◆♦♥✲❡q✉✐❧✐❜r✐✉♠ ❛❧♠♦st✲st❛t✐♦♥❛r② st❛t❡s ❛♥❞ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r ❣❛♣♣❡❞ ♥♦♥✲✐♥t❡r❛❝t✐♥❣ q✉❛♥t✉♠ s②st❡♠s ●✐♦✈❛♥♥❛ ▼❛r❝❡❧❧✐ ❥♦✐♥t ✇♦r❦s ✇✐t❤ ❉✳ ▼♦♥❛❝♦ ✭ ❘♦♠❛ ❚r❡ ✱ ❘♦♠❛✮✱ ●✳ P❛♥❛t✐ ✭ ▲❛ ❙❛♣✐❡♥③❛ ✱ ❘♦♠❛✮✱ ❛♥❞ ❙✳ ❚❡✉❢❡❧ ✭ ❊❜❡r❤❛r❞✲❑❛r❧s ✱ ❚ü❜✐♥❣❡♥✮ ❬▼❛▼♦P❚❪✿ s♦♦♥ ♦♥ ❛r❳✐✈ ❛♥❞ ❬▼❛❚❪✿ ✐♥ ♣r♦❣r❡ss ❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❘♦♠❛ ❚r❡✱ ✶✾t❤ ❙❡♣t❡♠❜❡r✱ ✷✵✶✾
■♥tr♦❞✉❝t✐♦♥ ❇❛rr② ❙✐♠♦♥✿ ❋✐❢t❡❡♥ ♣r♦❜❧❡♠s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ✭✶✾✽✹✮ ✹✳ ❚r❛♥s♣♦rt ❚❤❡♦r② ✿ ❆t s♦♠❡ ❧❡✈❡❧✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞✐✣❝✉❧t② ♦❢ tr❛♥s♣♦rt t❤❡♦r② ✐s t❤❛t ✐t ✐s ❛ st❡❛❞② st❛t❡ r❛t❤❡r t❤❛♥ ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠✱ s♦ t❤❛t t❤❡ ♣♦✇❡r❢✉❧ ❢♦r♠❛❧✐s♠ ♦❢ ❡q✉✐❧✐❜r✐✉♠ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s ✐s ✉♥❛✈❛✐❧❛❜❧❡✱ ❛♥❞ ♦♥❡ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② ✇❛② ♦❢ ♣r❡❝✐s❡❧② ✐❞❡♥t✐❢②✐♥❣ t❤❡ st❡❛❞② st❛t❡ ❛♥❞ t❤❡r❡❜② ❝♦♠♣✉t✐♥❣ t❤✐♥❣s ✐♥ ✐t✳ ✳ ✳ ✳ Pr♦❜❧❡♠ ✹ ❇ ✭❑✉❜♦ ❋♦r♠✉❧❛✮ ✿ ❊✐t❤❡r ❥✉st✐❢② ❑✉❜♦✬s ❢♦r♠✉❧❛ ✐♥ ❛ q✉❛♥t✉♠ ♠♦❞❡❧✱ ♦r ❡❧s❡ ✜♥❞ ❛♥ ❛❧t❡r♥❛t❡ t❤❡♦r② ♦❢ ❝♦♥❞✉❝t✐✈✐t②✳
▲✐♥❡❛r r❡s♣♦♥s❡ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ q✉❛♥t✉♠ s②st❡♠s✱ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r♠❛❧✐s♠ ❛♥s✇❡rs t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿
▲✐♥❡❛r r❡s♣♦♥s❡ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ q✉❛♥t✉♠ s②st❡♠s✱ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r♠❛❧✐s♠ ❛♥s✇❡rs t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿ ◗✶✮ ❍♦✇ ❞♦❡s ❛ s②st❡♠ ❞❡s❝r✐❜❡❞ ❜② ❛ ❍❛♠✐❧t♦♥✐❛♥ H ✵ t❤❛t ✐s ✐♥✐t✐❛❧❧② ✐♥ ❛♥ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ Π ✵ r❡s♣♦♥❞ t♦ ❛ s♠❛❧❧ st❛t✐❝ ♣❡rt✉r❜❛t✐♦♥ ε V ❄ ( H ✵ , Π ✵ , ε V ) ρ ε − → ❤❡r❡ ρ ε ❞❡♥♦t❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥
▲✐♥❡❛r r❡s♣♦♥s❡ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ q✉❛♥t✉♠ s②st❡♠s✱ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r♠❛❧✐s♠ ❛♥s✇❡rs t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿ ◗✶✮ ( H ✵ , Π ✵ , ε V ) − → ρ ε ❤❡r❡ ρ ε ❞❡♥♦t❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥
▲✐♥❡❛r r❡s♣♦♥s❡ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ q✉❛♥t✉♠ s②st❡♠s✱ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r♠❛❧✐s♠ ❛♥s✇❡rs t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿ ◗✶✮ ( H ✵ , Π ✵ , ε V ) − → ρ ε ❤❡r❡ ρ ε ❞❡♥♦t❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥ ◗✷✮ ❲❤❛t ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡ ♦❢ ❛♥ ♦❜s❡r✈❛❜❧❡ A ❝❛✉s❡❞ ❜② t❤❡ ♣❡rt✉r❜❛t✐♦♥ ε V ❛t t❤❡ ❧❡❛❞✐♥❣ ♦r❞❡r ✐♥ ✐ts str❡♥❣t❤ ε ≪ ✶ ❄ ( H ✵ , Π ✵ , ε V ) Re T ( A ρ ε ) − Re T ( A Π ✵ ) = : ε · LR A + o ( ε ) − → ❤❡r❡ T ( · ) ❞❡♥♦t❡s ❛ tr❛❝❡✲❧✐❦❡ ❢✉♥❝t✐♦♥❛❧ ❛♥❞ LR A ✐s ❝❛❧❧❡❞ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥t ❢♦r A
▲✐♥❡❛r r❡s♣♦♥s❡ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ q✉❛♥t✉♠ s②st❡♠s✱ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r♠❛❧✐s♠ ❛♥s✇❡rs t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿ ◗✶✮ ( H ✵ , Π ✵ , ε V ) − → ρ ε ❤❡r❡ ρ ε ❞❡♥♦t❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥ ◗✷✮ ( H ✵ , Π ✵ , ε V ) ReTr( A ρ ε ) − ReTr( A Π ✵ ) = : ε · G A + o ( ε ) − → ❤❡r❡ A ✐s ❛♥ ✐♥t❡♥s✐✈❡ ♦❜s❡r✈❛❜❧❡✱ Tr( · ) ✐s t❤❡ st❛♥❞❛r❞ tr❛❝❡ ❛♥❞ G A ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♥❞✉❝t❛♥❝❡ ❢♦r A
▲✐♥❡❛r r❡s♣♦♥s❡ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ q✉❛♥t✉♠ s②st❡♠s✱ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❢♦r♠❛❧✐s♠ ❛♥s✇❡rs t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿ ◗✶✮ ( H ✵ , Π ✵ , ε V ) − → ρ ε ❤❡r❡ ρ ε ❞❡♥♦t❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥ ◗✷✮ ( H ✵ , Π ✵ , ε V ) Re τ ( A ρ ε ) − Re τ ( A Π ✵ ) = : ε · σ A + o ( ε ) − → ❤❡r❡ A ✐s ❛♥ ❡①t❡♥s✐✈❡ ♦❜s❡r✈❛❜❧❡✱ τ ( · ) ✐s t❤❡ tr❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❛♥❞ σ A ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♥❞✉❝t✐✈✐t② ❢♦r A
▲❡t t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✶ ✵ ✵ ✵ ❚❤❡♥✱ ✵ ♦r ❢♦r ❛♥② ✵ ✐s ✏t❤❡ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥✑ ✳ ❆ ♠♦❞❡❧ ❢♦r t❤❡ s✇✐t❝❤✐♥❣ ♣r♦❝❡ss ▲❡t H ε ( t ) : = H ✵ + ε f ( t ) V , t ∈ I ✱ ✇❤❡r❡ [ − ✶ , ✵ ] ⊂ I ⊂ R ✐s ❝♦♠♣❛❝t ✐♥t❡r✈❛❧ ❛♥❞ ε ≪ ✶✳ f ✶ t − ✶
▲❡t t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✶ ✵ ✵ ✵ ❚❤❡♥✱ ✵ ♦r ❢♦r ❛♥② ✵ ✐s ✏t❤❡ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥✑ ✳ ❆ ♠♦❞❡❧ ❢♦r t❤❡ s✇✐t❝❤✐♥❣ ♣r♦❝❡ss ▲❡t H ε ( η t ) : = H ✵ + ε f ( η t ) V , η t ∈ I ✱ ✇❤❡r❡ [ − ✶ , ✵ ] ⊂ I ⊂ R ✐s ❝♦♠♣❛❝t ✐♥t❡r✈❛❧✱ ε ≪ ✶ ❛♥❞ η ≪ ✶✳ f ✶ t − ✶
❆ ♠♦❞❡❧ ❢♦r t❤❡ s✇✐t❝❤✐♥❣ ♣r♦❝❡ss ▲❡t H ε ( η t ) : = H ✵ + ε f ( η t ) V , η t ∈ I ✱ ✇❤❡r❡ [ − ✶ , ✵ ] ⊂ I ⊂ R ✐s ❝♦♠♣❛❝t ✐♥t❡r✈❛❧✱ ε ≪ ✶ ❛♥❞ η ≪ ✶✳ f ✶ t − ✶ ▲❡t ρ ( t ) t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤② ♣r♦❜❧❡♠ � i d d t ρ ( t ) = [ H ε ( η t ) , ρ ( t )] ρ ( t ✵ ) = Π ✵ ∀ t ✵ ≤ − ✶ / η . ❚❤❡♥✱ ρ ( ✵ ) ♦r ρ ( t ) ❢♦r ❛♥② t ≥ ✵ ✐s ✏t❤❡ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ st❛t❡ ρ ε ♦❢ t❤❡ s②st❡♠ ❛❢t❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❤❛s ❜❡❡♥ t✉r♥❡❞ ♦♥✑ ✳
❑✉❜♦✬s ❢♦r♠✉❧❛ ❇② t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ♦♥❡ ♦❜t❛✐♥s t❤❛t ρ ε : = ρ ( ✵ ) � ✵ d t f ( η t ) e i tH ✵ [ V , Π ✵ ] e − i tH ✵ + R ε , η , f , ρ ε = Π ✵ − i ε −∞
❑✉❜♦✬s ❢♦r♠✉❧❛ ❇② t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ♦♥❡ ♦❜t❛✐♥s t❤❛t ρ ε : = ρ ( ✵ ) � ✵ d t f ( η t ) e i tH ✵ [ V , Π ✵ ] e − i tH ✵ + R ε , η , f , ρ ε = Π ✵ − i ε −∞ ❛♥❞ t❤✉s σ η , f + τ ( AR ε , η , f ) τ ( A ρ ε ) = τ ( A Π ✵ ) + ε · � ✇✐t❤ � ✵ σ η , f : = − i d t f ( η t ) τ ( A e i tH ✵ [ V , Π ✵ ] e − i tH ✵ ) . � −∞
Recommend
More recommend