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  1. ❊❳P❖◆❊◆❚■❆▲ ❖P❊❘❆❚❖❘❙ ✐♥ ♣❤②s✐❝❛❧ t❤❡♦r✐❡s ❇♦❣❞❛♥ ▼✐❡❧♥✐❦ ❉❡♣t♦✳ ❞❡ ❋ís✐❝❛✱ ❈❡♥tr♦ ❞❡ ■♥✈❡st✐❣❛❝✐ó♥ ② ❞❡ ❊st✉❞✐♦s ❆✈❛♥③❛❞♦s ❞❡❧ ■P◆ ❈■◆❱❊❙❚❆❱ ❆✳P✳ ✶✹✲✼✹✵✱ ▼é①✐❝♦✱ ❉❋ ✵✼✵✵✵✱ ▼❊❳■❈❖ ❜♦❣❞❛♥❅✜s✳❝✐♥✈❡st❛✈✳♠① ❊❞✐t❡❞ ❜② ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❇✐❛➟②st♦❦✱ ❈✐♦➟❦♦✇s❦✐❡❣♦ ✶▼✱ ✶✺✲✷✹✺ ❇✐❛➟②st♦❦✱ P♦❧❛♥❞ ❛❧✐♥❛✳❞♦❜r♦❣♦✇s❦❛❅✉✇❜✳❡❞✉✳♣❧ ❱ ❙❝❤♦♦❧ ♦♥ ●❡♦♠❡tr② ❛♥❞ P❤②s✐❝s ✹ ❏✉❧② ✕ ✾ ❏✉❧② ✷✵✶✻ ❇✐❛➟♦✇✐❡➺❛✱ P♦❧❛♥❞ ✶

  2. ❆❜str❛❝t ■♥ t❤❡ ♣r❡s❡♥t ❞❛② q✉❛♥t✉♠ t❤❡♦r✐❡s t❤❡ ✜✲ ♥✐t❡ ♦r ✐♥✜♥✐t❡ ♣r♦❞✉❝ts ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦♣✲ ❡r❛t✐♦♥s e a 1 e a 2 . . . e a n ✭❛♥❞ t❤❡✐r ❝♦♥t✐♥✉♦✉s ❡q✉✐✈✲ ❛❧❡♥ts✮ ❛r❡ ♦❢ t❤❡ ❦♥♦✇♥ ✐♠♣♦rt❛♥❝❡✱ ❜✉t t❤❡ ♣r♦❜❧❡♠ ♦❢ ❤♦✇ t♦ r❡♣r❡s❡♥t t❤❡♠ ❜② ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ ♦♣❡r❛t✐♦♥ e Ω ✇❤❡r❡ Ω ✐s t❤❡ ✧♣❤❛s❡ ♦♣❡r❛t♦r✧ ♣r❡s❡♥ts s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞✐✣❝✉❧✲ t✐❡s✳ ❚❤❡ r❡♣♦rt ❜❡❧♦✇ ♣r❡s❡♥ts t❤❡ ❛❧❣♦r✐t❤♠s ✇❤✐❝❤ ♠❛❦❡ t❤✐s t❛s❦ s✐❣♥✐✜❝❛♥t❧② ❡❛s✐❡r✳ ■♥ s♦♠❡ ❝❛s❡s ❧✐❦❡ t❤❡ ✶❉ ♦s❝✐❧❧❛t♦r ✇✐t❤ t✐♠❡ ❞❡✲ ♣❡♥❞❡♥t ❡❧❛st✐❝ ❢♦r❝❡ t❤❡② ❧❡❛❞ t♦ ✐♥t❡r❡st✐♥❣ ❡①❛❝t s♦❧✉t✐♦♥s✳ ■♥ s♦♠❡ ♦t❤❡r ♠♦r❡ ❞✐♠❡♥✲ s✐♦♥❛❧ ❝❛s❡s t❤❡② tr❛❞✉❝❡ t❤❡♠s❡❧✈❡s ✐♥t♦ t❤❡ ✐♠♣♦rt❛♥t ♥♦♥✲❧✐♥❡❛r ♠❛tr✐① ❡q✉❛t✐♦♥s✳ ✷

  3. ❈♦♥t❡♥ts ■◆❚❘❖❉❯❈❚■❖◆ ✹ ✶ ❚❍❊ ❉■❙❈❘❊❚❊ ❈❆❙❊ ✻ ✶✳✶ ❆♥ ❛✉①✐❧✐❛r② s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❚❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷ ❚❍❊ ❈❖◆❚■◆❯❖❯❙ ❈❆❙❊ ✶✺ ✷✳✶ ❚❤❡ ❧✐♠✐t✐♥❣ ♣r♦❝❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❆♣♣r♦①✐♠❛t✐♦♥s ♦❢ ▼❛❣♥✉s ❛♥❞ ❲✐❧❝♦① ✳ ✶✼ ✷✳✸ ❚❤❡ s❡❛r❝❤ ❢♦r ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥s ✳ ✳ ✳ ✶✽ ✸ ❱❆❘■❆❇▲❊ ❖❙❈■▲▲❆❚❖❘❙ ✷✺ ✸✳✶ ❈❧❛ss✐❝❛❧✕◗✉❛♥t✉♠ ❡q✉✐✈❛❧❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷ ❑✐❝❦ ♦♣❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸ ❚❤❡ ❣❡♥❡r❛❧ ♦♣❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹ ❚❤❡ ♠❛♥✐♣✉❧❛t✐♦♥ ❜② t✐♠❡ ❞❡♣❡♥❞❡♥t ♠❛❣✲ ♥❡t✐❝ ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸

  4. ■◆❚❘❖❉❯❈❚■❖◆ ■♥ t❤❡ ❝✉st♦♠❛r② ❝❛❧❝✉❧❛t✐♦♥ tr❡♥❞s ♦❢ ◗▼ s♦♠❡ ✜♥❡ ❛❧❣❡❜r❛✐❝ ♣r♦❜❧❡♠s ❢♦r ♥♦♥✲❝♦♠♠✉t✐♥❣ ❡①♣♦♥❡♥ts ✈❡r② s❡❧❞♦♠ ❛♣♣❡❛r✳ ❚❤❡ r❡♣♦rt ❜❡❧♦✇ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s e a e b ❢♦r t❤❡ ♥♦♥✲❝♦♠♠✉t✐♥❣ a, b ✱ ❛❧s♦ ❢♦r ♠✉❧t✐♣❧❡ ❡q✉✐✈❛❧❡♥ts e a 1 e a 2 . . . e a n ✇❤❡r❡ t❤❡ a k = − iH k δ k r❡♣r❡s❡♥t ❞✐st✐♥❝t✱ ♥♦♥✲❝♦♠♠✉t✐♥❣ ❡✈♦❧✉t✐♦♥ st❡✲ ♣s✱ ❛s ✇❡❧❧ ❛s t♦ t❤❡✐r ❧✐♠✐t✐♥❣ ❝❛s❡ ✐✳❡✳ t❤❡ ❡✈♦❧✉t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② t❤❡ − iH ( t ) dt ✇❤❡r❡ H ( t ) ✐s ❛ t✐♠❡ ❞❡♣❡♥✲ ❞❡♥t ❍❛♠✐❧t♦♥✐❛♥✳ ■♥ ❛❧❧ t❤❡s❡ ❝❛s❡s✱ t❤❡ q✉❡st✐♦♥ ✐s✱ ❤♦✇ t♦ ❡①♣r❡ss ✐t ❜② t❤❡ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ e Ω( t ) ✱ ✇❤❡r❡ Ω( t ) ✐s t❤❡ ✧♣❤❛s❡ ♦♣❡r❛t♦r✧❄ ❚❤❡ ❛tt❡♠♣ts t♦ s♦❧✈❡ t❤❡ ❧❛st ♣r♦❜❧❡♠ ❜② ✐t❡r❛t✲ ✐♥❣ t❤❡ ♥♦♥✲❧✐♥❡❛r ❡q✉❛t✐♦♥ ❢♦r Ω ❢❛✐❧❡❞ ❞✉❡ t♦ t❤❡ ❢❛st ✐♥❝r❡❛s✐♥❣ ❝♦♠♣❧✐❝❛t✐♦♥ ♦❢ ❡❛❝❤ st❡♣✳ ✭▼❛❣♥✉s ✇r✐t❡s ❛❜♦✉t t❤❡ ✧❝♦♠❜✐♥❛t♦r✐❛❧ ♠❡ss✧✮✳ ❍♦✇❡✈❡r✱ ✇❤♦ ✐s ✐♥✲ t❡r❡st❡❞ ✐♥ ❛ s②♠❜♦❧✐❝ ❜✉t s✐♠♣❧❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜✲ ❧❡♠ ✭❣✐✈✐♥❣ ❡①♣❧✐❝❛t✐✈❡❧② ❛❧❧ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ st❡♣s ✐♥ ❢♦r♠ ♦❢ ♠✉❧t✐❝♦♠♠✉t❛t♦r ❡①♣r❡ss✐♦♥s✮✱ ❝❛♥ s❡❡ ✐t ✐♥ ❙❡❝✲ t✐♦♥ ✷✳✸ ♦❢ t❤✐s r❡♣♦rt✳ ■♥ ❝❛s❡s ✇❤❡♥ t❤❡ ♦♣❡r❛t✐♦♥ ❡①♣♦♥❡♥ts r❡♣r❡s❡♥t t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛✱ t❤❡ t❡r♠s ♦❢ t❤❡ ✐♥✜✲ ♥✐t❡ ♠✉❧t✐♣❧❡ ❝♦♠♠✉t❛t♦r s❡r✐❡s st❛rt t♦ r❡♣❡❛t t❤❡♠✲ ✹

  5. s❡❧✈❡s✱ s✉♠♠✐♥❣ ✉♣ t♦ s♦♠❡ ❝❧♦s❡❞ ♠❛tr✐① ❡①♣r❡ss✐♦♥s✳ ❙♦ ✐t ❤❛♣♣❡♥s ❢♦r q✉❛♥t✉♠ s②st❡♠s ✇✐t❤ ❍❛♠✐❧t♦♥✐❛♥s q✉❛❞r❛t✐❝ ✐♥ t❤❡ ❝❛♥♦♥✐❝❛❧ ♦♣❡r❛t♦rs q 1 , ..., q n , p 1 , ...p n ✳ ■♥ t❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ ✶❉ t✐♠❡ ❞❡♣❡♥❞❡♥t ♦s❝✐❧❧❛t♦r ❍❛♠✐❧t♦♥✐❛♥s H ( t ) = p 2 2 + β ( t ) q 2 2 ✇✐t❤ t❤❡ ✈❛r✐❛❜❧❡ ❡❧❛s✲ t✐❝ ❢♦r❝❡✱ t❤❡ ❡✈♦❧✉t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥t 2 × 2 s②♠♣❧❡❝t✐❝ ♠❛tr✐① ✇❤✐❝❤ ✐♥ t❤❡ s②♠♠❡✲ tr② ✐♥t❡r✈❛❧s ♦❢ β ❛❧❧♦✇s t❤❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥s✱ ♦✛❡r✐♥❣ t❤❡ s♦❢t ✐♠✐t❛t✐♦♥s ♦❢ t❤❡ ♦s❝✐❧❧❛t♦r δ ✲❦✐❝❦s ♦r t❤❡ ❞✐s✲ t♦rt❡❞ ❝❛s❡s ♦❢ t❤❡ ❢r❡❡ ❡✈♦❧✉t✐♦♥ ✭s❡❡ ❙❡❝t✐♦♥ ✸✳✷✲✸✳✹ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❛❜✐❧✐t② ♠❛♣s✮✳ ❈❡rt❛✐♥ ♠❛♥② ❞✐♠❡♥s✐♦♥❛❧ ♠♦❞❡❧s ❧❡❛❞ ❛❧s♦ t♦ s♦♠❡ ✐♥t❡r❡st✐♥❣ ♥♦♥✲ ❧✐♥❡❛r ♠❛tr✐① ❡q✉❛t✐♦♥s✱ ✇✐t❤ ♣♦ss✐❜❧❡ ♣❤②s✐❝❛❧ ✐♠♣♦r✲ t❛♥❝❡✱ t❤♦✉❣❤ ✐♥ ❣❡♥❡r❛❧✱ t❤❡② ❝❛♥♥♦t ❜❡ r❡s♦❧✈❡❞ ✇✐t❤✲ ♦✉t t❤❡ ❤❡❧♣ ♦❢ ❝♦♠♣✉t❡rs✳ ■♥ s❡✈❡r❛❧ ♣❧❛❝❡s ♦✉r r❡♣♦rt ♦✛❡rs ♥♦ ❞❡t❛✐❧s✱ ❜✉t ♦♥❧② ❤✐♥ts ❛♥❞ r❡❢❡r❡♥❝❡s ❢♦r ✐♥t❡r✲ ❡st❡❞ r❡❛❞❡rs✳ ✺

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