Bessel potensiali

Matematikada Bessel potentsiali Riesz potentsialiga oʻxshash potentsialdir (Fridrix Vilgelm Bessel nomi bilan atalgan), lekin cheksizlikda yaxshiroq parchalanish xususiyatlariga ega.

Agar s musbat haqiqiy qismga ega kompleks son boʻlsa, s tartibli Bessel potensiali operatori:

$(I-\Delta )^{-s/2}$

bu yerda Δ – Laplas operatori va kasr quvvati Furye transformlari yordamida aniqlanadi.

Yukava potensiali Bessel potentsiallarining $s=2$ 3 oʻlchovli fazoda xususiy holatlaridir

Furye fazosida koʻrinishi

Bessel potentsiali Furye oʻzgarishlariga koʻpaytirish orqali taʼsir qiladi: har bir $\xi \in \mathbb {R} ^{d}$uchun:

${\mathcal {F}}((I-\Delta )^{-s/2}u)(\xi )={\frac {{\mathcal {F}}u(\xi )}{(1+4\pi ^{2}\vert \xi \vert ^{2})^{s/2}}}.$

Integral koʻrinishlari

$s>0$ boʻlganda, Bessel potensiali $\mathbb {R} ^{d}$da quyidagicha ifodalanishi mumkin:

$(I-\Delta )^{-s/2}u=G_{s}\ast u,$

bu yerda Bessel yadrosi $G_{s}$ $x\in \mathbb {R} ^{d}\setminus \{0\}$ uchun integral formula boʻyicha ifodalanadi

$G_{s}(x)={\frac {1}{(4\pi )^{s/2}\Gamma (s/2)}}\int _{0}^{\infty }{\frac {e^{-{\frac {\pi \vert x\vert ^{2}}{y}}-{\frac {y}{4\pi }}}}{y^{1+{\frac {d-s}{2}}}}}\,\mathrm {d} y.$

Bu yerda $\Gamma$ Gamma funksiyasini bildiradi. Bessel yadrosi orqali $x\in \mathbb {R} ^{d}\setminus \{0\}$ ham ifodalanishi mumkin:

$G_{s}(x)={\frac {e^{-\vert x\vert }}{(2\pi )^{\frac {d-1}{2}}2^{\frac {s}{2}}\Gamma ({\frac {s}{2}})\Gamma ({\frac {d-s+1}{2}})}}\int _{0}^{\infty }e^{-\vert x\vert t}{\Big (}t+{\frac {t^{2}}{2}}{\Big )}^{\frac {d-s-1}{2}}\,\mathrm {d} t.$

Ushbu oxirgi ifodani oʻzgartirilgan Bessel funksiyasi nuqtai nazaridan qisqaroq yozish mumkin, shu sababli potensial oʻz nomini oladi:

$G_{s}(x)={\frac {1}{2^{(s-2)/2}(2\pi )^{d/2}\Gamma ({\frac {s}{2}})}}K_{(d-s)/2}(\vert x\vert )\vert x\vert ^{(s-d)/2}.$

Asimptotiklar

Kelib chiqishida birida $\vert x\vert \to 0$,

$G_{s}(x)={\frac {\Gamma ({\frac {d-s}{2}})}{2^{s}\pi ^{s/2}\vert x\vert ^{d-s}}}(1+o(1))\quad {\text{ if }}0<s<d,$

$G_{d}(x)={\frac {1}{2^{d-1}\pi ^{d/2}}}\ln {\frac {1}{\vert x\vert }}(1+o(1)),$

$G_{s}(x)={\frac {\Gamma ({\frac {s-d}{2}})}{2^{s}\pi ^{s/2}}}(1+o(1))\quad {\text{ if }}s>d.$

Xususan, $0<s<d$ boʻlganda Bessel potensiali Riesz potensiali kabi asimptotik tarzda oʻzini tutadi.

Cheksizlikda, $\vert x\vert \to \infty$,

$G_{s}(x)={\frac {e^{-\vert x\vert }}{2^{\frac {d+s-1}{2}}\pi ^{\frac {d-1}{2}}\Gamma ({\frac {s}{2}})\vert x\vert ^{\frac {d+1-s}{2}}}}(1+o(1)).$

Shuningdek qarang:

• Riesz potensiali
• Sobolev fazosi
• Shredinger kasr tenglamasi
• Yukava potensiali

Manbalar

• Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics, 250-jild (2nd-nashr), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8