❋❧♦q✉❡t ❙♦❧✉t✐♦♥s ❙♣❡❝tr❛❧ ❚❤❡♦r② ♦❢ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ●❛♣s ❇❛rr② ❙✐♠♦♥ ❙♣❡❝tr✉♠ ■❇▼ Pr♦❢❡ss♦r ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❚❤❡♦r❡t✐❝❛❧ P❤②s✐❝s P♦t❡♥t✐❛❧ ❚❤❡♦r② ❈❛❧✐❢♦r♥✐❛ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣② P❛s❛❞❡♥❛✱ ❈❆✱ ❯✳❙✳❆✳ ▲❡❝t✉r❡ ✼✿ P❡r✐♦❞✐❝ ❖P❘▲
❙♣❡❝tr❛❧ ❚❤❡♦r② ♦❢ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ▲❡❝t✉r❡ ✺✿ ❑✐❧❧✐♣✕❙✐♠♦♥ ❚❤❡♦r❡♠ ♦♥ [ − 2 , 2] ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ▲❡❝t✉r❡ ✻✿ ❙③❡❣➤ ❆s②♠♣t♦t✐❝s ❛♥❞ ❙❤♦❤❛t✲◆❡✈❛✐ ❢♦r ●❛♣s [ − 2 , 2] ❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r② ▲❡❝t✉r❡ ✼✿ P❡r✐♦❞✐❝ ❖P❘▲ ▲❡❝t✉r❡ ✽✿ ❋✐♥✐t❡ ●❛♣ ■s♦s♣❡❝tr❛❧ ❚♦r✉s
❘❡❢❡r❡♥❝❡s ❬❖P❯❈❪ ❇✳ ❙✐♠♦♥✱ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ♦♥ t❤❡ ❯♥✐t ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s ❈✐r❝❧❡✱ P❛rt ✶✿ ❈❧❛ss✐❝❛❧ ❚❤❡♦r② ✱ ❆▼❙ ❈♦❧❧♦q✉✐✉♠ ❙❡r✐❡s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ✺✹✳✶ ✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ▼❛tr✐❝❡s ✷✵✵✺✳ ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ●❛♣s ❬❖P❯❈✷❪ ❇✳ ❙✐♠♦♥✱ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ♦♥ t❤❡ ❯♥✐t ❙♣❡❝tr✉♠ ❈✐r❝❧❡✱ P❛rt ✷✿ ❙♣❡❝tr❛❧ ❚❤❡♦r② ✱ ❆▼❙ ❈♦❧❧♦q✉✐✉♠ ❙❡r✐❡s✱ P♦t❡♥t✐❛❧ ❚❤❡♦r② ✺✹✳✷ ✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✺✳ ❬❙③❚❤♠❪ ❇✳ ❙✐♠♦♥✱ ❙③❡❣➤✬s ❚❤❡♦r❡♠ ❛♥❞ ■ts ❉❡s❝❡♥❞❛♥ts✿ ❙♣❡❝tr❛❧ ❚❤❡♦r② ❢♦r L 2 P❡rt✉r❜❛t✐♦♥s ♦❢ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ✱ ▼✳ ❇✳ P♦rt❡r ▲❡❝t✉r❡s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❏✱ ✷✵✶✶✳
❋❧♦q✉❡t ❙♦❧✉t✐♦♥s ❚❤❡ ❧❡❝t✉r❡ t✐t❧❡ ✐s ❛ ❜✐t ♦❢ ❛ ♠✐s♥♦♠❡r ✐♥ t❤❛t ✇❡✬❧❧ ♠❛✐♥❧② ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s ❞✐s❝✉ss ✇❤♦❧❡ ❧✐♥❡ ♣❡r✐♦❞✐❝ ❏❛❝♦❜✐ ♠❛tr✐❝❡s ❛❧t❤♦✉❣❤ t❤❡ P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❤❛❧❢✲❧✐♥❡ ♦❜❥❡❝ts ✇✐❧❧ ❡♥t❡r ❛ ❧♦t ✐♥ ❢✉t✉r❡ ❧❡❝t✉r❡s✳ ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ❙♦ { a n , b n } ∞ ●❛♣s n = −∞ ❛r❡ t✇♦✲s✐❞❡❞ s❡q✉❡♥❝❡s ✇✐t❤ s♦♠❡ p > 0 ❙♣❡❝tr✉♠ ✐♥ Z s♦ t❤❛t P♦t❡♥t✐❛❧ ❚❤❡♦r② a n + p = a n b n + p = b n ❋♦r z ∈ C ✜①❡❞✱ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ s♦❧✉t✐♦♥s { u n } ∞ n =0 ♦❢ a n u n +1 + b n u n + a n − 1 u n − 1 = zu n
❋❧♦q✉❡t ❙♦❧✉t✐♦♥s t❤❛t ❛❧s♦ ♦❜❡② ❢♦r s♦♠❡ λ ∈ C ✭ λ = e iθ , θ ∈ C ✮ ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s u n + p = λu n P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❙✉❝❤ s♦❧✉t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❋❧♦q✉❡t s♦❧✉❛t✐♦♥s ❛s t❤❡② ❛r❡ ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ❛♥❛❧♦❣s ♦❢ s♦❧✉t✐♦♥s ♦❢ ❖❉❊✱ ❡s♣❡❝✐❛❧❧② ❍✐❧❧✬s ❡q✉❛t✐♦♥ ●❛♣s ′′ + V u = zu ✱ V ( x + p ) = V ( x ) ✳ − u ❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r② ❚❤❡ ❛♥❛❧②s✐s ♦❢ s✉❝❤ s♦❧✉t✐♦♥s ✐s ❛ ❞❡❧✐❣❤t❢✉❧ ❛♠❛❧❣❛♠ ♦❢ t❤r❡❡ t♦♦❧s✱ t❤❡ ✜rst ♦❢ ✇❤✐❝❤ ✐s ❥✉st t❤❡ ❢❛❝t t❤❛t t❤❡ s❡t ♦❢ ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✐s t✇♦✲❞✐♠❡♥s✐♦♥❛❧✳ ❚❤✉s✱ t❤❡r❡ ❛r❡✱ ❢♦r z ✜①❡❞✱ ❛t ♠♦st t✇♦ ❞✐✛❡r❡♥t λ ✬s ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✳ ■❢ λ 1 ✱ λ 2 ❛r❡ t✇♦ s✉❝❤ λ ✬s✱ t❤❡✐r ❲r♦♥s❦✐❛♥ ✐s ♥♦♥✲③❡r♦ s♦ ❝♦♥st❛♥❝② ♦❢ t❤❡ ❲r♦♥s❦✐❛♥ ✐♠♣❧✐❡s λ 1 λ 2 = 1 .
P❡r✐♦❞✐❝ ❇✳❈✳ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ✭t✇✐st❡❞✮ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❏❛❝♦❜✐ ♠❛tr✐① ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s J per ,λ ✐s p × p ✳ ■t ✐s t❤❡ ✜♥✐t❡ ❏❛❝♦❜✐ ♠❛tr✐① ✇✐t❤ 1 p ❛♥❞ p 1 P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ♠❛tr✐① ❡❧❡♠❡♥ts ❛❞❞❡❞✿ ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t J jj = b j , J j j +1 = a j , J j j − 1 = a j − 1 ●❛♣s ❙♣❡❝tr✉♠ J 1 p = a p λ − 1 , J p 1 = a p λ P♦t❡♥t✐❛❧ ❚❤❡♦r② n = −∞ ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥✱ u 0 = λ − 1 u p ✱ ■❢ { u n } ∞ u = { u n } p n =1 ❤❛s J per ,λ � u p +1 = λu 1 s♦ � u = z � u ✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ � u s♦❧✈❡s t❤✐s✱ t❤❡ ✉♥✐q✉❡ u ✇✐t❤ u n + p = λu n u = { u n } ∞ ❛♥❞ � n =1 ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥✳
P❡r✐♦❞✐❝ ❇✳❈✳ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤✐s ✐♠♣❧✐❡s ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s ❋♦r ❛♥② λ ✱ t❤❡r❡ ❛r❡ ❛t ♠♦st p z ✬s ✇❤✐❝❤ ❤❛✈❡ ❛ P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ❢♦r t❤❛t λ ✳ ✭❲❡✬❧❧ s❡❡ s♦♦♥ t❤❛t ✐❢ ▼❛tr✐❝❡s λ � = ± 1 ✱ t❤❡r❡ ❛r❡ ❡①❛❝t❧② p ✳✮ ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ■❢ λ = e iθ ✱ θ ∈ R ✱ λ � = ± 1 ✱ t❤❡r❡ ❛r❡ ♣r❡❝✐s❡❧② p ❞✐st✐♥❝t ●❛♣s ❙♣❡❝tr✉♠ z ✬s ❛❧❧ r❡❛❧✱ ❢♦r ✇❤✐❝❤ t❤❡r❡ ❛r❡ ❋❧♦q✉❡t s♦❧✉t✐♦♥s ✇✐t❤ P♦t❡♥t✐❛❧ ❚❤❡♦r② t❤❛t λ ✳ ❚❤❡ r❡❛❧✐t② ❝♦♠❡s ❢r♦♠ ❤❡r♠✐❝✐t② ♦❢ J per ,λ ✳ ■❢ λ � = ± 1 ✱ ¯ λ � = λ ✳ ■❢ u ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ❢♦r λ ✱ s✐♥❝❡ z ✐s u ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ❢♦r ¯ r❡❛❧✱ ¯ λ s♦ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❢♦r t❤❛t z ✳ ❚❤✉s✱ ❢♦r λ ∈ ∂ D \ {± 1 } ✱ J per ,λ ❤❛s p ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡❛❝❤ s✐♠♣❧❡✳
❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ❚❤❡ t❤✐r❞ t♦♦❧ ❝♦♥❝❡r♥s t❤❡ p ✲st❡♣ tr❛♥s❢❡r ♠❛tr✐①✳ ❋❧♦q✉❡t ❙♦❧✉t✐♦♥s T p ( z )( u 1 a 0 u 0 ) = λ ( u 1 a 0 u 0 ) ✐s ❡q✉✐✈❛❧❡♥t t♦ ( u 1 a 0 u 0 ) ❣❡♥❡r❛t✐♥❣ ❛ P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❋❧♦q✉❡t s♦❧✉t✐♦♥ ✦ ✭◆♦t❡✿ a 0 ♠❛② ♥♦t ❜❡ ✶✳✮ ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t ■♥ t❡r♠s ♦❢ t❤❡ ❖P✬s ❢♦r { a n , b n } ∞ n =1 ✱ ●❛♣s � � ❙♣❡❝tr✉♠ p p ( z ) − q p ( z ) P♦t❡♥t✐❛❧ ❚❤❡♦r② T p ( z ) = a p p p − 1 ( z ) − a p q p − 1 ( z ) ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t✱ ∆( z ) ✱ ✐s ❞❡✜♥❡❞ ❜② � � ∆( z ) = ❚r T p ( z ) = p p ( z ) − a p q p − 1 ( z ) ✐s ❛ ✭r❡❛❧✮ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ❡①❛❝t❧② p ✳
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