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❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚r❛❝❡ ❛s②♠♣t♦t✐❝s ❢♦r ❢r❛❝t✐♦♥❛❧ ❙❝❤rö❞✐♥❣❡r ❖♣❡r❛t♦rs

▲✉✐s ❆❝✉♥❛

P✉r❞✉❡ ❯♥✐✈❡rs✐t②

❋❡❜r✉❛r② ✶✾✱ ✷✵✶✸

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿▼✳❑❛❝✱ ✶✾✻✻✱ ❈❛♥ ✇❡ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❛ ❞r✉♠❄ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✱ ✇✐t❤ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❤♦❧❡s ❛♥❞ ♣✐❡❝❡ s♠♦♦t❤ ❜♦✉♥❞❛r②✳❚❤✐♥❦✐♥❣ ♦❢ ❉ ❛s ❛ ❞r✉♠ ❛♥❞ ✵ < λ✶ < λ✷ ≤✳❡t❝✱ ❛s ✐ts ❢✉♥❞❛♠❡♥t❛❧ t♦♥❡s✱ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❉❄ ❚❤❡ s❡t

♥

✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ❢ ❢ ✐♥ ❉ ❢ ✵ ✐♥ ❉ ❚❤❡♦r❡♠ ✭▼❝❦❡❛♥✱❙✐♥❣❡r ✶✾✻✼✮ ❩❉ t

♥ ✶

❡

t

♥

❉ ✹ t ❉ ✹ ✹ t ✶ ✷ ✶ ✻ ✶ ❤ t✶ ✷ t ✵ ❤❂ ♥✉♠❜❡r ♦❢ ❤♦❧❡s✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿▼✳❑❛❝✱ ✶✾✻✻✱ ❈❛♥ ✇❡ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❛ ❞r✉♠❄ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✱ ✇✐t❤ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❤♦❧❡s ❛♥❞ ♣✐❡❝❡ s♠♦♦t❤ ❜♦✉♥❞❛r②✳❚❤✐♥❦✐♥❣ ♦❢ ❉ ❛s ❛ ❞r✉♠ ❛♥❞ ✵ < λ✶ < λ✷ ≤✳❡t❝✱ ❛s ✐ts ❢✉♥❞❛♠❡♥t❛❧ t♦♥❡s✱ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❉❄ ❚❤❡ s❡t {λ♥} ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ∆❢ = λ❢ ✐♥ ❉, ❢ = ✵ ✐♥ ∂❉ ❚❤❡♦r❡♠ ✭▼❝❦❡❛♥✱❙✐♥❣❡r ✶✾✻✼✮ ❩❉ t

♥ ✶

❡

t

♥

❉ ✹ t ❉ ✹ ✹ t ✶ ✷ ✶ ✻ ✶ ❤ t✶ ✷ t ✵ ❤❂ ♥✉♠❜❡r ♦❢ ❤♦❧❡s✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿▼✳❑❛❝✱ ✶✾✻✻✱ ❈❛♥ ✇❡ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❛ ❞r✉♠❄ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✱ ✇✐t❤ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❤♦❧❡s ❛♥❞ ♣✐❡❝❡ s♠♦♦t❤ ❜♦✉♥❞❛r②✳❚❤✐♥❦✐♥❣ ♦❢ ❉ ❛s ❛ ❞r✉♠ ❛♥❞ ✵ < λ✶ < λ✷ ≤✳❡t❝✱ ❛s ✐ts ❢✉♥❞❛♠❡♥t❛❧ t♦♥❡s✱ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❉❄ ❚❤❡ s❡t {λ♥} ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ∆❢ = λ❢ ✐♥ ❉, ❢ = ✵ ✐♥ ∂❉ ❚❤❡♦r❡♠ ✭▼❝❦❡❛♥✱❙✐♥❣❡r ✶✾✻✼✮ ❩❉ t

♥ ✶

❡

t

♥

❉ ✹ t ❉ ✹ ✹ t ✶ ✷ ✶ ✻ ✶ ❤ t✶ ✷ t ✵ ❤❂ ♥✉♠❜❡r ♦❢ ❤♦❧❡s✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿▼✳❑❛❝✱ ✶✾✻✻✱ ❈❛♥ ✇❡ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❛ ❞r✉♠❄ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ t❤❡ ♣❧❛♥❡✱ ✇✐t❤ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❤♦❧❡s ❛♥❞ ♣✐❡❝❡ s♠♦♦t❤ ❜♦✉♥❞❛r②✳❚❤✐♥❦✐♥❣ ♦❢ ❉ ❛s ❛ ❞r✉♠ ❛♥❞ ✵ < λ✶ < λ✷ ≤✳❡t❝✱ ❛s ✐ts ❢✉♥❞❛♠❡♥t❛❧ t♦♥❡s✱ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ❤❡❛r t❤❡ s❤❛♣❡ ♦❢ ❉❄ ❚❤❡ s❡t {λ♥} ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ∆❢ = λ❢ ✐♥ ❉, ❢ = ✵ ✐♥ ∂❉ ❚❤❡♦r❡♠ ✭▼❝❦❡❛♥✱❙✐♥❣❡r ✶✾✻✼✮ ❩❉(t) =

∞

• ♥=✶

❡−tλ♥ = |❉| ✹πt − |∂❉| ✹(✹πt)✶/✷ − ✶ ✻(✶ − ❤) + O(t✶/✷), t ↓ ✵. ❤❂ ♥✉♠❜❡r ♦❢ ❤♦❧❡s✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿ ❉ ❜♦✉♥❞❡❞ ♦♣❡♥ ❝♦♥♥❡❝t❡❞✳ ❙✉♣♣♦s❡ ❉ ❤❛s t❡♠♣❡r❛t✉r❡ ✶ ❛t t✐♠❡ t = ✵✱ ✇❤✐❧❡ R✷ − ❉ ✐s ❤❡❧❞ ❛t t❡♠♣❡r❛t✉r❡ ✵ ❢♦r ❛❧❧ ♣♦s✐t✐✈❡ t✐♠❡❀ ✇❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ❛s t ↓ ✵ ♦❢ ◗❉(t)✱ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ ❉ ❛t t✐♠❡ t❄ ◗❉ t

❉ ✉ t ① ❞①✒ ✇❤❡r❡ ✉ t ①

✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ ✉ t ① ✉ t ✐♥ ❉ ✵ ✉ ✵ ① ✶ ✐♥ ❉ ❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱ ●❛❧❧✱✾✹ ✮ ◗❉ t ❉ ✷

✶ ✷

❉ t✶ ✷ ✶ ❤ t t✸ ✷ t ✵ ▲❡t ♣ ✷

❉

t ① ② ❜❡ t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t✐❡s ♦❢ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❑✐❧❧❡❞ ♦✉ts✐❞❡ ❉ ❛♥❞

❉

✐♥❢ s ✵ ❇t ❉❝

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿ ❉ ❜♦✉♥❞❡❞ ♦♣❡♥ ❝♦♥♥❡❝t❡❞✳ ❙✉♣♣♦s❡ ❉ ❤❛s t❡♠♣❡r❛t✉r❡ ✶ ❛t t✐♠❡ t = ✵✱ ✇❤✐❧❡ R✷ − ❉ ✐s ❤❡❧❞ ❛t t❡♠♣❡r❛t✉r❡ ✵ ❢♦r ❛❧❧ ♣♦s✐t✐✈❡ t✐♠❡❀ ✇❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ❛s t ↓ ✵ ♦❢ ◗❉(t)✱ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ ❉ ❛t t✐♠❡ t❄ ◗❉(t) =

• ❉ ✉(t, ①)❞①✒ ✇❤❡r❡ ✉(t, ①) ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢

∆✉(t, ①) = ∂✉ ∂t ✐♥ ❉ × (✵, +∞), ✉(✵, ①) = ✶ ✐♥ ❉ ❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱ ●❛❧❧✱✾✹ ✮ ◗❉ t ❉ ✷

✶ ✷

❉ t✶ ✷ ✶ ❤ t t✸ ✷ t ✵ ▲❡t ♣ ✷

❉

t ① ② ❜❡ t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t✐❡s ♦❢ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❑✐❧❧❡❞ ♦✉ts✐❞❡ ❉ ❛♥❞

❉

✐♥❢ s ✵ ❇t ❉❝

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

◗✉❡st✐♦♥✿ ❉ ❜♦✉♥❞❡❞ ♦♣❡♥ ❝♦♥♥❡❝t❡❞✳ ❙✉♣♣♦s❡ ❉ ❤❛s t❡♠♣❡r❛t✉r❡ ✶ ❛t t✐♠❡ t = ✵✱ ✇❤✐❧❡ R✷ − ❉ ✐s ❤❡❧❞ ❛t t❡♠♣❡r❛t✉r❡ ✵ ❢♦r ❛❧❧ ♣♦s✐t✐✈❡ t✐♠❡❀ ✇❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ❛s t ↓ ✵ ♦❢ ◗❉(t)✱ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ ❉ ❛t t✐♠❡ t❄ ◗❉(t) =

• ❉ ✉(t, ①)❞①✒ ✇❤❡r❡ ✉(t, ①) ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢

∆✉(t, ①) = ∂✉ ∂t ✐♥ ❉ × (✵, +∞), ✉(✵, ①) = ✶ ✐♥ ❉ ❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱ ●❛❧❧✱✾✹ ✮ ◗❉(t) = |❉| − ✷π−✶/✷|∂❉|t✶/✷ + π(✶ − ❤)t + O(t✸/✷), t ↓ ✵ ▲❡t ♣(✷)

❉ (t, ①, ②) ❜❡ t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t✐❡s ♦❢ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❑✐❧❧❡❞

♦✉ts✐❞❡ ❉ ❛♥❞ τ❉ = ✐♥❢ {s > ✵ : ❇t ∈ ❉❝}

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❘❡♠❛r❦ ♣(✷)

❉ (t, ①, ②) = ∞

• ♥=✶

❡−tλ♥φ♥(①)φ♥(②) ❩❉ t

❉

♣ ✷

❉

t ① ① ❞①

♥ ✶

❡

t

♥

✉ t ① P①

❉

t

❉

♣ ✷

❉

t ① ② ❞② ◗❉ t

♥ ✶

❡

t

♥

❉ ♥ ① ❞① ✷ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❘❡♠❛r❦ ♣(✷)

❉ (t, ①, ②) = ∞

• ♥=✶

❡−tλ♥φ♥(①)φ♥(②) ❩❉(t) =

• ❉

♣(✷)

❉ (t, ①, ①)❞① = ∞

• ♥=✶

❡−tλ♥ ✉ t ① P①

❉

t

❉

♣ ✷

❉

t ① ② ❞② ◗❉ t

♥ ✶

❡

t

♥

❉ ♥ ① ❞① ✷ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❘❡♠❛r❦ ♣(✷)

❉ (t, ①, ②) = ∞

• ♥=✶

❡−tλ♥φ♥(①)φ♥(②) ❩❉(t) =

• ❉

♣(✷)

❉ (t, ①, ①)❞① = ∞

• ♥=✶

❡−tλ♥ ✉(t, ①) = P①(τ❉ > t) =

• ❉

♣(✷)

❉ (t, ①, ②)❞②

◗❉ t

♥ ✶

❡

t

♥

❉ ♥ ① ❞① ✷ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❘❡♠❛r❦ ♣(✷)

❉ (t, ①, ②) = ∞

• ♥=✶

❡−tλ♥φ♥(①)φ♥(②) ❩❉(t) =

• ❉

♣(✷)

❉ (t, ①, ①)❞① = ∞

• ♥=✶

❡−tλ♥ ✉(t, ①) = P①(τ❉ > t) =

• ❉

♣(✷)

❉ (t, ①, ②)❞②

◗❉(t) =

∞

• ♥=✶

❡−tλ♥

• ❉

φ♥(①)❞① ✷

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❘❡♠❛r❦ ♣(✷)

❉ (t, ①, ②) = ∞

• ♥=✶

❡−tλ♥φ♥(①)φ♥(②) ❩❉(t) =

• ❉

♣(✷)

❉ (t, ①, ①)❞① = ∞

• ♥=✶

❡−tλ♥ ✉(t, ①) = P①(τ❉ > t) =

• ❉

♣(✷)

❉ (t, ①, ②)❞②

◗❉(t) =

∞

• ♥=✶

❡−tλ♥

• ❉

φ♥(①)❞① ✷

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❉❡✜♥✐t✐♦♥ ❚❤❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ α✲♣r♦❝❡ss✱ ✵ < α < ✷✱ ❳ =

• ❳t, t ≥ ✵, P①, ① ∈ R❞

✐s ❛ ▲é✈② ♣r♦❝❡ss ✇✐t❤ ❊①(❡✐ξ·(❳t−❳✵)) = ❡−t|ξ|α, ① ∈ R❞, ξ ∈ R❞.

❉❡✜♥✐t✐♦♥ ▲❡t ✵ ✷✳ ❙t ✐s ❛♥ ✷✲s✉❜♦r❞✐♥❛t♦r ✐❢ ✐t ✐s ❛ ▲❡✈② ♣r♦❝❡ss✱ st❛rt❡❞ ❛t ✵✱ str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱ ✇✐t❤ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❊ ❡

❙t

❡

t

✷

❚❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t✐❡s ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜②

✷ t

❲❤❡♥ ✷✱ ❙t t✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❉❡✜♥✐t✐♦♥ ❚❤❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ α✲♣r♦❝❡ss✱ ✵ < α < ✷✱ ❳ =

• ❳t, t ≥ ✵, P①, ① ∈ R❞

✐s ❛ ▲é✈② ♣r♦❝❡ss ✇✐t❤ ❊①(❡✐ξ·(❳t−❳✵)) = ❡−t|ξ|α, ① ∈ R❞, ξ ∈ R❞.

❉❡✜♥✐t✐♦♥ ▲❡t ✵ < α < ✷✳ ❙t ✐s ❛♥ α/✷✲s✉❜♦r❞✐♥❛t♦r ✐❢ ✐t ✐s ❛ ▲❡✈② ♣r♦❝❡ss✱ st❛rt❡❞ ❛t ✵✱ str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱ ✇✐t❤ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❊[❡−λ❙t] = ❡−tλα/✷. ❚❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t✐❡s ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② η(α/✷)

t

. ❲❤❡♥ α = ✷✱ ❙t = t✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ✭❈❛✉❝❤② Pr♦❝❡ss✱ α = ✶✮ ▲❡t {❲t}t≥✵ ❜❡ ❛ st❛♥❞❛r❞ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡♥ ❙t = ✐♥❢

• s > ✵ : ❲s =

t √ ✷

• .

■t ✐s ❛❧s♦ ❦♥♦✇♥ t❤❛t ✐ts ❞❡♥s✐t② ✐s ❣✐✈❡♥ ❜② η(✶/✷)

t

(s) = t ✷√π s−✸/✷❡−t✷/✹s. ▲❡t ❇t ❜❡ ❛ ❞✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✐♥

✶ P① ✱ ❛♥❞ ❙t ❛♥

✷ s✉❜♦r❞✐♥❛t♦r ✐♥

✷ P ✳ ❉❡✜♥❡ ✐♥ ✶① ✷ P①

P①①P ✱ ❳t ✇✶ ✇✷ ❇❙t ✇✷ ✇✶ ✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ✭❈❛✉❝❤② Pr♦❝❡ss✱ α = ✶✮ ▲❡t {❲t}t≥✵ ❜❡ ❛ st❛♥❞❛r❞ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡♥ ❙t = ✐♥❢

• s > ✵ : ❲s =

t √ ✷

• .

■t ✐s ❛❧s♦ ❦♥♦✇♥ t❤❛t ✐ts ❞❡♥s✐t② ✐s ❣✐✈❡♥ ❜② η(✶/✷)

t

(s) = t ✷√π s−✸/✷❡−t✷/✹s. ▲❡t ❇t ❜❡ ❛ ❞✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✐♥ (Ω✶, P①)✱ ❛♥❞ ❙t ❛♥ α/✷ s✉❜♦r❞✐♥❛t♦r ✐♥ (Ω✷, P)✳ ❉❡✜♥❡ ✐♥ (Ω = Ω✶①Ω✷, P① = P①①P)✱ ❳t(✇✶, ✇✷) = ❇❙t(✇✷)(✇✶) ✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ✭❈❛✉❝❤② Pr♦❝❡ss✱ α = ✶✮ ▲❡t {❲t}t≥✵ ❜❡ ❛ st❛♥❞❛r❞ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡♥ ❙t = ✐♥❢

• s > ✵ : ❲s =

t √ ✷

• .

■t ✐s ❛❧s♦ ❦♥♦✇♥ t❤❛t ✐ts ❞❡♥s✐t② ✐s ❣✐✈❡♥ ❜② η(✶/✷)

t

(s) = t ✷√π s−✸/✷❡−t✷/✹s. ▲❡t ❇t ❜❡ ❛ ❞✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✐♥ (Ω✶, P①)✱ ❛♥❞ ❙t ❛♥ α/✷ s✉❜♦r❞✐♥❛t♦r ✐♥ (Ω✷, P)✳ ❉❡✜♥❡ ✐♥ (Ω = Ω✶①Ω✷, P① = P①①P)✱ ❳t(✇✶, ✇✷) = ❇❙t(✇✷)(✇✶) ✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❋✐❣✉r❡✿ ❝❛♣t✐♦♥t❡①t

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♣(α)

t

(①) = ✶ (✷π)❞

• R❞ ❡−✐①·ξ❡−t|ξ|α❞ξ =

∞

✵

✶ (✹πs)❞/✷ ❡

−|①|✷ ✹s

ηα/✷

t

(s) ❞s, = ❊[♣(✷)

❙t (①)]

♣t ① t

❞

♣✶ t

✶

① t

❞

♣✶ ✵ t

❞ ❞

❞ ✷

❞

♣t ✵ ♣t ✵ ✵ ❛♥❞ ❈

✶ ❞

t

❞

t ① ② ❞ ♣t ① ② ❈

❞

t

❞

t ① ② ❞

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♣(α)

t

(①) = ✶ (✷π)❞

• R❞ ❡−✐①·ξ❡−t|ξ|α❞ξ =

∞

✵

✶ (✹πs)❞/✷ ❡

−|①|✷ ✹s

ηα/✷

t

(s) ❞s, = ❊[♣(✷)

❙t (①)]

♣(α)

t

(①) = t−❞/α♣(α)

✶ (t−✶/α①) ≤ t−❞/α♣α ✶ (✵) =

= t−❞/α ω❞Γ(❞/α) (✷π)❞α = ♣α

t (✵) (= ♣α t (✵, ✵))

❛♥❞ ❈

✶ ❞

t

❞

t ① ② ❞ ♣t ① ② ❈

❞

t

❞

t ① ② ❞

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♣(α)

t

(①) = ✶ (✷π)❞

• R❞ ❡−✐①·ξ❡−t|ξ|α❞ξ =

∞

✵

✶ (✹πs)❞/✷ ❡

−|①|✷ ✹s

ηα/✷

t

(s) ❞s, = ❊[♣(✷)

❙t (①)]

♣(α)

t

(①) = t−❞/α♣(α)

✶ (t−✶/α①) ≤ t−❞/α♣α ✶ (✵) =

= t−❞/α ω❞Γ(❞/α) (✷π)❞α = ♣α

t (✵) (= ♣α t (✵, ✵))

❛♥❞ ❈ −✶

α,❞

• t−❞/α ∧

t |① − ②|❞+α

• ≤ ♣(α)

t

(① − ②) ≤ ❈α,❞

• t−❞/α ∧

t |① − ②|❞+α

• ,

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❋✐❣✉r❡✿ ❘✕s♠♦♦t❤ ❜♦✉♥❞❛r② ❞♦♠❛✐♥s

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱✽✼✮ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✇✐t❤ ❘✲s♠♦♦t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥

• ❩t(❉) −

|❉| (✹πt)❞/✷ + |∂❉| ✹(✹πt)(❞−✶)/✷

• ≤

❞ ✹ π❞/✷ |❉| t(❞−✷)/✷❘✷ , ❢♦r ❛❧❧ t > ✵ ❞ ✷ ❧✐♠

t ✵ ❩t ❉

❉ ✹ t ❉ ✹ ✹ t ✶ ✷ ✶ ❤ ✻ ❚❤❡ ✐❞❡❛ ✐s t♦ ✉s❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❇t ❇✶

t

❇❞

t

✇❤❡r❡ ❇✐

t ❛r❡ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s ❛♥❞ t❤❡

♣r♦♣❡rt✐❡s ♦❢ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❜r✐❞❣❡✳ ❘❡♠❛r❦ ❳t ❛♥❞ ❨t ❳ ✶

t

❳ ❞

t ✱ ❳ ✐ t ✱ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t

✕st❛❜❧❡ ♣r♦❝❡ss❡s ❛r❡ t✇♦ ❞✐✛❡r❡♥t ▲❡✈② ♣r♦❝❡ss❡s s✐♥❝❡ ❊① ❡✐

❨t ❨✵

❡

t

❞ ❥ ✶ ❥

①

❞ ✶ ❞ ❞ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱✽✼✮ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✇✐t❤ ❘✲s♠♦♦t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥

• ❩t(❉) −

|❉| (✹πt)❞/✷ + |∂❉| ✹(✹πt)(❞−✶)/✷

• ≤

❞ ✹ π❞/✷ |❉| t(❞−✷)/✷❘✷ , ❢♦r ❛❧❧ t > ✵ ❞ = ✷, ❧✐♠

t↓✵ ❩t(❉) − |❉|

✹πt + |∂❉| ✹(✹πt)✶/✷ = ✶ − ❤ ✻ ❚❤❡ ✐❞❡❛ ✐s t♦ ✉s❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❇t ❇✶

t

❇❞

t

✇❤❡r❡ ❇✐

t ❛r❡ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s ❛♥❞ t❤❡

♣r♦♣❡rt✐❡s ♦❢ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❜r✐❞❣❡✳ ❘❡♠❛r❦ ❳t ❛♥❞ ❨t ❳ ✶

t

❳ ❞

t ✱ ❳ ✐ t ✱ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t

✕st❛❜❧❡ ♣r♦❝❡ss❡s ❛r❡ t✇♦ ❞✐✛❡r❡♥t ▲❡✈② ♣r♦❝❡ss❡s s✐♥❝❡ ❊① ❡✐

❨t ❨✵

❡

t

❞ ❥ ✶ ❥

①

❞ ✶ ❞ ❞ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱✽✼✮ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✇✐t❤ ❘✲s♠♦♦t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥

• ❩t(❉) −

|❉| (✹πt)❞/✷ + |∂❉| ✹(✹πt)(❞−✶)/✷

• ≤

❞ ✹ π❞/✷ |❉| t(❞−✷)/✷❘✷ , ❢♦r ❛❧❧ t > ✵ ❞ = ✷, ❧✐♠

t↓✵ ❩t(❉) − |❉|

✹πt + |∂❉| ✹(✹πt)✶/✷ = ✶ − ❤ ✻ ❚❤❡ ✐❞❡❛ ✐s t♦ ✉s❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❇t = (❇✶

t , ..., ❇❞ t ),

✇❤❡r❡ ❇✐

t ❛r❡ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s ❛♥❞ t❤❡

♣r♦♣❡rt✐❡s ♦❢ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❜r✐❞❣❡✳ ❘❡♠❛r❦ ❳t ❛♥❞ ❨t ❳ ✶

t

❳ ❞

t ✱ ❳ ✐ t ✱ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t

✕st❛❜❧❡ ♣r♦❝❡ss❡s ❛r❡ t✇♦ ❞✐✛❡r❡♥t ▲❡✈② ♣r♦❝❡ss❡s s✐♥❝❡ ❊① ❡✐

❨t ❨✵

❡

t

❞ ❥ ✶ ❥

①

❞ ✶ ❞ ❞ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ✭❱❛♥ ❞❡♥ ❇❡r❣✱✽✼✮ ▲❡t ❉ ❜❡ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✇✐t❤ ❘✲s♠♦♦t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥

• ❩t(❉) −

|❉| (✹πt)❞/✷ + |∂❉| ✹(✹πt)(❞−✶)/✷

• ≤

❞ ✹ π❞/✷ |❉| t(❞−✷)/✷❘✷ , ❢♦r ❛❧❧ t > ✵ ❞ = ✷, ❧✐♠

t↓✵ ❩t(❉) − |❉|

✹πt + |∂❉| ✹(✹πt)✶/✷ = ✶ − ❤ ✻ ❚❤❡ ✐❞❡❛ ✐s t♦ ✉s❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❇t = (❇✶

t , ..., ❇❞ t ),

✇❤❡r❡ ❇✐

t ❛r❡ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s ❛♥❞ t❤❡

♣r♦♣❡rt✐❡s ♦❢ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❜r✐❞❣❡✳ ❘❡♠❛r❦ ❳t ❛♥❞ ❨t = (❳ ✶

t , ..., ❳ ❞ t )✱ ❳ ✐ t ✱ ♦♥❡✕❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥t α✕st❛❜❧❡ ♣r♦❝❡ss❡s

❛r❡ t✇♦ ❞✐✛❡r❡♥t ▲❡✈② ♣r♦❝❡ss❡s s✐♥❝❡ ❊①(❡✐ξ·(❨t−❨✵)) = ❡

−t

❞

• ❥=✶

|ξ❥ |α

, ① ∈ R❞, ξ = (ξ✶, ..., ξ❞) ∈ R❞.

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡ s❡t {λ♥} ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ (−∆)

α ✷ ❢ = λ❢ ✐♥ ❉,

❢ = ✵ ✐♥ ∂❉ ❚❤❡♦r❡♠ ✭❇✳ ❚✳ ❑✉❧❝③②❝❦✐ ❵✵✽✱ ❘✲s♠♦♦t❤ ❞♦♠❛✐♥s✳❙❛♠❡ ❛s ✈❛♥ ❞❡♥ ❇❡r❣ ❢♦r α = ✷✮ ▲❡t ❩❉(t) =

• ❉ ♣(α)

❉ (t, ①, ①)❞①✳ ❚❤❡♥

• ❩❉(t) − ❈✶(α, ❞)

t❞/α |❉| + ❈✷(α, ❞) t(❞−✶)/α |∂❉|

• ≤

❈✸ |❉| t(❞−✷)/α❘✷ , t > ✵. ✇❤❡r❡ ❈✶ = ♣(α)

✶ (✵) ❛♥❞ ❈✷ =

+∞

✵

r (α)

❍ (✶, (q, ✵, ...✵), (q, ✵, ...✵))❞q✱ ❍

✉♣♣❡r✕❤❛❧❢ ♣❧❛♥❡ ✐♥ R❞✳ ❖♣❡♥ ♣r♦❜❧❡♠✿ ❲❤❡♥ ❞❂✷✱ ❧✐♠

t ✵ ❩❉ t

❈✶ ✷ t✷ ❉ ❈✷ ✷ t✶

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡ s❡t {λ♥} ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ (−∆)

α ✷ ❢ = λ❢ ✐♥ ❉,

❢ = ✵ ✐♥ ∂❉ ❚❤❡♦r❡♠ ✭❇✳ ❚✳ ❑✉❧❝③②❝❦✐ ❵✵✽✱ ❘✲s♠♦♦t❤ ❞♦♠❛✐♥s✳❙❛♠❡ ❛s ✈❛♥ ❞❡♥ ❇❡r❣ ❢♦r α = ✷✮ ▲❡t ❩❉(t) =

• ❉ ♣(α)

❉ (t, ①, ①)❞①✳ ❚❤❡♥

• ❩❉(t) − ❈✶(α, ❞)

t❞/α |❉| + ❈✷(α, ❞) t(❞−✶)/α |∂❉|

• ≤

❈✸ |❉| t(❞−✷)/α❘✷ , t > ✵. ✇❤❡r❡ ❈✶ = ♣(α)

✶ (✵) ❛♥❞ ❈✷ =

+∞

✵

r (α)

❍ (✶, (q, ✵, ...✵), (q, ✵, ...✵))❞q✱ ❍

✉♣♣❡r✕❤❛❧❢ ♣❧❛♥❡ ✐♥ R❞✳ ❖♣❡♥ ♣r♦❜❧❡♠✿ ❲❤❡♥ ❞❂✷✱ ❧✐♠

t↓✵ ❩❉(t) − ❈✶(α, ✷)

t✷/α |❉| + ❈✷(α, ✷) t✶/α =?,

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

▲❡t ❍α = −∆α/✷ ❜❡ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ α✕st❛❜❧❡ ♣r♦❝❡ss ❛♥❞ ❍❱ = −∆α/✷ + ❱ ✱ ❱ ❜♦✉♥❞❡❞ ❛♥❞ ✐♥t❡❣r❛❜❧❡ ♣♦t❡♥t✐❛❧✳ ❚❤❡ ❤❡❛t ❦❡r♥❡❧ ♦❢ ❍❱ ✿ ♣❍ t ① ② ♣t ① ② ❊ t

① ② ❡

t ✵ ❱ ❳s ❞s

✇❤❡r❡ ❊ t

① ② ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ st❛❜❧❡ ♣r♦❝❡ss ✭❜r✐❞❣❡✮

st❛rt✐♥❣ ❛t ① ❛♥❞ ❝♦♥❞✐t✐♦♥❡❞ t♦ ❜❡ ❛t ② ❛t t✐♠❡ t✳ ■♥t❡r❡st❡❞ ✐♥ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ♦❢ ❚r ❡

t❍❱

❡

t❍

❞

♣❍ t ① ① ♣t ① ① ❞① ❛s t ✵✳ ❚r ❡

t❍❱

❡

t❍

r❡♣r❡s❡♥ts ❛♥ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ ❣❛✐♥ ♦r ❧♦ss ❤❡❛t t❤r♦✉❣❤t t❤❡ ❞✐❛❣♦♥❛❧ ✐♥

✷❞ ❛t t✐♠❡ t✱ s♦ t❤❛t ❢♦r s♠❛❧❧ t ✇❡ q✉❛♥t✐❢② ❤♦✇ ❢❛st t❤❡

❞✐❛❣♦♥❛❧ ✇❛s ♦✈❡r❤❡❛t❡❞ ♦r ❝♦♦❧❡❞ ❞♦✇♥ ❜② t❤❡ ♣♦t❡♥t❛✐❧ ❱✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

▲❡t ❍α = −∆α/✷ ❜❡ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ α✕st❛❜❧❡ ♣r♦❝❡ss ❛♥❞ ❍❱ = −∆α/✷ + ❱ ✱ ❱ ❜♦✉♥❞❡❞ ❛♥❞ ✐♥t❡❣r❛❜❧❡ ♣♦t❡♥t✐❛❧✳ ❚❤❡ ❤❡❛t ❦❡r♥❡❧ ♦❢ ❍❱ ✿ ♣(α)

❍ (t, ①, ②) = ♣(α) t

(①, ②)❊ t

①,②

• ❡−

t

✵ ❱ (❳s)❞s

, ✇❤❡r❡ ❊ t

①,② ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ st❛❜❧❡ ♣r♦❝❡ss ✭❜r✐❞❣❡✮

st❛rt✐♥❣ ❛t ① ❛♥❞ ❝♦♥❞✐t✐♦♥❡❞ t♦ ❜❡ ❛t ② ❛t t✐♠❡ t✳ ■♥t❡r❡st❡❞ ✐♥ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ♦❢ ❚r ❡

t❍❱

❡

t❍

❞

♣❍ t ① ① ♣t ① ① ❞① ❛s t ✵✳ ❚r ❡

t❍❱

❡

t❍

r❡♣r❡s❡♥ts ❛♥ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ ❣❛✐♥ ♦r ❧♦ss ❤❡❛t t❤r♦✉❣❤t t❤❡ ❞✐❛❣♦♥❛❧ ✐♥

✷❞ ❛t t✐♠❡ t✱ s♦ t❤❛t ❢♦r s♠❛❧❧ t ✇❡ q✉❛♥t✐❢② ❤♦✇ ❢❛st t❤❡

❞✐❛❣♦♥❛❧ ✇❛s ♦✈❡r❤❡❛t❡❞ ♦r ❝♦♦❧❡❞ ❞♦✇♥ ❜② t❤❡ ♣♦t❡♥t❛✐❧ ❱✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

▲❡t ❍α = −∆α/✷ ❜❡ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ α✕st❛❜❧❡ ♣r♦❝❡ss ❛♥❞ ❍❱ = −∆α/✷ + ❱ ✱ ❱ ❜♦✉♥❞❡❞ ❛♥❞ ✐♥t❡❣r❛❜❧❡ ♣♦t❡♥t✐❛❧✳ ❚❤❡ ❤❡❛t ❦❡r♥❡❧ ♦❢ ❍❱ ✿ ♣(α)

❍ (t, ①, ②) = ♣(α) t

(①, ②)❊ t

①,②

• ❡−

t

✵ ❱ (❳s)❞s

, ✇❤❡r❡ ❊ t

①,② ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ st❛❜❧❡ ♣r♦❝❡ss ✭❜r✐❞❣❡✮

st❛rt✐♥❣ ❛t ① ❛♥❞ ❝♦♥❞✐t✐♦♥❡❞ t♦ ❜❡ ❛t ② ❛t t✐♠❡ t✳ ■♥t❡r❡st❡❞ ✐♥ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ♦❢ ❚r(❡−t❍❱ − ❡−t❍α) =

• R❞
• ♣(α)

❍ (t, ①, ①) − ♣(α) t

(①, ①)

• ❞①,

❛s t ↓ ✵✳ ❚r ❡

t❍❱

❡

t❍

r❡♣r❡s❡♥ts ❛♥ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ ❣❛✐♥ ♦r ❧♦ss ❤❡❛t t❤r♦✉❣❤t t❤❡ ❞✐❛❣♦♥❛❧ ✐♥

✷❞ ❛t t✐♠❡ t✱ s♦ t❤❛t ❢♦r s♠❛❧❧ t ✇❡ q✉❛♥t✐❢② ❤♦✇ ❢❛st t❤❡

❞✐❛❣♦♥❛❧ ✇❛s ♦✈❡r❤❡❛t❡❞ ♦r ❝♦♦❧❡❞ ❞♦✇♥ ❜② t❤❡ ♣♦t❡♥t❛✐❧ ❱✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

▲❡t ❍α = −∆α/✷ ❜❡ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ α✕st❛❜❧❡ ♣r♦❝❡ss ❛♥❞ ❍❱ = −∆α/✷ + ❱ ✱ ❱ ❜♦✉♥❞❡❞ ❛♥❞ ✐♥t❡❣r❛❜❧❡ ♣♦t❡♥t✐❛❧✳ ❚❤❡ ❤❡❛t ❦❡r♥❡❧ ♦❢ ❍❱ ✿ ♣(α)

❍ (t, ①, ②) = ♣(α) t

(①, ②)❊ t

①,②

• ❡−

t

✵ ❱ (❳s)❞s

, ✇❤❡r❡ ❊ t

①,② ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ st❛❜❧❡ ♣r♦❝❡ss ✭❜r✐❞❣❡✮

st❛rt✐♥❣ ❛t ① ❛♥❞ ❝♦♥❞✐t✐♦♥❡❞ t♦ ❜❡ ❛t ② ❛t t✐♠❡ t✳ ■♥t❡r❡st❡❞ ✐♥ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ♦❢ ❚r(❡−t❍❱ − ❡−t❍α) =

• R❞
• ♣(α)

❍ (t, ①, ①) − ♣(α) t

(①, ①)

• ❞①,

❛s t ↓ ✵✳ ❚r(❡−t❍❱ − ❡−t❍α) r❡♣r❡s❡♥ts ❛♥ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ ❣❛✐♥ ♦r ❧♦ss ❤❡❛t t❤r♦✉❣❤t t❤❡ ❞✐❛❣♦♥❛❧ ✐♥ R✷❞ ❛t t✐♠❡ t✱ s♦ t❤❛t ❢♦r s♠❛❧❧ t ✇❡ q✉❛♥t✐❢② ❤♦✇ ❢❛st t❤❡ ❞✐❛❣♦♥❛❧ ✇❛s ♦✈❡r❤❡❛t❡❞ ♦r ❝♦♦❧❡❞ ❞♦✇♥ ❜② t❤❡ ♣♦t❡♥t❛✐❧ ❱✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

▼✳ ✈❛♥ ❞❡♥ ❇❡r❣ ✭✶✾✾✶✮✕❋♦r α = ✷ ❱ ∈ ▲∞(R❞) ∩ ▲✶(R❞)∩ ❍ö❧❞❡r ✵ < γ ≤ ✶✿ |❱ (①) − ❱ (②)| ≤ ▼|① − ②|γ ❚r(❡−t❍ − ❡−t❍✷) = ✶ (✹πt)❞/✷

• −t
• R❞ ❱ (①)❞① + ✶

✷t✷

• R❞ |❱ (①)|✷❞① + O(t✸)
• ❋♦r ❛❧❧ ✵ < γ < α❀

❊✵ [|❳✶|γ] < +∞. ❘✳❇✳ ❙✳ ❨♦❧❝✉ ✭✷✵✶✶✮✿ ❍ = ∆α/✷ + ❱ ✱ ❍✵ = ∆α/✷✱ ✵ < α ≤ ✷✳ ❱ ∈ ▲∞(R❞) ∩ ▲✶(R❞)∩ ❍ö❧❞❡r✱ ✵ < γ ≤ ✶ ❛♥❞ γ < α✳

• ❚r(❡−t❍❱ − ❡−t❍✵)
• ♣(α)

t

(✵) + t

• R❞ ❱ (①)❞① − ✶

✷t✷

• R❞ |❱ (①)|✷❞①
• ≤ ❈α,γ,❞❱ ✶
• ❱ ✷

∞❡t❱ ∞t✸ + ▼tγ/α+✷

, ∀t > ✵. ⇒ ❚r(❡−t❍❱ − ❡−t❍✵) ♣(α)

t

(✵) = −t

• R❞ ❱ (①)❞① + ✶

✷t✷

• R❞ |❱ (①)|✷❞① + O(tγ/α+✷)

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ❋♦r ❱ ∈ S(R❞)✱ t❤❡r❡ ✐s ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦♥ t❤❡ ❡①♣❛♥s✐♦♥ ❚r(❡−t❍❱ − ❡−t❍✷) ♣(✷)

t (✵)

+ t

• R❞ ❱ (θ)❞θ − t✷

✷!

• R❞ ❱ ✷(θ)❞θ

+ t✸ ✸!

• R❞ ❱ ✸(θ)❞θ + t✸

✶✷

• R❞ |∇❱ (θ)|✷❞θ = O(t✹),

✭✶✮ ❛s t ↓ ✵✳ ◆♦t❡✿

• R❞ |∇❱ (θ)|✷❞θ =
• R❞ −∆❱ (θ)❱ (θ)❞θ = E(❱ , ❱ ),

✇❤✐❝❤ ✐s t❤❡ ❉✐r✐❝❤❧❡t ❢♦r♠ ♦❢ ❱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❛♣❧❛❝✐❛♥✳ ❚❤❡ ❡①♣❡❝t❡❞ t❡r♠ ✇❛s

❞

❱

✷❞

❞

✷❱

❱ ❞ ❱ ❱ ✇❤✐❝❤ ✐s t❤❡ ❉✐r✐❝❤❧❡t ❢♦r♠ ♦❢ ❱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❋r❛❝t✐♦♥❛❧ ▲❛♣❧❛❝✐❛♥

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ❋♦r ❱ ∈ S(R❞)✱ t❤❡r❡ ✐s ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦♥ t❤❡ ❡①♣❛♥s✐♦♥ ❚r(❡−t❍❱ − ❡−t❍✷) ♣(✷)

t (✵)

+ t

• R❞ ❱ (θ)❞θ − t✷

✷!

• R❞ ❱ ✷(θ)❞θ

+ t✸ ✸!

• R❞ ❱ ✸(θ)❞θ + t✸

✶✷

• R❞ |∇❱ (θ)|✷❞θ = O(t✹),

✭✶✮ ❛s t ↓ ✵✳ ◆♦t❡✿

• R❞ |∇❱ (θ)|✷❞θ =
• R❞ −∆❱ (θ)❱ (θ)❞θ = E(❱ , ❱ ),

✇❤✐❝❤ ✐s t❤❡ ❉✐r✐❝❤❧❡t ❢♦r♠ ♦❢ ❱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❛♣❧❛❝✐❛♥✳ ❚❤❡ ❡①♣❡❝t❡❞ t❡r♠ ✇❛s

• R❞ |ξ|α| ˆ

❱ (ξ)|✷❞ξ =

• R❞ −∆α/✷❱ (θ)❱ (θ)❞θ = Eα(❱ , ❱ ),

✇❤✐❝❤ ✐s t❤❡ ❉✐r✐❝❤❧❡t ❢♦r♠ ♦❢ ❱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❋r❛❝t✐♦♥❛❧ ▲❛♣❧❛❝✐❛♥

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ✭❇❛ñ✉❡❧♦s✱ ❙á ❇❛rr❡t♦✱✾✺✮ ❋♦r ❛♥② ✐♥t❡❣❡r ❏ ≥ ✶✱ ❚r(❡−t❍❱ − ❡−t❍✷) ♣(✷)

t (✵)

=

❏

• ❧=✶

❝❧(❱ )t❧ + O(t❏+✶), ✭✷✮ ❛s t ↓ ✵✱ ✇✐t❤ ❝✶(❱ ) = −

• R❞ ❱ (θ)❞θ, ❝❧(❱ ) = (−✶)❧

❥+♥=❧ ❥≥✷

❈ (✷)

♥,❥ (❱ ), ❈❞,✷ = (✷π)❞,

❈ (✷)

♥,❥ (❱ ) =

❈❞,✷ (✷π)❥❞♥!

• ■❥
• R(❥−✶)❞
• ▲(✷)

❥ (λ, θ)

♥ ❱ (−

❥−✶

• ✐=✶

θ✐)

❥−✶

• ✐=✶
• ❱ (θ✐)❞θ✐❞λ✐❞λ❥,

▲(✷)

❥ (λ, θ) = ❥−✶

• ❦=✶

(λ❦ − λ❦+✶)

• ❦
• ✐=✶

θ✐

• ✷

−

• ❥−✶
• ❦=✶

(λ❦ − λ❦+✶)

❦

• ✐=✶

θ✐

• ✷

.

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

▼❛✐♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ s❝❛tt❡r✐♥❣ t❤❡♦r② ❢♦r ❍❱ = −∆ + ❱ ✭✏s❝❛tt❡r✐♥❣ ❜② ♣♦t❡♥t✐❛❧s✧✮✿ ❯♥❞❡r ✏t❤❡s❡ ❛ss✉♠♣t✐♦♥s✧ ♠❡r♦♠♦r♣❤✐❝ ❡①t❡♥s✐♦♥ ♦❢ (❍❱ − λ✷)−✶ ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② ♣♦❧❡s✳ ✭Pr❡❝✐s❡ st❛t❡♠❡♥t ✐♥ ❚❤❡♦r❡♠ ✹✳✶ ✐♥s ❘❇ ✫ ❙á ❇❛rr❡t♦ ✶✾✾✺✮ ❊①❛♠♣❧❡ ❋♦r ❞ ≥ ✷✱ ❚r(❡−t❍❱ − ❡−t❍α) ♣(α)

t

(✵) + t

• R❞ ❱ (θ)❞θ − t✷

✷!

• R❞ ❱ ✷(θ)❞θ + t✸

✸!

• R❞ ❱ ✸(θ)❞θ

=

• O(t✷+ ✷

α ),

✐❢ α ∈ (✶, ✷), O(t✹), ✐❢ α ∈ (✵, ✶], ❛s t ↓ ✵✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ❋♦r ❛❧❧ ❞ ≥ ✶ ❛♥❞ ✸

✷ < α < ✷✱ ✇❡ ❤❛✈❡

❚r(❡−t❍❱ − ❡−t❍α) ♣(α)

t

(✵) +t

• R❞ ❱ (θ)❞θ − t✷

✷!

• R❞ ❱ ✷(θ)❞θ

+t✸ ✸!

• R❞ ❱ ✸(θ)❞θ + L❞,α t✷+ ✷

α

• R❞ |∇❱ (θ)|✷ ❞θ = O(t✹),

❛s t ↓ ✵✱ ✇❤❡r❡ L❞,α = π❞/✷ (✷π)❞♣(α)

✶ (✵)

✶

✵

λ✶

✵

❊

• ❙∗

✶−✇❙∗ ✇

(❙∗

✶−✇ + ❙∗ ✇)✶+ ❞

✷

• ❞✇❞λ✶.

❞

✶ ✶✷ ❛s ✷✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ❋♦r ❛❧❧ ❞ ≥ ✶ ❛♥❞ ✸

✷ < α < ✷✱ ✇❡ ❤❛✈❡

❚r(❡−t❍❱ − ❡−t❍α) ♣(α)

t

(✵) +t

• R❞ ❱ (θ)❞θ − t✷

✷!

• R❞ ❱ ✷(θ)❞θ

+t✸ ✸!

• R❞ ❱ ✸(θ)❞θ + L❞,α t✷+ ✷

α

• R❞ |∇❱ (θ)|✷ ❞θ = O(t✹),

❛s t ↓ ✵✱ ✇❤❡r❡ L❞,α = π❞/✷ (✷π)❞♣(α)

✶ (✵)

✶

✵

λ✶

✵

❊

• ❙∗

✶−✇❙∗ ✇

(❙∗

✶−✇ + ❙∗ ✇)✶+ ❞

✷

• ❞✇❞λ✶.

L❞,α → ✶ ✶✷ ❛s α ↑ ✷✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❊①❛♠♣❧❡ ✭ ❞ ≥ ✹✱ α = ✶ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ − √ ∆✮ ❚r(❡−❍❱ t − ❡−❍✶t) ❈❞t−❞ + t

• R❞ ❱ (θ)❞θ − t✷

✷!

• R❞ ❱ ✷(θ)❞θ

+t✸ ✸!

• R❞ ❱ ✸(θ)❞θ − t✹

✹!

• R❞ ❱ ✹(θ)❞θ + ✹!L❞,✶
• R❞ |∇❱ (θ)|✷❞θ
• +t✺

✺!

• R❞ ❱ ✺(θ)❞θ − ✺!M❞,✶
• R❞ ❱ (θ) |∇❱ (θ)|✷ ❞θ
• = O(t✻),

❛s t ↓ ✵✳

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t ▼ ≥ ✶ ✐s ❛♥ ✐♥t❡❣❡r s❛t✐s❢②✐♥❣ ▼ < ❞+α

✷ ✳

✭❛✮ ●✐✈❡♥ ❏ ≥ ✷✱ t❤❡r❡ ❡①✐sts ❛ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ ❘(α)

❏+✶(t)✱ ✵ < t < ✶✱ s✉❝❤ t❤❛t

❚r(❡−t❍❱ − ❡−t❍α) ♣(α)

t

(✵) = −t

• R❞ ❱ (θ)❞θ+

❏

• ❥=✷

▼−✶

• ♥=✵

(−✶)♥+❥❈ (α)

♥,❥ (❱ )t

✷♥ α +❥ + tΦ(α) ❏+✶(▼)❘(α)

❏+✶(t)

✇❤❡r❡ Φ(α)

❏+✶(▼) = ♠✐♥

• ❏ + ✶, ✷ + ✷▼

α

• , ❈❞,α =

π❞/✷ ♣(α)

✶

(✵)✱ ❛♥❞ t❤❡ ❝♦♥st❛♥ts

❈ (α)

♥,❥ (❱ ) ❛r❡ ❣✐✈❡♥ ❜②

❈ (α)

♥,❥ (❱ ) =

❈❞,α (✷π)❥❞♥!

• ■❥
• R(❥−✶)❞ ❊
• ❙−❞/✷

✶, α

✷

• ▲(α)

❥

(λ, θ) ♥

• ❱ (−

❥−✶

• ✐=✶

θ✐)

❥−✶

• ✐=✶
• ❱ (θ✐)×

❞θ✐❞λ✐❞λ❥,

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❚❤❡♦r❡♠ ✭❈♦♥t✐♥✉❡❞✮ ▲(α)

❥

(λ, θ) =

❥−✶

• ❦=✶

❙∗

λ❦−λ❦+✶

• ❦
• ✐=✶

θ✐

• ✷

− ✶ ❙✶, α

✷

|

❥−✶

• ❦=✶

❙∗

λ❦−λ❦+✶ ❦

• ✐=✶

θ✐|✷, ▼♦r❡♦✈❡r✱ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❙∗

λ✶−λ✷✱ ❙∗ λ✷−λ✸✱✳✳✳✱❙∗ λ❥−✶−λ❥ ✱ ❙∗ ✶−(λ✶−λ❥ ) ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ s❛t✐s❢②

❙∗

✶−(λ✶−λ❥ ) + ❥−✶

• ❦=✶

❙∗

λ❦−λ❦+✶ = ❙✶, α

✷

❛♥❞ ❙∗

❧ D

= ❙❧ ❢♦r ❛♥② ❧ ∈ {✶ − (λ✶ − λ❥), λ❦ − λ❦+✶}❥−✶

❦=✶✳

✭❜✮ ❋♦r ❛♥② ❥ ≥ ✷ ❛♥❞ ✶ ≤ ♥ ≤ ▼✱ ❧✐♠

α↑✷ ❈ (α) ♥,❥ (❱ ) = ❈ (✷) ♥,❥ (❱ ). ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❘❡♠❛r❦ ❋♦r ❛❧❧ η < α/✷✿ ❊

• ❙η

✶, α

✷

• = Γ(✶ − ✷η

α )

Γ(✶ − η) . ❋♦r ❛❧❧ ❝ > ✵ ❛♥❞ t > ✵✱ ❡−t❝|ξ|α = ❊

• ❡−t

✷ α ❙❝ |ξ|✷

.

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

✭❛✮ ❚r(❡−t❍❱ − ❡−t❍α) = −t♣(α)

t

(✵) ❱ (✵)+ t✷ (✷π)✷❞ ✶

✵

λ✶

✵

• R✷❞ ❡−t(✶−(λ✶−λ✷))|ξ|α−t(λ✶−λ✷)|ξ−θ✶|α ˆ

❱ (−θ✶) ˆ ❱ (θ✶)❞θ✶❞λ✷❞λ✶❞ + ❘❡♠❛✐♥❞❡r ✭❜✮ ❙✶ ❙ ✶

✶ ✷ ✶ ✷

❙

✶ ✷

❙

✶ ✷

❙✶

✶ ✷

❙

✶ ✷ ❞ ❡

t ✶

✶ ✷

t

✶ ✷ ✶

❞

❞ ❊

❡

t

✷

❙✶

✶ ✷ ✷

❙

✶ ✷ ✶ ✷

❞ ❈❞ ♣t ✵ ❊ ❙

❞ ✷ ✶

✷

❡

t✷ ▲✷

✇❤❡r❡ ▲✷ ❙

✶ ✷❙✶ ✶ ✷

❙

✶ ✷

❙✶

✶ ✷

✶ ✷ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

✭❛✮ ❚r(❡−t❍❱ − ❡−t❍α) = −t♣(α)

t

(✵) ❱ (✵)+ t✷ (✷π)✷❞ ✶

✵

λ✶

✵

• R✷❞ ❡−t(✶−(λ✶−λ✷))|ξ|α−t(λ✶−λ✷)|ξ−θ✶|α ˆ

❱ (−θ✶) ˆ ❱ (θ✶)❞θ✶❞λ✷❞λ✶❞ + ❘❡♠❛✐♥❞❡r ✭❜✮ ❙✶ =

• ❙ ✶−(λ✶−λ✷) +(λ✶−λ✷) − ❙λ✶−λ✷
• + ❙λ✶−λ✷ = ❙∗

✶−(λ✶−λ✷) + ❙∗ λ✶−λ✷.

• R❞ ❡−t(✶−(λ✶−λ✷))|ξ|α−t(λ✶−λ✷)|ξ−θ✶|α❞ξ =
• R❞ ❊
• ❡

−t

✷ α

• ❙∗

✶−(λ✶−λ✷)|ξ|✷+❙∗ λ✶−λ✷ |ξ−θ✶|✷

❞ξ = ❈❞,α♣(α)

t

(✵)❊

• ❙−❞/✷

✶, α

✷

❡−t✷/α▲(α)

✷

(λ,θ)

• .

✇❤❡r❡ ▲(α)

✷ (λ, θ) =

❙∗

λ✶−λ✷❙∗ ✶−(λ✶−λ✷)

❙∗

λ✶−λ✷ + ❙∗ ✶−(λ✶−λ✷)

|θ✶|✷

▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❆❧❧ r❡❞✉❝❡s t♦ ✜♥❞✐♥❣ ❛♥ ❡①♣❛♥s✐♦♥ ❢♦r ✭❝✮ ❍(t) = ❊

• ❙−❞/✷

✶, α

✷

❡−t✷/α▲(α)

✷

(λ,θ)

• ✭❞✮

❍ t ❊ ❙

❞ ✷ ✶

✷

t✷ ❊ ❙

✶ ✷❙✶ ✶ ✷

❙

✶ ✷

❙✶

✶ ✷

✶

❞ ✷

✶ ✷

❘❡♠❛✐♥❞❡r ✭❡✮ ❋♦r ♥

❞ ✷ ✱

❊ ❙

❞ ✷ ✶ ✷

▲❥

♥

❊ ❙♥

❞ ✷ ✶ ✷ ❥ ✶ ❦ ✶ ❦ ✷ ♥ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❆❧❧ r❡❞✉❝❡s t♦ ✜♥❞✐♥❣ ❛♥ ❡①♣❛♥s✐♦♥ ❢♦r ✭❝✮ ❍(t) = ❊

• ❙−❞/✷

✶, α

✷

❡−t✷/α▲(α)

✷

(λ,θ)

• ✭❞✮

❍(t) = ❊

• ❙−❞/✷

✶, α

✷

• −t✷/α❊

  ❙∗

λ✶−λ✷❙∗ ✶−(λ✶−λ✷)

(❙∗

λ✶−λ✷ + ❙∗ ✶−(λ✶−λ✷))✶+ ❞

✷

  |θ✶|✷+❘❡♠❛✐♥❞❡r ✭❡✮ ❋♦r ♥

❞ ✷ ✱

❊ ❙

❞ ✷ ✶ ✷

▲❥

♥

❊ ❙♥

❞ ✷ ✶ ✷ ❥ ✶ ❦ ✶ ❦ ✷ ♥ ▲✉✐s ❆❝✉♥❛

❛♣♣❧✐❝❛t✐♦♥s ❙✉❜♦r❞✐♥❛t♦rs ❚✐♠❡ ❇✳▼ ❙✐♠✐❧✐t✐❡s ❚r❛❝❡

❆❧❧ r❡❞✉❝❡s t♦ ✜♥❞✐♥❣ ❛♥ ❡①♣❛♥s✐♦♥ ❢♦r ✭❝✮ ❍(t) = ❊

• ❙−❞/✷

✶, α

✷

❡−t✷/α▲(α)

✷

(λ,θ)

• ✭❞✮

❍(t) = ❊

• ❙−❞/✷

✶, α

✷

• −t✷/α❊

  ❙∗

λ✶−λ✷❙∗ ✶−(λ✶−λ✷)

(❙∗

λ✶−λ✷ + ❙∗ ✶−(λ✶−λ✷))✶+ ❞

✷

  |θ✶|✷+❘❡♠❛✐♥❞❡r ✭❡✮ ❋♦r ♥ ≤ ❞+α

✷ ✱

❊

• ❙−❞/✷

✶,α/✷

• ▲(α)

❥

(λ, θ) ♥ ≤ ❊

• ❙♥−❞/✷

✶,α/✷

❥−✶

• ❦=✶

|γ❦|✷ ♥ .

▲✉✐s ❆❝✉♥❛