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❊❳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♦❧❛♥❞

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❆❜str❛❝t

■♥ t❤❡ ♣r❡s❡♥t ❞❛② q✉❛♥t✉♠ t❤❡♦r✐❡s t❤❡ ✜✲ ♥✐t❡ ♦r ✐♥✜♥✐t❡ ♣r♦❞✉❝ts ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦♣✲ ❡r❛t✐♦♥s ea1ea2 . . . ean ✭❛♥❞ t❤❡✐r ❝♦♥t✐♥✉♦✉s ❡q✉✐✈✲ ❛❧❡♥ts✮ ❛r❡ ♦❢ t❤❡ ❦♥♦✇♥ ✐♠♣♦rt❛♥❝❡✱ ❜✉t t❤❡ ♣r♦❜❧❡♠ ♦❢ ❤♦✇ t♦ r❡♣r❡s❡♥t t❤❡♠ ❜② ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ ♦♣❡r❛t✐♦♥ eΩ ✇❤❡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✳

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❈♦♥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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

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■◆❚❘❖❉❯❈❚■❖◆ ■♥ 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 eaeb ❢♦r t❤❡ ♥♦♥✲❝♦♠♠✉t✐♥❣ a, b✱ ❛❧s♦ ❢♦r ♠✉❧t✐♣❧❡ ❡q✉✐✈❛❧❡♥ts ea1ea2 . . . ean ✇❤❡r❡ t❤❡ ak = −iHkδk 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✐❛❧ eΩ(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❤❡♠✲

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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 q1, ..., qn, p1, ...pn✳ ■♥ t❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ ✶❉ t✐♠❡ ❞❡♣❡♥❞❡♥t ♦s❝✐❧❧❛t♦r ❍❛♠✐❧t♦♥✐❛♥s H(t) = p2

2 + β(t)q2 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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✶ ❚❍❊ ❉■❙❈❘❊❚❊ ❈❆❙❊

✶✳✶ ❆♥ ❛✉①✐❧✐❛r② s♣❛❝❡

❆ s✐♠♣❧❡ ❡①♣♦♥❡♥t✐❛❧ str✉❝t✉r❡ ✇❛s ❝♦♥s✐❞❡r❡❞ ✐♥ ✼✵✲ t✐❡t❤ ❜② ❏❡r③② P❧❡❜❛➠s❦✐✱ ✇❤♦ ✇❛s ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs ❣❡♥❡r❛t❡❞ ❜② ❛ ❝♦♥t✐♥✉♦✉s ❢❛♠✐❧② ♦❢ t✐♠❡ ❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥s✱ ❡❛❝❤ t✇♦ ❝♦♠♠✉t✐♥❣ t♦ ❛ ♥✉♠❜❡r✳ ❙✉r♣r✐s✐♥❣❧②✱ ♦♥❡ ♦❢ t❤❡ ♠♦st ♥❛✐✈❡ s♦❧✉✲ t✐♦♥ ✇❛s ♦❜t❛✐♥❡❞ ❞✉r✐♥❣ t❤❡ ❞✐s❝✉ss✐♦♥s ❛t t❤❡ ❲❛rs❛✇ ■♥st✐t✉t❡ ♦❢ ❚❤❡♦r❡t✐❝❛❧ P❤②s✐❝s ✐♥ ✶✾✺✼✱ ❜② ❝♦♥s✐❞❡r✲ ✐♥❣ ❥✉st t✇♦ ❡①♣♦♥❡♥ts ❝♦♠♠✉t✐♥❣ t♦ ❛ ♥✉♠❜❡r✳ ❚❤❡ s♦❧✉t✐♦♥ ✇❛s ♦❜t❛✐♥❡❞ ❛ ❜✐t ♠②st❡r✐♦✉s❧②✱ ❜② r❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ♣r♦❜❧❡♠ ✐♥ t✇♦ ✐❞❡♥t✐❝❛❧ ❝♦♣✐❡s ♦❢ t❤❡ ❍✐❧❜❡rt s♣❛❝❡✳ ❙✉♣♣♦s❡✱ ✇❡ ❤❛✈❡ t✇♦ ♦♣❡r❛t♦rs ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ H✱ ❝♦♠♠✉t✐♥❣ t♦ ❛ ♥✉♠❜❡r α ∈ C✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s t♦ ❡①♣r❡ss eaeb ❛s ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ eΩ✳ ❚❤❡♥ ❝♦♥✲ s✐❞❡r ❛ t✇✐♥ ❝♦♣② H′ ♦❢ t❤❡ s❛♠❡ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ t❤❡ t✇✐♥ ❝♦♣✐❡s a′, b′ ♦❢ t❤❡ ♦♣❡r❛t♦rs a, b ❝♦♠♠✉t✐♥❣ t♦ t❤❡ s❛♠❡ ♥✉♠❜❡r [a′, b′] = α✳ ◆♦✇ ❞❡✜♥❡ t❤❡ t❡♥s♦r ♣r♦❞✉❝t H⊗H′✳ ❚❤❡ ♦♣❡r❛t♦rs a, b, a′, b′ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ♦♣❡r❛✲ t♦rs ❛❝t✐♥❣ ✐♥ ✇❤♦❧❡ H⊗H′✱ ❜♦t❤ a, b tr❛♥s❢♦r♠✐♥❣ ♦♥❧②

✻

t❤❡ ❝♦♠♣♦♥❡♥t H ✇✐t❤♦✉t ❛✛❡❝t✐♥❣ H′ ❛♥❞ ✐♥✈❡rs❡❧②✱ a′, b′ tr❛♥s❢♦r♠✐♥❣ H′ ✇✐t❤♦✉t ❛✛❡❝t✐♥❣ H✳ ❍❡♥❝❡✱ t❤❡ a, b ❛♥❞ t❤❡✐r ❢✉♥❝t✐♦♥s ❝♦♠♠✉t❡ ✇✐t❤ a′, b′✳ ■t ✐s ❛❧s♦ ❡s✲ s❡♥t✐❛❧ t❤❛t ✐♥ t❤❡ ❝♦♠♠✉t❛t♦r [a+b′, b+a′] t✇♦ ❝♦♠♠✉✲ t❛t♦rs [a, b] = α ❛♥❞ [b′, a′] = −α ❝❛♥❝❡❧✱ s♦ t❤❡ (a+b′) ❛♥❞ (b+a′) ❝♦♠♠✉t❡✳ ◆♦✇ ❝♦♥s✐❞❡r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♣r♦❞✉❝ts✿

✼

❇② ♠✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ❜② e−(a+b) ❢r♦♠ t❤❡ ❧❡❢t ❛♥❞ ❜② e−a′e−b′ ❢r♦♠ t❤❡ r✐❣❤t ♦♥❡ ♦❜t❛✐♥s✿ ❚❤❡ ♦♥❧② ♦♣❡r❛t♦r ✇❤✐❝❤ ❛❝ts ✐♥ H ✇✐t❤♦✉t t♦✉❝❤✐♥❣ H′ ❜✉t s✐♠✉❧t❛♥❡♦✉s❧② ❛❝ts ✐♥ H′ ✇✐t❤♦✉t ❛✛❡❝t✐♥❣ H ✐s ❥✉st ❛ ♥✉♠❜❡r✳ ❍❡♥❝❡✿ e−(a+b)eaeb = κ ∈ C. ■❢ ❤♦✇❡✈❡r ea ❛♥❞ eb ❣❡♥❡r❛t❡ t❤❡ ✉♥✐t❛r② ♦♣❡r❛t✐♦♥s ✐♥ H✱ t❤❡♥ κ ♠✉st ❛❧s♦ ❜❡ ❛ ✉♥✐t❛r② ♦♣❡r❛t♦r✱ ✐♠♣❧②✐♥❣ κ = eiφ✳ ❍❡♥❝❡✿ eaeb = eiφea+b. ❚❤❡ r❡s✉❧t ✐s ❡❛s✐❧② ✭t❤♦✉❣❤ ❥✉st s②♠❜♦❧✐❝❛❧❧②✮ ❡①t❡♥❞❡❞ t♦ ❛♥② ♥✉♠❜❡r ♦❢ ♦♣❡r❛t♦rs ♦r t♦ t❤❡ q✉❛♥t✉♠ ❡✈♦❧✉t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥s ❍✭t✮ ❝♦♠♠✉t✐♥❣ t♦ ♥✉♠❜❡rs [H(t), H(t′)] = α(t, t′)✳ ■t ✐s ✇♦rt❤ ♥♦t✐❝✐♥❣ t❤❛t t❤❡ ❛✉①✐❧✐❛r② str✉❝t✉r❡s ✇❡r❡ ✉s❡❞ ❛s ❛ ❧❡❣✐t✐♠❛t❡ t♦♦❧ t♦ ♣r♦✈❡ s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧

✽

❢❛❝ts ♥♦t ♦♥❧② ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❚❤❡ ❛♥❛❧♦❣♦✉s t❡❝❤♥✐q✉❡s ❛r❡ r❡❝❡♥t❧② ✉s❡❞ ❜② ❙✳▲✳ ❲♦r♦♥♦✇✲ ✐❝③ ❜② ❛ss♦❝✐❛t✐♥❣ t❤❡ ❍❡✐s❡♥❜❡r❣ ✇✐t❤ ✬❛♥t✐✲❍❡✐s❡♥❜❡r❣✬ ❞❡s❝r✐♣t✐♦♥s ✐♥ ❤✐s r❡s❡❛r❝❤ ♦♥ q✉❛♥t✉♠ ❣r♦✉♣s✳ P❧❡❜❛➠s❦✐✱ ♠❡❛♥✇❤✐❧❡✱ ❝♦♥s✐❞❡r❡❞ t❤❡ ❛r❣✉♠❡♥t ❡①✲ tr❡♠❡❧② ♣❡❝✉❧✐❛r✱ ❛♥❞ ❤❡ ✇❛♥t❡❞ ♠♦r❡ s❡❝✉r✐t②✳ ❲❡ ❤❛✈❡ s❤♦✇♥ t❤❡ ❧❡♠♠❛ t♦ ■✇♦ ❇✐❛➟②♥✐❝❦✐✲❇✐r✉❧❛✱ ✇❤♦ ❢♦✉♥❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ t❤♦✉❣❤ str❛♥❣❡✱ ✇❛s ❝♦rr❡❝t✳ ❇✉t ❧❛t❡r ♦♥✱ ❤❡ ❢♦✉♥❞ ❛❧s♦ t❤❛t t❤❡ ✇❤♦❧❡ r❡s✉❧t ✇❛s ❥✉st ❛♥ ✐♥❝♦♠♣❧❡t❡ ❢♦r♠ ♦❢ t❤❡ ✈❡r② ♦❧❞ ♣r♦❜❧❡♠ ♦❢ ❇❛❦❡r✕❈❛♠♣❜❡❧❧✕❍❛✉s❞♦r✛ ✭❇❈❍✮ ❢♦r♠✉❧❛✳ ❙♦ ✐t ✇❛s ✐♥❞❡❡❞✱ ❛♥❞ t♦ ❝❤❡❝❦ ✐t ❢♦r ❛ ♣❛✐r ✭♦r ❢❛♠✐❧②✮ ♦❢ ♦♣❡r❛t♦rs ❝♦♠♠✉t✐♥❣ t♦ ❛ ♥✉♠❜❡r✱ ❛ s✐♠♣❧❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇♦r❦s t❤❡ ❜❡st✳

✶✳✷ ❚❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥

❙✉♣♣♦s❡ a, b ❛r❡ ❡❧❡♠❡♥ts ♦❢ ❛ ❝❡rt❛✐♥ t♦♣♦❧♦❣✐❝❛❧ ❛❧❣❡✲ ❜r❛ A ✇✐t❤ s♦♠❡ ❡❧❡♠❡♥ts a, b ❝♦♠♠✉t✐♥❣ t♦ ❛ ♥✉♠❜❡r [a, b] = α ∈ C✳ ❚❤❡♥ ❝♦♥s✐❞❡r t❤❡ ✶✲♣❛r❛♠❡t❡r ❢❛♠✐❧✐❡s eλa ❛♥❞ eλb ✭λ ∈ R✮✳ ❆ss✉♠❡ t❤❡② ❛r❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ◆♦✇✱ ❛♣♣❧② t❤❡ ❞❡r✐✈❛t✐✈❡

d dλ ❛ss✉♠✐♥❣

t❤❛t ✐t ✐s ❧✐♥❡❛r ❛♥❞ ✇✐t❤ ♦r❞✐♥❛r② ♣r♦♣❡rt✐❡s ✇❤❡♥ ❛❝t✲ ✐♥❣ ♦♥ ♣r♦❞✉❝ts✳ ❇♦t❤ eλa ❛♥❞ eλb ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✱ ♦❜❡②✐♥❣ t❤❡ ♦❜✈✐♦✉s r✉❧❡s

d dλeλa = aeλa = eλaa ✭s✐♠✐✲ ✾

❧❛r❧② ❢♦r b✮✳ ◆♦✇ ❝♦♥s✐❞❡r t❤❡ ♣r♦❞✉❝t U(λ) = eλaeλbe−λ(a+b). ■ts ❞❡r✐✈❛t✐✈❡ ✐s✿ dU dλ = eλa(a+b)eλbe−λ(a+b)−eλaeλb(a+b)e−λ(a+b). ✭✶✳✶✮ ❚❤✐s ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ ❜② ❛♥ ♦❜✈✐♦✉s ❧❡♠♠❛✳ ❚❤❡ ❢♦r✲ ♠✉❧❛ ♦❢ ❇❛❦❡r ❬✶❪✿ eλBAe−λB = A + λ[B, A] + λ2 2! [B, [B, A]] + . . . ✐s ♦❜t❛✐♥❡❞ ❜② ❞❡✈❡❧♦♣✐♥❣ ❢♦r♠❛❧❧② t❤❡ ❧❡❢t s✐❞❡ ✐♥t♦ t❤❡ ❚❛②❧♦r s❡r✐❡s ✐♥ λ✳ ■t ❜❡❝♦♠❡s s♣❡❝✐❛❧❧② ❡❧❡♠❡♥t❛r② ✐❢ ♦♥❧② ❢❡✇ ♠✉❧t✐❝♦♠♠✉t❛t♦rs ❞♦♥✬t ✈❛♥✐s❤✳ ❚❤✐s ❤❛♣♣❡♥s ♣r❡❝✐s❡❧② ❢♦r B = b ❛♥❞ A = a✱ ✇❤❡♥ ❛❧r❡❛❞② t❤❡ ✜rst ❝♦♠♠✉t❛t♦r ✐s ❛ ♥✉♠❜❡r [b, a] = −α ❛♥❞ ❝♦♠♠✉t❡s ✇✐t❤ ❛❧❧ t❤❡ r❡st✳ ❙♦✿ eλbae−λb = a − λα ✐♠♣❧②✐♥❣ t❤❡ ♣❡r♠✉t❛t✐♦♥ r✉❧❡s eλba = (a − λα)eλb ✭❛♥❞ t❤❡ s✐♠✐❧❛r ♦♥❡ ❜② ✐♥t❡r❝❤❛♥❣✐♥❣ a ❛♥❞ b ❛♥❞ ❝❤❛♥❣✲ ✐♥❣ t❤❡ s✐❣♥ ♦❢ α✮✳ ❇② ❡♠♣❧♦②✐♥❣ ✐t t♦ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ❢♦r♠✉❧❛ ✭✶✳✶✮ ♦♥❡ ❝❛♥ ✐♥t❡r❝❤❛♥❣❡ eλb ✇✐t❤ (a+b)

✶✵

♦❜t❛✐♥✐♥❣ t♦ t❤❡ r✐❣❤t ♦❢ eλa t❤❡ s✉♠ ♦❢ ✺ t❡r♠s ✐♥ ✇❤✐❝❤ a+b ❝❛♥❝❡❧s ✇✐t❤ −(a+b) ❧❡❛✈✐♥❣ ✐♥ ♣❧❛❝❡ ♦♥❧② t❤❡ ♥✉✲ ♠❡r❛❧ t❡r♠ −λα✱ ❝♦♠♠✉t✐♥❣ ✇✐t❤ ❡✈❡r②t❤✐♥❣✳ ❍❡♥❝❡✿ dU(λ) dλ = λαU(λ) ✐s ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐❧② s♦❧✈❡❞✿ U(λ) = e

λ2 2 αU(0) = e λ2 2 [a,b]U(0)

✐♠♣❧②✐♥❣✿ eλaeλb = e

λ2 2 αeλ(a+b) = eλ(a+b)+λ2 2 [a,b],

✐✳❡✳ t❤❡ ✜rst ❛♣♣r♦①✐♠❛t✐♦♥ st❡♣ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❇❛❦❡r✲ ❈❛♠♣❜❡❧❧✲❍❛✉s❞♦r✛ ❢♦r♠✉❧❛ ✖ ❡①♣❧❛✐♥✐♥❣ t❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢ t❤❡ ♣❤❛s❡ ❢❛❝t♦r iφ ✐♥ ♦✉r ♣r❡✈✐♦✉s ❛r❣✉♠❡♥t✳ ❚❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ❢♦r t❤❡ ♠✉❧t✐♣❧❡ ♦r ❝♦♥t✐♥✉♦✉s ❡①♣♦♥❡♥ts ❝♦♠♠✉t✐♥❣ ❛❧✇❛②s t♦ t❤❡ ♥✉♠❜❡rs ❝❛♥ ❜❡ r❡❛❞✐❧② ♦❜✲ t❛✐♥❡❞✳ ■♥ t❤❡ s✐♠✐❧❛r ❢♦r♠❛❧ ✇❛②✱ ♦♥❡ ❝❛♥ s❤♦✇ ❛❧s♦ t❤❛t ❡✈❡♥ ✐❢ [a, b] ✐s ♥♦t ❛ ♥✉♠❜❡r✱ ❜✉t ❜♦t❤ ❞♦✉❜❧❡ ❝♦♠♠✉✲ t❛t♦rs ❛r❡✿ [a, [a, b]] ∈ C ❛♥❞ [b, [b, a]] ∈ C✱ t❤❡♥ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿ eλaeλb = eλ(a+b)+λ2

2 [a,b]+λ3 12 ([a,[a,b]]+[b,[b,a]]).

✶✶

❚❤❡ ❝❛s❡s ✇❤❡♥ ❛❧❧ ♥✲t❤ ♦r❞❡r ❝♦♠♠✉t❛t♦rs ❛r❡ ♥✉♠✲ ❜❡rs✱ ♦r ❛ ❣❡♥❡r❛❧ ❡①♣♦♥❡♥t ✐♥ ❢♦r♠ ♦❢ ❛♥ ✐♥✜♥✐t❡ ♠✉❧✲ t✐❝♦♠♠✉t❛t♦r s❡r✐❡s r❡q✉✐r❡ ❛❧r❡❛❞② t❤❡ ✉s❡ ♦❢ t❤❡ ✧♣♦✲ ❧❛r✐③❛t✐♦♥ ❞❡r✐✈❛t✐✈❡✧ ♦❢ ❍❛✉s❞♦r✛ ❬✷✱ ✶✱ ✸❪✳ ❚❤❡ r❡❧❛t❡❞ ❝❤❛❧❧❡♥❣❡ ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ❡①♣♦♥❡♥✲ t✐❛❧ ♣r♦❞✉❝ts✳ ❆s ✐♥t❡r❡st✐♥❣ ♠✐❣❤t ❜❡ ❛ ❞✉❛❧ ❩❛ss❡♥❤❛✉s ♣r♦❜❧❡♠ ❬✹❪ ♦❢ ❤♦✇ t♦ ❞❡❝♦♠♣♦s❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ eλ(a+b) ✐♥t♦ t❤❡ ♣r♦❞✉❝t ♦❢ s✐♠♣❧❡r ♦♥❡s✳ ❖♥❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❞❡❝♦♠✲ ♣♦s✐t✐♦♥s ✐s eλ(a+b) = eλaeλbe−λ2

2 [a,b]e λ3 6 (2[b,[a,b]]+[a,[a,b]]) . . . .

❚❤❡ s❛♠❡ r❡s✉❧t ♦❢ ❋r✐❡❞r✐❝❤s ❬✻❪ ✐♠♣❧✐❡s ❛s ✇❡❧❧ t❤❛t ❛❧❧ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r ❡①♣♦♥❡♥ts ✐♥ t❤✐s ✐♥✜♥✐t❡ ♣r♦❞✉❝t ❛r❡ ▲✐❡ ❡❧❡♠❡♥ts ✭❛♥❞ s♦✱ ❝❛♥ ❜❡ ❛❧✇❛②s ✇r✐tt❡♥ ❡✳❣✳ ✐♥ ❉②♥❦✐♥✬s ♠✉❧t✐❝♦♠♠✉t❛t♦r ♥♦t❛t✐♦♥✮✳ ❲❍❨ ❆▲▲ ❚❍■❙ ❈❆◆ ❇❊ ❖❋ ■◆❚❊❘❊❙❚ ❋❖❘ P❍❨❙■❈❙❄

✶✷

✶✸

✶✹

✷ ❚❍❊ ❈❖◆❚■◆❯❖❯❙ ❈❆❙❊

✷✳✶ ❚❤❡ ❧✐♠✐t✐♥❣ ♣r♦❝❡ss

❙♦♠❡ ✐♥t❡r❡st✐♥❣ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ tr❛❞✐t✐♦♥❛❧ ❇❛❦❡r✕ ❈❛♠♣❜❡❧❧✕❍❛✉s❞♦r✛ ❢♦r♠✉❧❛ ❛r✐s❡ ❢♦r ✐♥❝r❡❛s✐♥❣ ♥✉♠✲ ❜❡r ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦♣❡r❛t♦rs ✇✐t❤ ✐♥✜♥✐t❡s✐♠❛❧❧② s♠❛❧❧ ❡①♣♦♥❡♥ts ❇✉t ❣♦♦❞ ♥❡✇s✦ ■❢ δk − → 0 t❤❡ r❡s✉❧ts s✉❣❣❡st t❤❡ ❞✐❢✲ ❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r U(t)✿

✶✺

❆ ❜❛❞ ♥❡✇s✿ ✐t ✐s ❖✳❑✳ t♦ r❡♣r❡s❡♥t ❯✭t✮ ❜✉t ❝❛♥ s✉❣❣❡st ✇r♦♥❣ ✐♥s♣✐r❛t✐♦♥s ❛❜♦✉t ✐ts ❡①♣♦♥❡♥t✦

✶✻

✷✳✷ ❆♣♣r♦①✐♠❛t✐♦♥s ♦❢ ▼❛❣♥✉s ❛♥❞ ❲✐❧❝♦①

❚♦ ♦❜t❛✐♥ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❡①♣♦♥❡♥t Ω✱ ▼❛❣✲ ♥✉s ❬✼❪ ❛♥❞ ❲✐❧❝♦① ❬✾❪ ✐♥tr♦❞✉❝❡❞ t❤❡ ♣❛r❛♠❡t❡r λ ✐♥t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ❇❛❦❡r✕❈❛♠♣❜❡❧❧✕❍❛✉s❞♦r✛ ♣r♦❜❧❡♠ dU dt (λ, t) = λA(t)U(λ, t), U(λ, 0) = 1. ❚❤❡② ❛ss✉♠❡ t❤❛t U(λ, t) = eΩ(λ,t)✳ ❚❤❡♥ t❤❡② ❛r❡ ❧♦♦❦✲ ✐♥❣ ❢♦r Ω ✐♥ ❢♦r♠ ♦❢ ❛ s②♠❜♦❧✐❝ s❡r✐❡s Ω(λ, t) =

∞

• n=1

λn∆n(t) ❛♥❞ tr✐❡❞ t♦ ✜♥❞ ✐t ❜② ✉s✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✐♥✲ s♣✐r❡❞ ❜② ❍❛✉s❞♦r✛ ❬✸❪ 1 eµΩdΩ dt e−µΩdµ = λA(t). ▼❛❣♥✉s ♦❜t❛✐♥ ∆1✱ ∆2 ❛♥❞ ∆3 ❛♥❞ ❝♦♥❝❧✉❞❡ t❤❡ r❡st ✐s ✧❝♦♠❜✐♥❛t♦r✐❛❧ ♠❡ss✧✳ ❲✐❧❝♦① ♦❜t❛✐♥❡❞ st✐❧❧ ∆4✳ ❆♠♦♥❣ t❤❡♠ ♦♥❧② ∆1 ❛♥❞ ∆2 ❛r❡ ❡❛s② t♦ ❣✉❡ss✿ ∆1(t) = t A(t1)dt1, ∆2(t) = 1 2 t dt1 t1 [A(t1), A(t2)]dt2. ❚❤❡ r❡♠❛✐♥✐♥❣ ∆3 ❛♥❞ ∆4 ❛r❡ ✐♥❞❡❡❞ ✐♥✈♦❧✈❡❞ ❡✈❡♥ ✐♥ ♠✉❧t✐ ❝♦♠♠✉t❛t♦r t❡r♠s ❬✺✱ ✾✱ ✶✶❪✳ ❨❡t✱ ♠✉❝❤ s✐♠♣❧❡r r❡s✉❧ts ❢♦❧❧♦✇ ❞✐r❡❝t❧② ❢r♦♠ U(λ, t)✳

✶✼

✷✳✸ ❚❤❡ s❡❛r❝❤ ❢♦r ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥s

❆♥ ❛✉t❤❡♥t✐❝ ❜r❡❛❦t❤r♦✉❣❤ ❝❛♠❡ ❢r♦♠ t❤❡ ❢♦r♠❛❧ s❡r✐❡s ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ U(t)✳ ✇❤❡r❡ ✭✷✳✶✮ ❛♥❞ θ(t) = 1 t ≥ 0 0 t < 0 , θij = θ(ti − tj).

✶✽

✇❤❡r❡ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t✐❝❡ t❤❛t t❤❡ ❢♦r♠✉❧❛ ❢♦r Z2 ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r Z ❜② ❛♥ ♦♣❡r❛t✐♦♥ ❳ ♦❢ ❞r♦♣♣✐♥❣ s♦♠❡ θ✬❛s✿

✶✾

■t t✉r♥s ♦✉t t❤❛t ❳ ✐s t❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇❤✐❝❤ ❛❝ts ♦♥ θ✿ ❍❡♥❝❡❢♦rt❤✱ t❤❡ ♦♣❡r❛t✐♦♥ ❳ ♣❡r♠✐ts t♦ ✇r✐t❡ ❞♦✇♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❛❧♦❣ ♦❢ ❇❛❦❡r✕❈❛♠♣❜❡❧❧✕❍❛✉s❞♦r✛ ❢♦r♠✉❧❛ ✇✐t❤ ❛❧❧ t❡r♠s ❧✐♥❡❛r ✐♥ Z✿

✷✵

✇❤❡r❡ Ln(t1, . . . , tn) ❛r❡ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ❦❡r♥❡❧s

✷✶

❙✐♥❝❡ ❡✈❡r② ♣r♦❞✉❝t θ2,1θ3,2 . . . θk,k−1 ❝♦♥t❛✐♥s ♦♥❧② t❤❡ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ θ✬❛s✱ t❤❡♥ t❤❡ ❛❧❧ ❤✐❣❤❡r ♦r❞❡r ❞❡r✐✈❛✲ t✐✈❡s dn dθn✱ ✇✐t❤ n > k✱ ✈❛♥✐s❤ ❛♥❞ ❡✈❡r② Ln(t1, . . . , tn) r❡❞✉❝❡s ✐ts❡❧❢ ✐♥t♦ ❛♥ ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✱ ✜♥✐t❡ ❝♦♠❜✐♥❛✲ t✐♦♥ ♦❢ θ✲♣r♦❞✉❝ts✳ ❋♦r ✐♥st❛♥❝❡✿ L1 =1, L2 =θ2,1 − 1 2, L3 =θ3,2θ2,1 − 1 2θ2,1 − 1 2θ3,2 + 1 3, L4 =θ4,3θ3,2θ2,1 − 1 2θ4,3θ3,2 − 1 2θ4,3θ2,1 − 1 2θ3,2θ2,1+ + 1 3θ2,1 + 1 3θ3,2 + 1 3θ4,3 − 1 4, etc. ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ❛♣♣❧② ❛❧s♦ t❤❡ ✐♥t❡❣r❛❧ ❡①♣r❡ss✐♦♥ e− d

dθ − 1

− d

dθ

= 1 e−µ d

dθ dµ.

❇② ✉s✐♥❣ t❤❡ ♣❡r♠✉t❛t✐♦♥ r✉❧❡ e−µ d

dθ θij = (θij − µ) e−µ d dθ

✷✷

♦♥❡ ♦❜t❛✐♥s✿ Ln(t1, . . . , tn) =

• e− d

dθ − 1

− d

dθ

• θ21 . . . θn,n−1 =

= 1 e−µ d

dθ (θ21 . . . θn,n−1) dµ =

= 1 (θ21 − µ) . . . (θn,n−1 − µ) dµ. ❆♥ ✐♥t❡r❡st✐♥❣ ❝♦♥s❡q✉❡♥❝❡ ✇❛s ❞❡r✐✈❡❞ ✐♥ ❬✶✶❪✿ ❆♠♣❧❡ ❞✐s❝✉ss✐♦♥s ♦❢ t❤❡ r❡s✉❧ts ✐♥ ❬✶✵✱ ✶✶❪ ✇❡r❡ ♦✛❡r❡❞ ❜② ❏✳ ❈③②➺ ❬✶✷❪ ❛♥❞ ■✳▼✳ ●❡❧❢❛♥❞ ❬✶✸❪✳

✷✸

❉✉❡ t♦ t❤❡ t❤❡♦r❡♠ ♦❢ ❋r✐❡❞r✐❝❤s✱ ❛❧❧ t❡r♠s ♦❢ t❤❡ ❡①✲ ♣❧✐❝✐t ❢♦r♠✉❧❛ ✭✷✳✶✮ ❛r❡ t❤❡ ▲✐❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❢r❡❡ ❛❧❣❡✲ ❜r❛ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♦♣❡r❛t♦rs A(t)✳ ❍♦✇❡✈❡r✱ ✐❢ ❛♣♣❧✐❡❞ t♦ ❆✭t✮ ♦❢ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❣r♦✉♣✱ t❤❡ ✐♥❝r❡❛s✐♥❣ ♠✉❧t✐❝♦♠♠✉t❛t♦r t❡r♠s ✇✐❧❧ s❤♦✇ r❡♣❡t✐t✐♦♥s ❧❡❛❞✐♥❣ t♦ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ♠❛tr✐① ❛❧❣❡❜r❛s✳ ❚❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ t❤✐s ♠❡❝❤❛♥✐s♠ ❛r❡ t❤❡ q✉❛♥t✉♠ t❤❡♦r✐❡s ♦❢ ♦s❝✐❧❧❛t♦rs ✇✐t❤ t✐♠❡ ❞❡♣❡♥❞❡♥t ❡❧❛st✐❝ ❢♦r❝❡s✳

✷✹

✸ ❱❆❘■❆❇▲❊ ❖❙❈■▲▲❆❚❖❘❙

✸✳✶ ❈❧❛ss✐❝❛❧✕◗✉❛♥t✉♠ ❡q✉✐✈❛❧❡♥❝❡

❇❡t✇❡❡♥ t❤❡ q✉❛♥t✉♠ s②st❡♠s ✇✐t❤ ❍❛♠✐❧t♦♥✐❛♥s q✉❛❞r❛t✐❝ ✐♥ ❝❛♥♦♥✐❝❛❧ ✈❛r✐❛❜❧❡s✱ t❤❡ s✐♠♣❧❡st s♦❧✉t✐♦♥s ❛r❡ ♦❜✲ t❛✐♥❡❞ ❢♦r 1D ♦s❝✐❧❧❛t♦rs ✭1 ♣♦s✐t✐♦♥ + 1 ♠♦♠❡♥t✉♠✮ ♣❡r♠✐tt✐♥❣ t♦ s✐♠♣❧✐❢② t❤❡ t❤❡♦r② ❞✉❡ t♦ ❛ ♥♦t❛❜❧❡ ♣❤❡✲ ♥♦♠❡♥♦♥✿ t❤❡ 2×2 ♠❛tr✐❝❡s ♦❢ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ (q, p) ✈❛r✐❛❜❧❡s ❛r❡ ❡①❛❝t❧② t❤❡ s❛♠❡✦

✷✺

◆♦t❡ t❤❛t t❤❡ t✇♦ ✉♥✐t❛r② ♦♣❡r❛t♦rs U1✱ U2 ✇❤✐❝❤ ❣❡♥✲ ❡r❛t❡ t❤❡ s❛♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q, p ♣❛✐rs ❝❛♥ ❞✐✛❡r ♦♥❧② ❜② ❛ C✲♥✉♠❜❡r ♣❤❛s❡✳ ■♥❞❡❡❞✿ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ♦♣❡r❛t♦rs U1 ❛♥❞ U2 ❡q✉✐✈❛❧❡♥t ✐♥ ❛♥② ♣❤②s✐❝❛❧ ❡①♣❡r✐♠❡♥t✳ ❆tt❡✦ ❚♦ ✉♥❞❡r❧✐♥❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r U = U(t, t0) t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✈♦❧✉t✐♦♥ ♠❛tr✐① ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ u = u(t, t0)✳

✷✻

❈♦♥s❡q✉❡♥❝❡s✿ ❚❤❡ ❝❧❛ss✐❝❛❧ ❝❛♥♦♥✐❝❛❧ tr❛♥s❢♦r♠❛✲ t✐♦♥s ❞❡t❡r♠✐♥❡ t❤❡ ✉♥✐t❛r② ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs✳ = ⇒ ❚❤❡ ❝❧❛ss✐❝❛❧ ♠♦t✐♦♥ ♣❡r♠✐ts r❡❝♦♥str✉❝t t❤❡ q✉❛♥t✉♠ ❡✈♦❧✉t✐♦♥ ❢♦r ❣❡♥❡r❛❧ t✐♠❡ ❞❡♣❡♥❞❡♥t ❡❧❛st✐❝ ❢♦r❝❡ −β(t)q✳

✸✳✷ ❑✐❝❦ ♦♣❡r❛t✐♦♥s

❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥s ♦❢ ❜♦t❤ ❝❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ♣r♦❜❧❡♠ r❡q✉✐r❡s t❤❡ ❝♦♠♣✉t❡r s♦❧✉t✐♦♥s ♦❢ t❤❡ ❝♦♠✲ ♠♦♥ 2×2 ❡✈♦❧✉t✐♦♥ ♠❛tr✐① u(t)✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r♦❜❧❡♠ ❛❞♠✐ts s♦♠❡ ❡①tr❡♠❡❧② s✐♠♣❧❡ ❡①❛❝t s♦❧✉t✐♦♥s✳

✷✼

❇♦t❤ ❛r❡ ❡❛s✐❧② ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❣❡♥❡r❛❧ ❧❛✇ ♦❢ ❇❛❦❡r ❬✶❪✿

✷✽

✷✾

❍❡r❡✱ t❤❡ s✐❞❡s ♦❢ t❤❡ ❤❡①❛❣♦♥ r❡♣r❡s❡♥t t❤❡ 6 ✐❞❡♥t✐❝❛❧ ❢r❡❡ ❡✈♦❧✉t✐♦♥ ✐♥t❡r✈❛❧s ❢♦r t❤❡ t✐♠❡ τ ❛♥❞ ❛❧❧ ✈❡rt✐❝❡s st❛♥❞ ❢♦r t❤❡ ♦s❝✐❧❧❛t♦r ♣♦t❡♥t✐❛❧ ❦✐❝❦s ✇✐t❤ 1

τ ❢♦r❝❡s✳ ❇②

t❤❡ s❛♠❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ 11 ♦♣❡r❛t♦rs ♦♥❧② ♠✉st ✐♥✈❡rt t❤❡ ❢r❡❡ ❡✈♦❧✉t✐♦♥ ❞✉r✐♥❣ t❤❡ t✐♠❡ τ✿

✸✵

✸✶

✸✷

✸✳✸ ❚❤❡ ❣❡♥❡r❛❧ ♦♣❡r❛t✐♦♥s

❚❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s♦❢t ♠❛♥✐♣✉❧❛t✐♦♥s ✐♥ ❣❡♥❡r❛❧ r❡✲ q✉✐r❡s t❤❡ ❝♦♠♣✉t❡r s♦❧✉t✐♦♥s ❢♦r t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥s✿ du dt (t, τ) = Λ(t)u(t, τ), du dτ (t, τ) = −u(t, τ)Λ(τ). ❆♥ ✐♥t❡r❡st✐♥❣ ♦❜s❡r✈❛t✐♦♥ ✇❛s✱ ❤♦✇❡✈❡r✱ t❤❛t ✐❢ β(t) ❛♥❞ t❤❡r❡❢♦r❡ Λ(t) ✐s s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ s♦♠❡ ❣✐✈❡♥ ♠♦♠❡♥t t = t0✱ t❤❡♥ s♦♠❡ r❡s✉❧ts ♦❢ t❤❡ ❡✈♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❡①❛❝t❧② ♣r❡❞✐❝t❡❞✳ ❆ss✉♠❡ ❢♦r s✐♠♣❧✐❝✐t② t0 = 0 ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❡✈♦❧✉t✐♦♥ ♠❛tr✐① u = u(t, −t) ✐♥ ❛ s②♠♠❡tr✐❝ ✐♥t❡r✈❛❧ [−t, t]✳ ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❢♦r u ❤❛s t❤❡ ❛♥t✐❝♦♠♠✉t❛t♦r ❢♦r♠✿ du dt = Λ(t)u + uΛ(t) ♦r ❡①♣❧✐❝✐t❧② du dt = u21 − βu12 ❚r u −β❚r u u21 − βu12

• =

= (u21 − βu12) 1 + ❚r u 1 −β 0

• .

❚❤❡r❡❢♦r❡✱ d dt (u12u21) = ❚r u (u21 − βu12) =

✸✸

= ❚r u1 2 d dt❚r u = 1 4 d dt (❚r u)2 ❛♥❞ ✐♥t❡❣r❛t✐♥❣ d dt

• u12u21 − 1

4 (❚r u)2

• = 0

⇓ u12u21 − 1 4 (❚r u)2 = C = const. ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(0, 0) = 1 ✐♠♣❧② C = −1 ❛♥❞ s♦ u12u21 = −1 4 (❚r u)2 − 1. ✭✸✳✶✮ ❍❡♥❝❡✱ ♦♥❡ ❛rr✐✈❡s ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳ ▲❡♠♠❛ ✸✳✶✳ ❲❤❡♥❡✈❡r t❤❡ ❡✈♦❧✉t✐♦♥ ♠❛tr✐① u(t) = u(t, −t) ❢♦r s②♠♠❡tr✐❝ β(t) r❡❛❝❤❡s t❤❡ t❤r❡s❤♦❧❞ ✈❛❧✲ ✉❡s ❚r u = ±2✱ ✭✸✳✶✮ ✐♠♣❧✐❡s t❤❛t ❡✐t❤❡r u12 ♦r u21 ✭♦r ❜♦t❤✮ ♠✉st ✈❛♥✐s❤ ❛♥❞ s✐♠✉❧t❛♥❡♦✉s❧② u11 = u22 = ±1✱ ❧❡❛❞✐♥❣ t♦ t❤❡ s♦❢t ❡✈♦❧✉t✐♦♥ ❝❛s❡s ✐♠✐t❛t✐♥❣ t❤❡ ♦s❝✐❧❧❛✲ t♦r ❦✐❝❦s✱ ✐♥❝✐❞❡♥ts ♦❢ ❞✐st♦rt❡❞ ❢r❡❡ ❡✈♦❧✉t✐♦♥✱ ♦r ❥✉st ♦♥❡ ♦❢ t❤❡ ❡✈♦❧✉t✐♦♥ ❧♦♦♣s✱ ❛❧❧ ♦❢ t❤❡♠ ✇✐t❤ ♦r ✇✐t❤♦✉t s✐♠✉❧t❛♥❡♦✉s ♣❛r✐t② tr❛♥s❢♦r♠❛t✐♦♥ ✭s❡❡ ❬✶✻✱ ✶✼❪✮✳

✸✳✹ ❚❤❡ ♠❛♥✐♣✉❧❛t✐♦♥ ❜② t✐♠❡ ❞❡♣❡♥❞❡♥t ♠❛❣♥❡t✐❝ ✜❡❧❞s

❚❤❡ ❛❜♦✈❡ r❡s✉❧ts ✇❡r❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r✲ ❛t♦rs ✐♥❞✉❝❡❞ ❜② ❤♦♠♦❣❡♥❡♦✉s ✈❛r✐❛❜❧❡ ♠❛❣♥❡t✐❝ ✜❡❧❞s

✸✹

B(t)✳ ■❢ t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ B(t) ✐s ♥♦t t♦♦ ✈✐♦✲ ❧❡♥t✱ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ st✐❧❧ ❤♦❧❞s ✉♣ t♦

1 c2✲t❡r♠s✱ ✭s❡❡ ❬✶✹✱ ✶✺✱ ✶✻✱ ✶✼✱ ✶✽❪✮ t❤❡ ♠♦t✐♦♥ ♦❢ ❝❤❛r❣❡❞

♣❛rt✐❝❧❡s ♦❜❡②s t❤❡ 2 ❞✐♠❡♥s✐♦♥❛❧ ♦s❝✐❧❧❛t♦rs ✇✐t❤ t✐♠❡✲ ❞❡♣❡♥❞❡♥❝❡ r❛❞✐❛❧ ❢♦r❝❡✳ ❚❤❡ st❛❜✐❧✐t② ❛♥❞ ✐♥st❛❜✐❧✐t② ❛r❡❛s ❢♦r ✜❡❧❞s ♦❢ t✇♦ ❢r❡q✉❡♥❝✐❡s ω ❛♥❞ 2ω ✇❡r❡ ❞❡✲ t❡r♠✐♥❡❞ ✐♥ ❬✶✻❪✱ ❧❡❛❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♣ ♦❢ ❆✳ ❘❛♠✐r❡③✿ ❚❤❡ ❘❛♠✐r❡③ ♠❛♣ ✐♥ ❬✶✻❪✳ ❚❤❡ ♣♦✐♥t ♦♥ t❤❡ ❜❧✉❡ ❛♥❞ r❡❞ st❛❜✐❧✐t② ❜♦r❞❡rs r❡♣r❡s❡♥t t❤❡ ✜❡❧❞ ♣❛r❛♠❡t❡rs ♣❡r✲ ♠✐tt✐♥❣ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢r❡❡ ❡✈♦❧✉t✐♦♥ ✇✐t❤ ♠♦❞✐✜❡❞

✸✺

✭❢❛st❡r✱ s❧♦✇❡r ♦r ✐♥✈❡rt❡❞✮ ❡✈♦❧✉t✐♦♥ t✐♠❡✱ ♦r t❤❡ s♦❢t❧② ❛❝❤✐❡✈❡❞ r❛❞✐❛❧ ♦s❝✐❧❧❛t♦r ♣✉❧s❡s✳ ❇❡❧♦✇✱ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ❛ ●❛✉ss✐❛♥ ✇❛✈❡ ♣❛❝❦❡t ♦❢ ❛ ❝❤❛r❣❡❞ ♣❛rt✐❝❧❡✱ ✐♥ ❛ ♣✉❧s❛t✐♥❣ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❬✶✻❪✳ ❚❤❡ ❝❡♥t❡rs ♣❡r❢♦r♠ ❞✐s♣❧❛❝❡♠❡♥ts ♦♣♣♦s✐t❡ t♦ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✳

✸✻

❚❤❡ s❡❛r❝❤ ❢♦r ✈❛r✐❛❜❧❡ ♦s❝✐❧❧❛t♦r ♣✉❧s❡s✱ ♣❡r♠✐tt✐♥❣ t♦ ❛❝❤✐❡✈❡ s♦♠❡ ♣❤②s✐❝❛❧❧② ✐♥t❡r❡st✐♥❣ r❡s✉❧t✱ ✐♥ s♣✐t❡ ♦❢ ✐ts ♥❛rr♦✇ s✉❜❥❡❝t✱ ✐s st✐❧❧ ❛♥ ♦♣❡♥ ❛r❡❛✳ ■♥ ♣❛r✲ t✐❝✉❧❛r✱ ②♦✉ ♠✐❣❤t ❜❡ ✐♥t❡r❡st❡❞ t♦ ❝♦♥s✉❧t ❬✶✾❪ ✭♥♦♥✲ ❤❡r♠✐t✐❛♥ ♣r♦❜❧❡♠s✮✱ ❛❧s♦ ❬✷✵❪ ✭♥♦♥✲❧✐♥❡❛r ❡q✉❛t✐♦♥s ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ♠♦❞❡❧s✮ ❬✷✶❪ ✭t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢♦r✲ ♠✉❧❛ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ♠❛tr✐❝❡s✱ t❤♦✉❣❤ t❤❡ ♣❤②s✲ ✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠✮✳ ❘❡❢❡r❡♥❝❡s ❬✶❪ ❍✳ ❋✳ ❇❛❦❡r✱ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✸✹✱ ✶✾✵✷✱ ✸✹✼❀ ✸✺✱ ✶✾✵✸✱ ✸✸✸❀ ✸✱ ✶✾✵✹✱ ✷✹✳ ❬✷❪ ❏✳ ❊✳ ❈❛♠♣❜❡❧❧✱ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✷✾✱ ✶✽✾✽✱ ✶✹✳ ❬✸❪ ❋✳ ❍❛✉s❞♦r✛✱ ❇❡r✳ ❱❡r❤❛♥❞❧✳ ❙❛❡❝❤s✳ ❆❦❛❞✳ ❲✐ss✳ ▲❡✐♣③✐❣✱ ▼❛t❤✳ ◆❛t✉r✇✐s✳ ❑❧✳✱ ✺✽✱ ✶✾✵✻✱ ✶✾✳ ❬✹❪ ❍✳ ❩❛ss❡♥❤❛✉s✱ ✉♥♣✉❜❧✐s❤❡❞✱ s❡❡ ▼❛❣♥✉s ❬✼❪✱ ❲✐❧❝♦① ❬✾❪ ❛♥❞ ♦t❤❡r ❛✉t❤♦rs✳ ❬✺❪ ❊✳ ❇✳ ❉②♥❦✐♥✱ ❉♦❦❧❛❞② ❆❦❛❞✳ ◆❛✉❦ ❯❙❙❘ ✭◆✳ ❙✳✮✱ ✺✼✱ ✶✾✹✼✱ ✸✷✸✳

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❬✻❪ ❑✳ ❖✳ ❋r✐❡❞r✐❝❤s✱ ❈♦♠♠✉♥✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳✱ ✻✱ ✶✾✺✸✱✶✳ ❬✼❪ ❲✳ ▼❛❣♥✉s✱ ❖♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ s♦❧✉t✐♦♥ ♦❢ ❞✐ ❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❢♦r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ❈♦♠✲ ♠✉♥✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳✱ ✼✱ ✶✾✺✹✱ ✻✹✾✳ ❬✽❪ ❏✳ P❧❡❜❛➠s❦✐✱ ❖♥ t❤❡ ❣❡♥❡r❛t♦rs ♦❢ ✉♥✐t❛r② ❛♥❞ ♣s❡✉❞♦✲♦rt❤♦❣♦♥❛❧ ❣r♦✉♣s✱ r❡♣♦rt ❈■❊❆✱ ▼❡①✲ ✐❝♦✱ ✶✾✻✻✳ ❬✾❪ ❘✳ ▼✳ ❲✐❧❝♦①✱ ❏✳ ▼❛t❤✳ P❤②s✳✱ ✽✱ ✶✾✻✼✱ ✾✻✷✳ ❬✶✵❪ ■✳ ❇✐❛➟②♥✐❝❦✐✲❇✐r✉❧❛✱ ❇✳ ▼✐❡❧♥✐❦ ❛♥❞ ❏✳ P❧❡✲ ❜❛➠s❦✐✱ ❆♥♥✳ ♦❢ P❤②s✳✱ ✺✶✱ ✶✾✻✾✱ ✶✽✼✳ ❬✶✶❪ ❇✳ ▼✐❡❧♥✐❦ ❛♥❞ ❏✳ P❧❡❜❛➠s❦✐✱ ❈♦♠❜✐♥❛t♦r✐❛❧ ❛♣♣r♦❛❝❤ t♦ ❇❛❦❡r✲❈❛♠♣❜❡❧❧✲❍❛✉s❞♦r✛ ❡①♣♦✲ ♥❡♥ts✱ ❆♥♥✳ ■♥st✳ ❍❡♥r✐ P♦✐♥❝❛ré✱ ❱♦❧✳ ❳■■✱ ✸✱ ✶✾✼✵✱ ♣✳ ✷✶✺✲✷✺✹✳ ❬✶✷❪ ❏✳ ❈③②➺✱ P❛r❛❞♦①❡s ♦❢ ♠❡❛s✉r❡s ❛♥❞ ❞✐♠❡♥✲ s✐♦♥s ♦r✐❣✐♥❛t❡❞ ✐♥ ❋❡❧✐① ❍❛✉s❞♦r ✐❞❡❛s✱ ❲♦r❧❞ ❙❝✐✳ ❙✐♥❣❛♣♦r❡✱ ✶✾✾✹✳ ❬✶✸❪ ■✳ ▼✳ ●❡❧❢❛♥❞✱ ❆❞✈✳ ▼❛t❤✳✱ ✶✶✷✱ ✷✶✽✱ ✶✾✾✺✳ ❬✶✹❪ ❇✳ ▼✐❡❧♥✐❦✱ ❏✳ ▼❛t❤✳ P❤②s✳✱ ✷✼✱ ✷✷✾✵✱ ✶✾✽✻✳ ❬✶✺❪ ❈✳ ❉✳❋❡r♥❛♥❞❡③ ❛♥❞ ❇✳ ▼✐❡❧♥✐❦✱ ❏✳ ▼❛t❤✳ P❤②s✳✱ ✸✺✱ ✷✷✾✵✱ ✶✾✾✹✳

✸✽

❬✶✻❪ ❇✳ ▼✐❡❧♥✐❦ ❛♥❞ ❆✳ ❘❛♠ír❡③✱ ▼❛❣♥❡t✐❝ ♦♣❡r❛✲ t✐♦♥s✿ ❛ ❧✐tt❧❡ ❢✉③③② ♠❡❝❤❛♥✐❝s❄✱ P❤②s✳ ❙❝r✳✱ ✽✹✱ ✷✵✶✶✱ ✵✹✺✵✵✽ ✭✶✼♣♣✮✳ ❬✶✼❪ ❇✳ ▼✐❡❧♥✐❦✱ ◗✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s✿ t❡❝❤♥✐❝❛❧ ♦r ❢✉♥❞❛♠❡♥t❛❧ ❝❤❛❧❧❡♥❣❡❄✱ ❏✳ P❤②s✳ ❆✿ ▼❛t❤✳ ❚❤❡♦r✳✱ ✹✻✱ ✷✵✶✸✱ ✸✽✺✸✵✶ ✭✷✸♣♣✮✳ ❬✶✽❪ ▲✳ ■♥❢❡❧❞ ❛♥❞ ❏✳ P❧❡❜❛➠s❦✐✱ ▼♦t✐♦♥ ❛♥❞ ❘❡❧✲ ❛t✐✈✐t②✱ ◆❡✇ ❨♦r❦✿ P❡r❣❛♠♦♥ ✴ P❛♥➧t✇♦✇❡ ❲②❞❛✇♥✐❝t✇♦ ◆❛✉❦♦✇❡✱ ✶✾✻✵✱ s❡❡ ❊■❍ ✭❊✐♥✲ st❡✐♥✱ ■♥❢❡❧❞✱ ❍♦✛♠❛♥✮ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❬✶✾❪ ❍ ❍❡r♥❛♥❞❡③✲❈♦r♦♥❛❞♦✱ ❉ ❑r❡❥↔✐➦í❦✱ P ❙✐❡❣❧✱ P❡r❢❡❝t tr❛♥s♠✐ss✐♦♥ s❝❛tt❡r✐♥❣ ❛s ❛ P❚✲ s②♠♠❡tr✐❝ s♣❡❝tr❛❧ ♣r♦❜❧❡♠✱ P❤②s✳ ▲❡tt✳ ❆✱ ✸✼✺ ✭✷✷✮✱ ✷✶✹✾✲✷✶✺✷✳ ❬✷✵❪ ❏✳ ❲❡✐ ❛♥❞ ❊✳ ◆♦r♠❛♥✱ ❖♥ ❣❧♦❜❛❧ r❡♣r❡s❡♥✲ t❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛s ❛ ♣r♦❞✉❝t ♦❢ ❡①♣♦♥❡♥t✐❛❧s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✭✹t❤ ❡❞✳✮✱ ✶✺✱ ♣♣✳ ✸✷✼✕✸✸✹✳ ❬✷✶❪ ❲✲❳✳ ▼❛✱ ❇✳ ❙❤❡❦❤t♠❛♥✱ ❉♦ t❤❡ ❝❤❛✐♥ r✉❧❡s ❢♦r ♠❛tr✐① ❢✉♥❝t✐♦♥s ❤♦❧❞ ✇✐t❤♦✉t ❝♦♠♠✉t❛t✐✈✲ ✐t②❄✱ ▲✐♥❡❛r ❛♥❞ ▼✉❧t✐❧✐♥❡❛r ❆❧❣❡❜r❛✱ ✺✽✱ ◆♦✳ ✶✱ ✷✵✶✵✱ ✼✾✵✽✼✳

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