srs stts r r - - PowerPoint PPT Presentation
srs stts r r rr rtrs rs tr t t s st
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❏♦✐♥t ✇✐t❤✿ ❆❧❡❦s❡② ❑♦st❡♥❦♦ ❛♥❞ ●❡r❛❧❞ ❚❡s❝❤❧
❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❱✐❡♥♥❛ ❆✲✶✵✾✵ ❱✐❡♥♥❛
❖❚■◆❉ ❱✐❡♥♥❛✱ ✶✾✳✶✷✳✷✵✶✻
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶ ✴ ✶✹
■♥
✷
✱
✷ ✷
✶
✷
✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡
✐
❛s ❄ ■♥
✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
❡
✐ ✷ ✷✳ ✵ ✵✿
✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ❡
✐
✵ ✵
✶ ✹ ✐ ❡✐
✷ ✹
❡✐
✷ ✹
❖❜✈✐♦✉s❧②✿ ❡
✐
✵ ✶
✶ ✹
✱ ✳ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡
✐
❛s ❄ ■♥
✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
❡
✐ ✷ ✷✳ ✵ ✵✿
✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ❡
✐
✵ ✵
✶ ✹ ✐ ❡✐
✷ ✹
❡✐
✷ ✹
❖❜✈✐♦✉s❧②✿ ❡
✐
✵ ✶
✶ ✹
✱ ✳ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥
✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
❡
✐ ✷ ✷✳ ✵ ✵✿
✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ❡
✐
✵ ✵
✶ ✹ ✐ ❡✐
✷ ✹
❡✐
✷ ✹
❖❜✈✐♦✉s❧②✿ ❡
✐
✵ ✶
✶ ✹
✱ ✳ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
✵ ✵✿
✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ❡
✐
✵ ✵
✶ ✹ ✐ ❡✐
✷ ✹
❡✐
✷ ✹
❖❜✈✐♦✉s❧②✿ ❡
✐
✵ ✶
✶ ✹
✱ ✳ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
H✵
✵✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿
❡−✐tH✵
✵f (x) =
✶ √ ✹π✐t
✹t
− ❡✐ (x+y)✷
✹t
❖❜✈✐♦✉s❧②✿ ❡
✐
✵ ✶
✶ ✹
✱ ✳ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
H✵
✵✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿
❡−✐tH✵
✵f (x) =
✶ √ ✹π✐t
✹t
− ❡✐ (x+y)✷
✹t
❖❜✈✐♦✉s❧②✿ ❡−✐tH✵L✶(R)→L∞(R) =
✶ √ ✹πt ✱ t → ∞✳
◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
H✵
✵✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿
❡−✐tH✵
✵f (x) =
✶ √ ✹π✐t
✹t
− ❡✐ (x+y)✷
✹t
❖❜✈✐♦✉s❧②✿ ❡−✐tH✵L✶(R)→L∞(R) =
✶ √ ✹πt ✱ t → ∞✳
◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿
✶
✶ ✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
H✵
✵✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿
❡−✐tH✵
✵f (x) =
✶ √ ✹π✐t
✹t
− ❡✐ (x+y)✷
✹t
❖❜✈✐♦✉s❧②✿ ❡−✐tH✵L✶(R)→L∞(R) =
✶ √ ✹πt ✱ t → ∞✳
◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿
✶
l ≥ − ✶
✷❄
✷
✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
H✵
✵✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿
❡−✐tH✵
✵f (x) =
✶ √ ✹π✐t
✹t
− ❡✐ (x+y)✷
✹t
❖❜✈✐♦✉s❧②✿ ❡−✐tH✵L✶(R)→L∞(R) =
✶ √ ✹πt ✱ t → ∞✳
◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿
✶
l ≥ − ✶
✷❄
✷
V = ✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
■♥ L✷(R+)✱ H = − d✷ dx✷ + l(l + ✶) x✷ + V (x) = Hl + V (x), l ≥ −✶ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡−✐Ht ❛s t → ∞❄ ■♥ L✷✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱
H✵
✵✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿
❡−✐tH✵
✵f (x) =
✶ √ ✹π✐t
✹t
− ❡✐ (x+y)✷
✹t
❖❜✈✐♦✉s❧②✿ ❡−✐tH✵L✶(R)→L∞(R) =
✶ √ ✹πt ✱ t → ∞✳
◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿
✶
l ≥ − ✶
✷❄
✷
V = ✵❄
✸
❖t❤❡r ❜✳❝✳❄
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹
❆♣♣❧✐❝❛t✐♦♥s t♦ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✿ ❡st✐♠❛t❡s ♦❢ ♣r❡✈✐♦✉s t②♣❡ ❧❡❛❞ t♦ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s ✭❙tr✐❝❤❛rt③ ✭✶✾✼✼✮✱ ✳✳✳✮ ■♥ ❝♦♥t❡①t ♦❢ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ♣r♦✈✐♥❣ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r s♦❧✐t♦♥s✭st❛♥❞✐♥❣ ✇❛✈❡s✮ ❘✐❝❤ ❤✐st♦r②✿ ❘❛✉❝❤ ✭✶✾✼✽✮✱ ❏❡♥s❡♥ ✫ ❑❛t♦ ✭✶✾✼✾✮✱ ✳✳✳✳✱ ❏♦✉r♥é ✫ ❙♦✛❡r ✫ ❙♦❣❣❡ ✭✶✾✾✶✮✱✳✳✳✱ ❙❝❤❧❛❣ ✭✷✵✵✸✮✱✳✳✳ ❋♦r ♦✉r ♦♣❡r❛t♦r ✿ ✵✿ ❲❡❞❡r ✭✷✵✵✸✮ ✵✱
✶ ✷✱ ❋r✐❡❞r✐❝❤s ❜✳❝✳✿ ❑♦✈❛r✐❦ ✫ ❚r✉❝ ✭✷✵✶✹✮✿
■♥tr♦❞✉❝❡
✶ ✷
✶ ✱ ✶ ✳ ❚❤❡♥ ❢♦r ✵ ✶
✷
✿ ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✸ ✴ ✶✹
❆♣♣❧✐❝❛t✐♦♥s t♦ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✿ ❡st✐♠❛t❡s ♦❢ ♣r❡✈✐♦✉s t②♣❡ ❧❡❛❞ t♦ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s ✭❙tr✐❝❤❛rt③ ✭✶✾✼✼✮✱ ✳✳✳✮ ■♥ ❝♦♥t❡①t ♦❢ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ♣r♦✈✐♥❣ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r s♦❧✐t♦♥s✭st❛♥❞✐♥❣ ✇❛✈❡s✮ ❘✐❝❤ ❤✐st♦r②✿ ❘❛✉❝❤ ✭✶✾✼✽✮✱ ❏❡♥s❡♥ ✫ ❑❛t♦ ✭✶✾✼✾✮✱ ✳✳✳✳✱ ❏♦✉r♥é ✫ ❙♦✛❡r ✫ ❙♦❣❣❡ ✭✶✾✾✶✮✱✳✳✳✱ ❙❝❤❧❛❣ ✭✷✵✵✸✮✱✳✳✳ ❋♦r ♦✉r ♦♣❡r❛t♦r ✿ ✵✿ ❲❡❞❡r ✭✷✵✵✸✮ ✵✱
✶ ✷✱ ❋r✐❡❞r✐❝❤s ❜✳❝✳✿ ❑♦✈❛r✐❦ ✫ ❚r✉❝ ✭✷✵✶✹✮✿
■♥tr♦❞✉❝❡
✶ ✷
✶ ✱ ✶ ✳ ❚❤❡♥ ❢♦r ✵ ✶
✷
✿ ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✸ ✴ ✶✹
❆♣♣❧✐❝❛t✐♦♥s t♦ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✿ ❡st✐♠❛t❡s ♦❢ ♣r❡✈✐♦✉s t②♣❡ ❧❡❛❞ t♦ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s ✭❙tr✐❝❤❛rt③ ✭✶✾✼✼✮✱ ✳✳✳✮ ■♥ ❝♦♥t❡①t ♦❢ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ♣r♦✈✐♥❣ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r s♦❧✐t♦♥s✭st❛♥❞✐♥❣ ✇❛✈❡s✮ ❘✐❝❤ ❤✐st♦r②✿ ❘❛✉❝❤ ✭✶✾✼✽✮✱ ❏❡♥s❡♥ ✫ ❑❛t♦ ✭✶✾✼✾✮✱ ✳✳✳✳✱ ❏♦✉r♥é ✫ ❙♦✛❡r ✫ ❙♦❣❣❡ ✭✶✾✾✶✮✱✳✳✳✱ ❙❝❤❧❛❣ ✭✷✵✵✸✮✱✳✳✳ ❋♦r ♦✉r ♦♣❡r❛t♦r ✿ ✵✿ ❲❡❞❡r ✭✷✵✵✸✮ ✵✱
✶ ✷✱ ❋r✐❡❞r✐❝❤s ❜✳❝✳✿ ❑♦✈❛r✐❦ ✫ ❚r✉❝ ✭✷✵✶✹✮✿
■♥tr♦❞✉❝❡
✶ ✷
✶ ✱ ✶ ✳ ❚❤❡♥ ❢♦r ✵ ✶
✷
✿ ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✸ ✴ ✶✹
❆♣♣❧✐❝❛t✐♦♥s t♦ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✿ ❡st✐♠❛t❡s ♦❢ ♣r❡✈✐♦✉s t②♣❡ ❧❡❛❞ t♦ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s ✭❙tr✐❝❤❛rt③ ✭✶✾✼✼✮✱ ✳✳✳✮ ■♥ ❝♦♥t❡①t ♦❢ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ♣r♦✈✐♥❣ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r s♦❧✐t♦♥s✭st❛♥❞✐♥❣ ✇❛✈❡s✮ ❘✐❝❤ ❤✐st♦r②✿ ❘❛✉❝❤ ✭✶✾✼✽✮✱ ❏❡♥s❡♥ ✫ ❑❛t♦ ✭✶✾✼✾✮✱ ✳✳✳✳✱ ❏♦✉r♥é ✫ ❙♦✛❡r ✫ ❙♦❣❣❡ ✭✶✾✾✶✮✱✳✳✳✱ ❙❝❤❧❛❣ ✭✷✵✵✸✮✱✳✳✳ ❋♦r ♦✉r ♦♣❡r❛t♦r ✿ ✵✿ ❲❡❞❡r ✭✷✵✵✸✮ ✵✱
✶ ✷✱ ❋r✐❡❞r✐❝❤s ❜✳❝✳✿ ❑♦✈❛r✐❦ ✫ ❚r✉❝ ✭✷✵✶✹✮✿
■♥tr♦❞✉❝❡
✶ ✷
✶ ✱ ✶ ✳ ❚❤❡♥ ❢♦r ✵ ✶
✷
✿ ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✸ ✴ ✶✹
❆♣♣❧✐❝❛t✐♦♥s t♦ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✿ ❡st✐♠❛t❡s ♦❢ ♣r❡✈✐♦✉s t②♣❡ ❧❡❛❞ t♦ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s ✭❙tr✐❝❤❛rt③ ✭✶✾✼✼✮✱ ✳✳✳✮ ■♥ ❝♦♥t❡①t ♦❢ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ♣r♦✈✐♥❣ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r s♦❧✐t♦♥s✭st❛♥❞✐♥❣ ✇❛✈❡s✮ ❘✐❝❤ ❤✐st♦r②✿ ❘❛✉❝❤ ✭✶✾✼✽✮✱ ❏❡♥s❡♥ ✫ ❑❛t♦ ✭✶✾✼✾✮✱ ✳✳✳✳✱ ❏♦✉r♥é ✫ ❙♦✛❡r ✫ ❙♦❣❣❡ ✭✶✾✾✶✮✱✳✳✳✱ ❙❝❤❧❛❣ ✭✷✵✵✸✮✱✳✳✳ ❋♦r ♦✉r ♦♣❡r❛t♦r H✿ l = ✵✿ ❲❡❞❡r ✭✷✵✵✸✮ ✵✱
✶ ✷✱ ❋r✐❡❞r✐❝❤s ❜✳❝✳✿ ❑♦✈❛r✐❦ ✫ ❚r✉❝ ✭✷✵✶✹✮✿
■♥tr♦❞✉❝❡
✶ ✷
✶ ✱ ✶ ✳ ❚❤❡♥ ❢♦r ✵ ✶
✷
✿ ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✸ ✴ ✶✹
❆♣♣❧✐❝❛t✐♦♥s t♦ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✿ ❡st✐♠❛t❡s ♦❢ ♣r❡✈✐♦✉s t②♣❡ ❧❡❛❞ t♦ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s ✭❙tr✐❝❤❛rt③ ✭✶✾✼✼✮✱ ✳✳✳✮ ■♥ ❝♦♥t❡①t ♦❢ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ♣r♦✈✐♥❣ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r s♦❧✐t♦♥s✭st❛♥❞✐♥❣ ✇❛✈❡s✮ ❘✐❝❤ ❤✐st♦r②✿ ❘❛✉❝❤ ✭✶✾✼✽✮✱ ❏❡♥s❡♥ ✫ ❑❛t♦ ✭✶✾✼✾✮✱ ✳✳✳✳✱ ❏♦✉r♥é ✫ ❙♦✛❡r ✫ ❙♦❣❣❡ ✭✶✾✾✶✮✱✳✳✳✱ ❙❝❤❧❛❣ ✭✷✵✵✸✮✱✳✳✳ ❋♦r ♦✉r ♦♣❡r❛t♦r H✿ l = ✵✿ ❲❡❞❡r ✭✷✵✵✸✮ V = ✵✱ l ≥ − ✶
✷✱ ❋r✐❡❞r✐❝❤s ❜✳❝✳✿ ❑♦✈❛r✐❦ ✫ ❚r✉❝ ✭✷✵✶✹✮✿
■♥tr♦❞✉❝❡ ν :=
✷ + l(l + ✶)✱ ρ(x) := ✶ + |x|✳ ❚❤❡♥ ❢♦r
s ∈ [✵, ✶
✷ + ν]✿
❡−✐tHlL✶(R+,ρs)→L∞(R+,ρ−s) = O(|t|−✶/✷−s).
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✸ ✴ ✶✹
❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿
✶ ✶ ✷ ✶
✶ ❧♦❣
✶ ✷
▲❡t
✷ ✷
✶
✷
❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ❢♦r ❛❧❧
✶ ✷
✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r
✶ ✷ ✶ ✷ ✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r ✶ ✷✳
❋♦r
✶ ✷ ✶ ✷ ✿
❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s ✱ ♣❛r❛♠❡tr✐③❡❞ ❜② ✵ ✭ ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r
✶ ✷✿ ♠✐♥
✐s s❡❧❢✲❛❞❥♦✐♥t ✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t
✷ ✷
✶
✷
❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ❢♦r ❛❧❧
✶ ✷
✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r
✶ ✷ ✶ ✷ ✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r ✶ ✷✳
❋♦r
✶ ✷ ✶ ✷ ✿
❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s ✱ ♣❛r❛♠❡tr✐③❡❞ ❜② ✵ ✭ ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r
✶ ✷✿ ♠✐♥
✐s s❡❧❢✲❛❞❥♦✐♥t ✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ❢♦r ❛❧❧
✶ ✷
✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r
✶ ✷ ✶ ✷ ✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r ✶ ✷✳
❋♦r
✶ ✷ ✶ ✷ ✿
❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s ✱ ♣❛r❛♠❡tr✐③❡❞ ❜② ✵ ✭ ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r
✶ ✷✿ ♠✐♥
✐s s❡❧❢✲❛❞❥♦✐♥t ✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r
✶ ✷ ✶ ✷ ✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r ✶ ✷✳
❋♦r
✶ ✷ ✶ ✷ ✿
❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s ✱ ♣❛r❛♠❡tr✐③❡❞ ❜② ✵ ✭ ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r
✶ ✷✿ ♠✐♥
✐s s❡❧❢✲❛❞❥♦✐♥t ✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
τ ✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r l ∈ [− ✶
✷, ✶ ✷)✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r l ≥ ✶ ✷✳
❋♦r
✶ ✷ ✶ ✷ ✿
❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s ✱ ♣❛r❛♠❡tr✐③❡❞ ❜② ✵ ✭ ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r
✶ ✷✿ ♠✐♥
✐s s❡❧❢✲❛❞❥♦✐♥t ✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
τ ✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r l ∈ [− ✶
✷, ✶ ✷)✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r l ≥ ✶ ✷✳
❋♦r l ∈ [− ✶
✷, ✶ ✷)✿ τ ❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s Hα✱ ♣❛r❛♠❡tr✐③❡❞ ❜②
α ∈ [✵, π) ✭α = ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r
✶ ✷✿ ♠✐♥
✐s s❡❧❢✲❛❞❥♦✐♥t ✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
τ ✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r l ∈ [− ✶
✷, ✶ ✷)✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r l ≥ ✶ ✷✳
❋♦r l ∈ [− ✶
✷, ✶ ✷)✿ τ ❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s Hα✱ ♣❛r❛♠❡tr✐③❡❞ ❜②
α ∈ [✵, π) ✭α = ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r l ≥ ✶
✷✿ H♠✐♥ = H ✐s s❡❧❢✲❛❞❥♦✐♥t
✵ ✱ ❛♥❞
✶
✵ ✐❢
✶ ✷ ✶ ✷
✭ ✵ ✐❢
✶ ✷ ✮✱
✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
τ ✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r l ∈ [− ✶
✷, ✶ ✷)✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r l ≥ ✶ ✷✳
❋♦r l ∈ [− ✶
✷, ✶ ✷)✿ τ ❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s Hα✱ ♣❛r❛♠❡tr✐③❡❞ ❜②
α ∈ [✵, π) ✭α = ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r l ≥ ✶
✷✿ H♠✐♥ = H ✐s s❡❧❢✲❛❞❥♦✐♥t
σac(H) = [✵, ∞)✱ σsc(H) = ∅ ❛♥❞ σpp(H) = {Ej}N
j=✶ ⊂ (−∞, ✵) ✐❢ l ∈ [− ✶ ✷, ✶ ✷) ✭⊂ (−∞, ✵] ✐❢ l ≥ ✶ ✷)✮✱
N < ∞✳ ❋♦r ❛♥ ❊✳❱✳ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ ✿ ❡
✐
❡
✐
❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
τ ✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r l ∈ [− ✶
✷, ✶ ✷)✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r l ≥ ✶ ✷✳
❋♦r l ∈ [− ✶
✷, ✶ ✷)✿ τ ❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s Hα✱ ♣❛r❛♠❡tr✐③❡❞ ❜②
α ∈ [✵, π) ✭α = ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r l ≥ ✶
✷✿ H♠✐♥ = H ✐s s❡❧❢✲❛❞❥♦✐♥t
σac(H) = [✵, ∞)✱ σsc(H) = ∅ ❛♥❞ σpp(H) = {Ej}N
j=✶ ⊂ (−∞, ✵) ✐❢ l ∈ [− ✶ ✷, ✶ ✷) ✭⊂ (−∞, ✵] ✐❢ l ≥ ✶ ✷)✮✱
N < ∞✳ ❋♦r ❛♥ ❊✳❱✳ Ej ≤ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ fj✿ ❡−✐tHfj = ❡−✐tEjfj ❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡ ❡
✐
✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✿ V ∈
l > − ✶
✷,
V ∈ L✶(R+, x(✶ + | ❧♦❣(x)|)), l = − ✶
✷.
▲❡t τ = − d✷
dx✷ + l(l+✶) x✷
+ V (x) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥✿ τ ✐s ❧✐♠✐t ♣♦✐♥t ❛t ∞ ❢♦r ❛❧❧ l ≥ − ✶
✷
τ ✐s ❧✐♠✐t ❝✐r❝❧❡ ❛t ✵ ❢♦r l ∈ [− ✶
✷, ✶ ✷)✱ ❛♥❞ ❧✐♠✐t ♣♦✐♥t ❢♦r l ≥ ✶ ✷✳
❋♦r l ∈ [− ✶
✷, ✶ ✷)✿ τ ❛❞♠✐ts s❡❧❢✲❛❞❥♦✐♥t r❡❛❧✐③❛t✐♦♥s Hα✱ ♣❛r❛♠❡tr✐③❡❞ ❜②
α ∈ [✵, π) ✭α = ✵✿ ❋r✐❡❞r✐❝❤s ❡①t❡♥s✐♦♥✮ ❋♦r l ≥ ✶
✷✿ H♠✐♥ = H ✐s s❡❧❢✲❛❞❥♦✐♥t
σac(H) = [✵, ∞)✱ σsc(H) = ∅ ❛♥❞ σpp(H) = {Ej}N
j=✶ ⊂ (−∞, ✵) ✐❢ l ∈ [− ✶ ✷, ✶ ✷) ✭⊂ (−∞, ✵] ✐❢ l ≥ ✶ ✷)✮✱
N < ∞✳ ❋♦r ❛♥ ❊✳❱✳ Ej ≤ ✵ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❊❋ fj✿ ❡−✐tHfj = ❡−✐tEjfj ⇒ ❲❛② ♦✉t✿ ❊st✐♠❛t❡s ♦❢ t②♣❡
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✹ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳
✷
✿ s♦❧✉t✐♦♥ ♦❢
✷
✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ ✿ ❏♦st s♦❧✉t✐♦♥ ♦❢
✷
✇✐t❤ ❡✐ ❛s ✳ ❏♦st ❢✉♥❝t✐♦♥✿
✷
✇❤❡r❡ ✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳
✷
✿ s♦❧✉t✐♦♥ ♦❢
✷
✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ ✿ ❏♦st s♦❧✉t✐♦♥ ♦❢
✷
✇✐t❤ ❡✐ ❛s ✳ ❏♦st ❢✉♥❝t✐♦♥✿
✷
✇❤❡r❡ ✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳
✷
✿ s♦❧✉t✐♦♥ ♦❢
✷
✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ ✿ ❏♦st s♦❧✉t✐♦♥ ♦❢
✷
✇✐t❤ ❡✐ ❛s ✳ ❏♦st ❢✉♥❝t✐♦♥✿
✷
✇❤❡r❡ ✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳ φ(k✷, x)✿ s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ ✿ ❏♦st s♦❧✉t✐♦♥ ♦❢
✷
✇✐t❤ ❡✐ ❛s ✳ ❏♦st ❢✉♥❝t✐♦♥✿
✷
✇❤❡r❡ ✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳ φ(k✷, x)✿ s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ f (k, x)✿ ❏♦st s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇✐t❤ f (k, x) ∼ ❡✐kx ❛s x → ∞✳ ❏♦st ❢✉♥❝t✐♦♥✿
✷
✇❤❡r❡ ✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳ φ(k✷, x)✿ s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ f (k, x)✿ ❏♦st s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇✐t❤ f (k, x) ∼ ❡✐kx ❛s x → ∞✳ ❏♦st ❢✉♥❝t✐♦♥✿ f (k) := W (f (k, .), φ(k✷, .)), F(k) := f (k) fl(k) = ClklW (f , φ), ✇❤❡r❡ ✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
▼❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ ❙t♦♥❡✬s ❢♦r♠✉❧❛✳ φ(k✷, x)✿ s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇❤✐❝❤ s❛t✐s✜❡s ❜✳❝✳ ♥❡❛r ✵✳ f (k, x)✿ ❏♦st s♦❧✉t✐♦♥ ♦❢ τf = k✷f ✇✐t❤ f (k, x) ∼ ❡✐kx ❛s x → ∞✳ ❏♦st ❢✉♥❝t✐♦♥✿ f (k) := W (f (k, .), φ(k✷, .)), F(k) := f (k) fl(k) = ClklW (f , φ), ✇❤❡r❡ W (f , g) = fg′ − f ′g✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✺ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ❡
✐
✷ ❡
✐
✷
✷ ✷ ✷ ✶ ✷
❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r l = ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r l = ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r l = ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ F(k) ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ k = ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢
✶ ✷ ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r l = ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ F(k) ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ k = ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢ − ✶
✷ ≤ l < ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
❙t♦♥❡✬s ❢♦r♠✉❧❛✰✐♥t❡❣r❛❧ r❡♣✳ ❢♦r r❡s♦❧✈❡♥t✰r❛♥❣❡ ♦❢ ❛❝✳ s♣❡❝tr✉♠ ⇒ [❡−✐tHPc(H)](x, y) = ✷ π ∞
−∞
❡−✐tk✷ φ(k✷, x)φ(k✷, y)k✷(l+✶) |F(k)|✷ dk ❉✐✣❝✉❧t✐❡s✿ ❙♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢r❡❡ ♦♣❡r❛t♦r Hl ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s
✶
❙❡❝♦♥❞ s♦❧✉t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ❛t ✵
✷
❙✐♠♣❧❡ ❣r♦✉♣ str✉❝t✉r❡ ❢♦r ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✭❛s ❢♦r l = ✵✮ ❜r❡❛❦s ❞♦✇♥ ❢♦r ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳
❚❤❡r❡❢♦r❡✿ P❡rt✉r❜❛t✐♦♥ t❤❡♦r②✱ t❤❡♦r② ♦❢ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❇❡ss❡❧ ❢✉♥❝t✐♦♥s✱ ❍❛♥❦❡❧ tr❛♥s❢♦r♠s✱✳✳✳✮✱ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭✈❛♥ ❞❡r ❈♦r♣✉t✬s ❧❡♠♠❛✱ ❲✐❡♥❡r ❛❧❣❡❜r❛s✱✳✳✳✮ ❋✉rt❤❡r ♣r♦❜❧❡♠s✿ ♣♦ss✐❜❧❡ ③❡r♦ ♦❢ F(k) ❛t t❤❡ ❡❞❣❡ ♦❢ ❛❝✲s♣❡❝tr✉♠ k = ✵✳ ■♥ t❤✐s ❝❛s❡✿
✶
✵ ✐s ❝❛❧❧❡❞ ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡ ✐❢ − ✶
✷ ≤ l < ✶ ✷
✷
✵ ✐s ❛♥ ❊❱✱ ✐❢ l ≥ ✶
✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✻ ✴ ✶✹
✷
❚❤❡♦r❡♠ ✶ ✭❑❚❚✮
▲❡t l > − ✶
✷ ❛♥❞ ❧❡t H ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❋r✐❡❞r✐❝❤s ❜✳❝✳ ❆ss✉♠❡ t❤❛t
✶
✵
|V (x)|dx < ∞, ❛♥❞ ∞
✶
x♠❛①(✷,l+✶)|V (x)|dx < ∞, ■❢ t❤❡r❡ ✐s ♥♦ r❡s♦♥❛♥❝❡ ❛t ✵ ✭♦r ♥♦ ❊❱✱ ✐❢ l ≥ ✶
✷✮✱ t❤❡♥
t → ∞.
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✼ ✴ ✶✹
✷
❚❡❝❤♥✐❝❛❧❧② ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❞✉❡ t♦ ❧♦❣❛r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ s♦❧✉t✐♦♥s ❛♥❞ ✳
❚❤❡♦r❡♠ ✷ ✭❍❑❚✷✮
▲❡t
✶ ✷ ❛♥❞ ❧❡t
❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❋r✐❡❞r✐❝❤s ❜✳❝✳ ❆ss✉♠❡ t❤❛t
✶ ✵
✶ ❧♦❣ ❛♥❞
✶
❧♦❣✷ ✶ ❛♥❞ s✉♣♣♦s❡ t❤❡r❡ ✐s ♥♦ r❡s♦♥❛♥❝❡ ❛t ✵✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡❝❛② ❤♦❧❞s ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✽ ✴ ✶✹
✷
❚❡❝❤♥✐❝❛❧❧② ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❞✉❡ t♦ ❧♦❣❛r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ s♦❧✉t✐♦♥s φ ❛♥❞ f ✳
❚❤❡♦r❡♠ ✷ ✭❍❑❚✷✮
▲❡t
✶ ✷ ❛♥❞ ❧❡t
❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❋r✐❡❞r✐❝❤s ❜✳❝✳ ❆ss✉♠❡ t❤❛t
✶ ✵
✶ ❧♦❣ ❛♥❞
✶
❧♦❣✷ ✶ ❛♥❞ s✉♣♣♦s❡ t❤❡r❡ ✐s ♥♦ r❡s♦♥❛♥❝❡ ❛t ✵✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡❝❛② ❤♦❧❞s ❡
✐
✶
✶ ✷
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✽ ✴ ✶✹
✷
❚❡❝❤♥✐❝❛❧❧② ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❞✉❡ t♦ ❧♦❣❛r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ s♦❧✉t✐♦♥s φ ❛♥❞ f ✳
❚❤❡♦r❡♠ ✷ ✭❍❑❚✷✮
▲❡t l = − ✶
✷ ❛♥❞ ❧❡t H ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❋r✐❡❞r✐❝❤s ❜✳❝✳ ❆ss✉♠❡ t❤❛t
✶
✵
(✶ − ❧♦❣(x))|V (x)|dx < ∞ ❛♥❞ ∞
✶
x ❧♦❣✷(✶ + x)|V (x)|dx < ∞, ❛♥❞ s✉♣♣♦s❡ t❤❡r❡ ✐s ♥♦ r❡s♦♥❛♥❝❡ ❛t ✵✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡❝❛② ❤♦❧❞s
t → ∞.
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✽ ✴ ✶✹
❉♦ ❞✐✛❡r❡♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝❤❛♥❣❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r❄ ❋♦r
✷✭◆❡✉♠❛♥♥ ❜✳❝✳✮✿ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛✈❛✐❧❛❜❧❡
❚❤❡♦r❡♠ ✸ ✭❍❑❚✶✱ ❈❛s❡ ✷✮
■❢ ✶ ✷ ✵ ✱ t❤❡♥ ❡
✐
✷ ✶
✶ ✷
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❡
✐
✷ ✶
♠❛① ✶ ♠✐♥ ✶ ✶ ✷
✇❤❡♥❡✈❡r ✵ ✶ ✷ ✳ ❚❤❡ ❧❛st ❡st✐♠❛t❡ ✐s s❤❛r♣✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✾ ✴ ✶✹
❉♦ ❞✐✛❡r❡♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝❤❛♥❣❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r❄ ❋♦r
✷✭◆❡✉♠❛♥♥ ❜✳❝✳✮✿ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛✈❛✐❧❛❜❧❡
❚❤❡♦r❡♠ ✸ ✭❍❑❚✶✱ ❈❛s❡ ✷✮
■❢ ✶ ✷ ✵ ✱ t❤❡♥ ❡
✐
✷ ✶
✶ ✷
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❡
✐
✷ ✶
♠❛① ✶ ♠✐♥ ✶ ✶ ✷
✇❤❡♥❡✈❡r ✵ ✶ ✷ ✳ ❚❤❡ ❧❛st ❡st✐♠❛t❡ ✐s s❤❛r♣✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✾ ✴ ✶✹
❉♦ ❞✐✛❡r❡♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝❤❛♥❣❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r❄ ❋♦r Hπ/✷
l
✭◆❡✉♠❛♥♥ ❜✳❝✳✮✿ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛✈❛✐❧❛❜❧❡
❚❤❡♦r❡♠ ✸ ✭❍❑❚✶✱ ❈❛s❡ ✷✮
■❢ ✶ ✷ ✵ ✱ t❤❡♥ ❡
✐
✷ ✶
✶ ✷
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❡
✐
✷ ✶
♠❛① ✶ ♠✐♥ ✶ ✶ ✷
✇❤❡♥❡✈❡r ✵ ✶ ✷ ✳ ❚❤❡ ❧❛st ❡st✐♠❛t❡ ✐s s❤❛r♣✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✾ ✴ ✶✹
❉♦ ❞✐✛❡r❡♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝❤❛♥❣❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r❄ ❋♦r Hπ/✷
l
✭◆❡✉♠❛♥♥ ❜✳❝✳✮✿ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛✈❛✐❧❛❜❧❡
❚❤❡♦r❡♠ ✸ ✭❍❑❚✶✱ ❈❛s❡ α = π/✷✮
■❢ l ∈ (−✶/✷, ✵]✱ t❤❡♥ ❡−✐tHπ/✷
l
L✶(R+)→L∞(R+) = O(|t|−✶/✷), t → ∞. ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❡−✐tHπ/✷
l
L✶(R+,♠❛①(x−l,✶))→L∞(R+,♠✐♥(xl,✶)) = O(|t|−✶/✷+l), t → ∞, ✇❤❡♥❡✈❡r l ∈ (✵, ✶/✷)✳ ❚❤❡ ❧❛st ❡st✐♠❛t❡ ✐s s❤❛r♣✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✾ ✴ ✶✹
❚❤❡♦r❡♠ ✹ ✭❍❑❚✶✱ ❈❛s❡ ✵ ✷ ✷ ✮
▲❡t ✶ ✷ ❛♥❞ ✵ ✷ ✷ ✳ ❚❤❡♥ ❡
✐
✶
✶ ✷
❢♦r ❛❧❧ ✶ ✷ ✵ ✱ ❛♥❞ ❡
✐
✶
♠❛① ✶ ♠✐♥ ✶ ✶ ✷
✇❤❡♥❡✈❡r ✵ ✶ ✷ ✳ ❚♦ s✉♠♠❛r✐③❡ ✇❤❛t ❤❛♣♣❡♥❡❞ s♦ ❢❛r✿ ❆❞❞✐t✐✈❡ ♣❡rt✉r❜❛t✐♦♥s ❞♦♥✬t ✐♥✢✉❡♥❝❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r✱ ❜✉t ❝❤❛♥❣❡s ✐♥ ❜✳❝✳ ❞♦✦
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✵ ✴ ✶✹
❚❤❡♦r❡♠ ✹ ✭❍❑❚✶✱ ❈❛s❡ α ∈ (✵, π/✷) ∪ (π/✷, π)✮
▲❡t |l| < ✶/✷ ❛♥❞ α ∈ (✵, π/✷) ∪ (π/✷, π)✳ ❚❤❡♥ ❡−✐tHα
l Pc(Hα
l )L✶(R+)→L∞(R+) = O(|t|−✶/✷),
t → ∞, ❢♦r ❛❧❧ l ∈ (−✶/✷, ✵]✱ ❛♥❞ ❡−✐tHα
l Pc(Hα
l )L✶(R+,♠❛①(x−l,✶))→L∞(R+,♠✐♥(xl,✶)) = O(|t|−✶/✷),
t → ∞, ✇❤❡♥❡✈❡r l ∈ (✵, ✶/✷)✳ ❚♦ s✉♠♠❛r✐③❡ ✇❤❛t ❤❛♣♣❡♥❡❞ s♦ ❢❛r✿ ❆❞❞✐t✐✈❡ ♣❡rt✉r❜❛t✐♦♥s ❞♦♥✬t ✐♥✢✉❡♥❝❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r✱ ❜✉t ❝❤❛♥❣❡s ✐♥ ❜✳❝✳ ❞♦✦
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✵ ✴ ✶✹
❚❤❡♦r❡♠ ✹ ✭❍❑❚✶✱ ❈❛s❡ α ∈ (✵, π/✷) ∪ (π/✷, π)✮
▲❡t |l| < ✶/✷ ❛♥❞ α ∈ (✵, π/✷) ∪ (π/✷, π)✳ ❚❤❡♥ ❡−✐tHα
l Pc(Hα
l )L✶(R+)→L∞(R+) = O(|t|−✶/✷),
t → ∞, ❢♦r ❛❧❧ l ∈ (−✶/✷, ✵]✱ ❛♥❞ ❡−✐tHα
l Pc(Hα
l )L✶(R+,♠❛①(x−l,✶))→L∞(R+,♠✐♥(xl,✶)) = O(|t|−✶/✷),
t → ∞, ✇❤❡♥❡✈❡r l ∈ (✵, ✶/✷)✳ ❚♦ s✉♠♠❛r✐③❡ ✇❤❛t ❤❛♣♣❡♥❡❞ s♦ ❢❛r✿ ❆❞❞✐t✐✈❡ ♣❡rt✉r❜❛t✐♦♥s ❞♦♥✬t ✐♥✢✉❡♥❝❡ ❞✐s♣❡rs✐✈❡ ❜❡❤❛✈✐♦r✱ ❜✉t ❝❤❛♥❣❡s ✐♥ ❜✳❝✳ ❞♦✦
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✵ ✴ ✶✹
✷
▼♦st ✐♠♣♦rt❛♥t ✭❛♥❞ ❞✐✣❝✉❧t✮ ♣❛rt✱ r❡s✉❧ts ❛r❡ ♦❢ ✐♠♣♦rt❛♥❝❡ ♦♥ t❤❡✐r ♦✇♥✿
❚❤❡♦r❡♠ ✺ ✭❑❚❚✮
▲❡t
✶ ✷ ❛♥❞
❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✳ ❚❤❡♥✿ ✵ ❢♦r ✵ ✶ ✶ ❛s ✳
✶ ♠✐♥ ✸ ✷ ✷
❛s ✵✳ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ ✵ ❛♥❞
✶
❛s ✳ ❆❞❞✐t✐♦♥❛❧❧②
✶ ✷
✿
✵ ✷
✵
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✶ ✴ ✶✹
✷
▼♦st ✐♠♣♦rt❛♥t ✭❛♥❞ ❞✐✣❝✉❧t✮ ♣❛rt✱ r❡s✉❧ts ❛r❡ ♦❢ ✐♠♣♦rt❛♥❝❡ ♦♥ t❤❡✐r ♦✇♥✿
❚❤❡♦r❡♠ ✺ ✭❑❚❚✮
▲❡t
✶ ✷ ❛♥❞
❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✳ ❚❤❡♥✿ ✵ ❢♦r ✵ ✶ ✶ ❛s ✳
✶ ♠✐♥ ✸ ✷ ✷
❛s ✵✳ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ ✵ ❛♥❞
✶
❛s ✳ ❆❞❞✐t✐♦♥❛❧❧②
✶ ✷
✿
✵ ✷
✵
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✶ ✴ ✶✹
✷
▼♦st ✐♠♣♦rt❛♥t ✭❛♥❞ ❞✐✣❝✉❧t✮ ♣❛rt✱ r❡s✉❧ts ❛r❡ ♦❢ ✐♠♣♦rt❛♥❝❡ ♦♥ t❤❡✐r ♦✇♥✿
❚❤❡♦r❡♠ ✺ ✭❑❚❚✮
▲❡t l > − ✶
✷ ❛♥❞ V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✳ ❚❤❡♥✿
✵ ❢♦r ✵ ✶ ✶ ❛s ✳
✶ ♠✐♥ ✸ ✷ ✷
❛s ✵✳ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ ✵ ❛♥❞
✶
❛s ✳ ❆❞❞✐t✐♦♥❛❧❧②
✶ ✷
✿
✵ ✷
✵
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✶ ✴ ✶✹
✷
▼♦st ✐♠♣♦rt❛♥t ✭❛♥❞ ❞✐✣❝✉❧t✮ ♣❛rt✱ r❡s✉❧ts ❛r❡ ♦❢ ✐♠♣♦rt❛♥❝❡ ♦♥ t❤❡✐r ♦✇♥✿
❚❤❡♦r❡♠ ✺ ✭❑❚❚✮
▲❡t l > − ✶
✷ ❛♥❞ V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss✳ ❚❤❡♥✿
F(k) = ✵ ❢♦r k ∈ R \ {✵} F(k) = ✶ + o(✶) ❛s |k| → ∞✳ |F(k)|−✶ ≤ O(|k|− ♠✐♥(l+✸/✷,✷)) ❛s k → ✵✳ F ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ k = ✵ ❛♥❞ |F ′(k)| ≤
C ✶+|k| ❛s |k| → ∞✳
❆❞❞✐t✐♦♥❛❧❧② V ∈ L✶(R+, x✷|V (x)|)✿ |F ′(k)| = O(|k|min(✵,✷l)), k → ✵.
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✶ ✴ ✶✹
✷
❚❤❡♦r❡♠ ✻ ✭❍❑❚✷✮
▲❡t ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss ❛♥❞
✶
❧♦❣✷ ✶ ✳ ❚❤❡♥✿ ✶ ✶ ❛s ✳
✶
✐ ✶ ❧♦❣
✷ ✷
✵ ✇❤❡r❡
✶ ❛♥❞ ✷ ❛r❡ ❝♦♥t✐♥✉♦✉s r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥
✳
✷ ✵
✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✵ ✐s ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡✳ ❚❤❡♥
✶ ✵ ✷ ❧♦❣ ✷
✵
✶ ✵
✵ ✐♥ t❤✐s ❝❛s❡✳ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ ✵ ❛♥❞ ✵✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✷ ✴ ✶✹
✷
❚❤❡♦r❡♠ ✻ ✭❍❑❚✷✮
▲❡t V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss ❛♥❞ ∞
✶ x ❧♦❣✷(✶ + x)|V (x)|dx < ∞✳
❚❤❡♥✿ ✶ ✶ ❛s ✳
✶
✐ ✶ ❧♦❣
✷ ✷
✵ ✇❤❡r❡
✶ ❛♥❞ ✷ ❛r❡ ❝♦♥t✐♥✉♦✉s r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥
✳
✷ ✵
✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✵ ✐s ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡✳ ❚❤❡♥
✶ ✵ ✷ ❧♦❣ ✷
✵
✶ ✵
✵ ✐♥ t❤✐s ❝❛s❡✳ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ ✵ ❛♥❞ ✵✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✷ ✴ ✶✹
✷
❚❤❡♦r❡♠ ✻ ✭❍❑❚✷✮
▲❡t V ❜❡❧♦♥❣ t♦ t❤❡ ▼❛r❝❤❡♥❦♦ ❝❧❛ss ❛♥❞ ∞
✶ x ❧♦❣✷(✶ + x)|V (x)|dx < ∞✳
❚❤❡♥✿ |F(k)| = ✶ + o(✶) ❛s |k| → ∞✳ F(k) = F✶(k) +
π ❧♦❣(k✷)
k → ✵, ✇❤❡r❡ F✶ ❛♥❞ F✷ ❛r❡ ❝♦♥t✐♥✉♦✉s r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ R✳ F✷(✵) = ✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢ k = ✵ ✐s ❛ ♣♦✐♥t ♦❢ r❡s♦♥❛♥❝❡✳ ❚❤❡♥ F(k) = F✶(✵) + O(k✷ ❧♦❣(−k✷)), k → ✵. F✶(✵) = ✵ ✐♥ t❤✐s ❝❛s❡✳ F ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r ❛❧❧ k = ✵ ❛♥❞ |F ′(k)| ≤ C
|k|,
k = ✵✳
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✷ ✴ ✶✹
❬❑❚❚❪❆✳ ❑♦st❡♥❦♦✱ ●✳ ❚❡s❝❤❧✱ ❛♥❞ ❏✳ ❚♦❧♦③❛✱ ❉✐s♣❡rs✐♦♥ ❡st✐♠❛t❡s ❢♦r s♣❤❡r✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥s✱ ❆♥♥✳ ❍❡♥r✐ P♦✐♥❝❛ré ✶✼✱ ✸✶✹✼✲✸✶✼✻ ✭✷✵✶✻✮ ❬❍❑❚✶❪▼✳ ❍♦❧③❧❡✐t♥❡r✱ ❆✳ ❑♦st❡♥❦♦✱ ❛♥❞ ●✳ ❚❡s❝❤❧✱ ❉✐s♣❡rs✐♦♥ ❡st✐♠❛t❡s ❢♦r s♣❤❡r✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥s✿ ❚❤❡ ❡✛❡❝t ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ❖♣✉s❝✉❧❛ ▼❛t❤✳ ✸✻✱ ✼✻✾✲✼✽✻ ✭✷✵✶✻✮✳ ❬❍❑❚✷❪▼✳ ❍♦❧③❧❡✐t♥❡r✱ ❆✳ ❑♦st❡♥❦♦✱ ❛♥❞ ●✳ ❚❡s❝❤❧✱ ❉✐s♣❡rs✐♦♥ ❡st✐♠❛t❡s ❢♦r s♣❤❡r✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥s ✇✐t❤ ❝r✐t✐❝❛❧ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠✱ ❆r❳✐✈✿ ✶✻✶✶✳✵✺✷✶✵
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✸ ✴ ✶✹
▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶✹ ✴ ✶✹