle \vert x\vert \to 0}
,
G
s
(
x
)
=
Γ
(
d
−
s
2
)
2
s
π
s
/
2
|
x
|
d
−
s
(
1
+
o
(
1
)
)
if
0
<
s
<
d
,
{\displaystyle G_{s}(x)={\frac {\Gamma ({\frac {d-s}{2}})}{2^{s}\pi ^{s/2}\vert x\vert ^{d-s}}}(1+o(1))\quad {\text{ if }}0<s<d,}
G
d
(
x
)
=
1
2
d
−
1
π
d
/
2
ln
1
|
x
|
(
1
+
o
(
1
)
)
,
{\displaystyle G_{d}(x)={\frac {1}{2^{d-1}\pi ^{d/2}}}\ln {\frac {1}{\vert x\vert }}(1+o(1)),}
G
s
(
x
)
=
Γ
(
s
−
d
2
)
2
s
π
s
/
2
(
1
+
o
(
1
)
)
if
s
>
d
.
{\displaystyle G_{s}(x)={\frac {\Gamma ({\frac {s-d}{2}})}{2^{s}\pi ^{s/2}}}(1+o(1))\quad {\text{ if }}s>d.}
Xususan,
0
<
s
<
d
{\displaystyle 0<s<d}
bo'lganda Bessel potensiali Riesz potensiali kabi asimptotik tarzda o'zini tutadi.
Cheksizlikda,
|
x
|
→
∞
{\displaystyle \vert x\vert \to \infty }
,
G
s
(
x
)
=
e
−
|
x
|
2
d
+
s
−
1
2
π
d
−
1
2
Γ
(
s
2
)
|
x
|
d
+
1
−
s
2
(
1
+
o
(
1
)
)
.
{\displaystyle G_{s}(x)={\frac {e^{-\vert x\vert }}{2^{\frac {d+s-1}{2}}\pi ^{\frac {d-1}{2}}\Gamma ({\frac {s}{2}})\vert x\vert ^{\frac {d+1-s}{2}}}}(1+o(1)).}
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