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)).}


Shuningdek qarang:




Manbalar




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