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❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❙♣❡❝tr❛❧ ❚❤❡♦r② ♦❢ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s

❇❛rr② ❙✐♠♦♥

■❇▼ Pr♦❢❡ss♦r ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❚❤❡♦r❡t✐❝❛❧ P❤②s✐❝s ❈❛❧✐❢♦r♥✐❛ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣② P❛s❛❞❡♥❛✱ ❈❆✱ ❯✳❙✳❆✳ ▲❡❝t✉r❡ ✼✿ P❡r✐♦❞✐❝ ❖P❘▲

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❙♣❡❝tr❛❧ ❚❤❡♦r② ♦❢ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s

▲❡❝t✉r❡ ✺✿ ❑✐❧❧✐♣✕❙✐♠♦♥ ❚❤❡♦r❡♠ ♦♥ [−2, 2] ▲❡❝t✉r❡ ✻✿ ❙③❡❣➤ ❆s②♠♣t♦t✐❝s ❛♥❞ ❙❤♦❤❛t✲◆❡✈❛✐ ❢♦r [−2, 2] ▲❡❝t✉r❡ ✼✿ P❡r✐♦❞✐❝ ❖P❘▲ ▲❡❝t✉r❡ ✽✿ ❋✐♥✐t❡ ●❛♣ ■s♦s♣❡❝tr❛❧ ❚♦r✉s

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❘❡❢❡r❡♥❝❡s

❬❖P❯❈❪ ❇✳ ❙✐♠♦♥✱ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ♦♥ t❤❡ ❯♥✐t ❈✐r❝❧❡✱ P❛rt ✶✿ ❈❧❛ss✐❝❛❧ ❚❤❡♦r②✱ ❆▼❙ ❈♦❧❧♦q✉✐✉♠ ❙❡r✐❡s ✺✹✳✶✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✺✳ ❬❖P❯❈✷❪ ❇✳ ❙✐♠♦♥✱ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ♦♥ t❤❡ ❯♥✐t ❈✐r❝❧❡✱ P❛rt ✷✿ ❙♣❡❝tr❛❧ ❚❤❡♦r②✱ ❆▼❙ ❈♦❧❧♦q✉✐✉♠ ❙❡r✐❡s✱ ✺✹✳✷✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✺✳ ❬❙③❚❤♠❪ ❇✳ ❙✐♠♦♥✱ ❙③❡❣➤✬s ❚❤❡♦r❡♠ ❛♥❞ ■ts ❉❡s❝❡♥❞❛♥ts✿ ❙♣❡❝tr❛❧ ❚❤❡♦r② ❢♦r L2 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 P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s

❚❤❡ ❧❡❝t✉r❡ t✐t❧❡ ✐s ❛ ❜✐t ♦❢ ❛ ♠✐s♥♦♠❡r ✐♥ t❤❛t ✇❡✬❧❧ ♠❛✐♥❧② ❞✐s❝✉ss ✇❤♦❧❡ ❧✐♥❡ ♣❡r✐♦❞✐❝ ❏❛❝♦❜✐ ♠❛tr✐❝❡s ❛❧t❤♦✉❣❤ t❤❡ ❤❛❧❢✲❧✐♥❡ ♦❜❥❡❝ts ✇✐❧❧ ❡♥t❡r ❛ ❧♦t ✐♥ ❢✉t✉r❡ ❧❡❝t✉r❡s✳ ❙♦ {an, bn}∞

n=−∞ ❛r❡ t✇♦✲s✐❞❡❞ s❡q✉❡♥❝❡s ✇✐t❤ s♦♠❡ p > 0

✐♥ Z s♦ t❤❛t an+p = an bn+p = bn ❋♦r z ∈ C ✜①❡❞✱ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ s♦❧✉t✐♦♥s {un}∞

n=0 ♦❢

anun+1 + bnun + an−1un−1 = zun

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s

t❤❛t ❛❧s♦ ♦❜❡② ❢♦r s♦♠❡ λ ∈ C ✭λ = eiθ, θ ∈ C✮ un+p = λun ❙✉❝❤ s♦❧✉t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❋❧♦q✉❡t s♦❧✉❛t✐♦♥s ❛s t❤❡② ❛r❡ ❛♥❛❧♦❣s ♦❢ s♦❧✉t✐♦♥s ♦❢ ❖❉❊✱ ❡s♣❡❝✐❛❧❧② ❍✐❧❧✬s ❡q✉❛t✐♦♥ −u

′′ + V u = zu✱ V (x + p) = V (x)✳

❚❤❡ ❛♥❛❧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✐♦♥✳ ■❢ λ1✱ λ2 ❛r❡ t✇♦ s✉❝❤ λ✬s✱ t❤❡✐r ❲r♦♥s❦✐❛♥ ✐s ♥♦♥✲③❡r♦ s♦ ❝♦♥st❛♥❝② ♦❢ t❤❡ ❲r♦♥s❦✐❛♥ ✐♠♣❧✐❡s λ1λ2 = 1.

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P❡r✐♦❞✐❝ ❇✳❈✳ ❏❛❝♦❜✐ ▼❛tr✐❝❡s

❚❤❡ ✭t✇✐st❡❞✮ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❏❛❝♦❜✐ ♠❛tr✐① Jper,λ ✐s p × p✳ ■t ✐s t❤❡ ✜♥✐t❡ ❏❛❝♦❜✐ ♠❛tr✐① ✇✐t❤ 1p ❛♥❞ p1 ♠❛tr✐① ❡❧❡♠❡♥ts ❛❞❞❡❞✿ Jjj = bj, Jj j+1 = aj, Jj j−1 = aj−1 J1p = apλ−1, Jp1 = apλ ■❢ {un}∞

n=−∞ ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥✱ u0 = λ−1up✱

up+1 = λu1 s♦ u = {un}p

n=1 ❤❛s Jper,λ

u = z u✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ u s♦❧✈❡s t❤✐s✱ t❤❡ ✉♥✐q✉❡ u ✇✐t❤ un+p = λun ❛♥❞ u = {un}∞

n=1 ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P❡r✐♦❞✐❝ ❇✳❈✳ ❏❛❝♦❜✐ ▼❛tr✐❝❡s

❚❤✐s ✐♠♣❧✐❡s ❋♦r ❛♥② λ✱ t❤❡r❡ ❛r❡ ❛t ♠♦st p z✬s ✇❤✐❝❤ ❤❛✈❡ ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ❢♦r t❤❛t λ✳ ✭❲❡✬❧❧ s❡❡ s♦♦♥ t❤❛t ✐❢ λ = ±1✱ t❤❡r❡ ❛r❡ ❡①❛❝t❧② p✳✮ ■❢ λ = eiθ✱ θ ∈ R✱ λ = ±1✱ t❤❡r❡ ❛r❡ ♣r❡❝✐s❡❧② p ❞✐st✐♥❝t z✬s ❛❧❧ r❡❛❧✱ ❢♦r ✇❤✐❝❤ t❤❡r❡ ❛r❡ ❋❧♦q✉❡t s♦❧✉t✐♦♥s ✇✐t❤ t❤❛t λ✳ ❚❤❡ r❡❛❧✐t② ❝♦♠❡s ❢r♦♠ ❤❡r♠✐❝✐t② ♦❢ Jper,λ✳ ■❢ λ = ±1✱ ¯ λ = λ✳ ■❢ u ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ❢♦r λ✱ s✐♥❝❡ z ✐s r❡❛❧✱ ¯ u ✐s ❛ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ❢♦r ¯ λ s♦ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❢♦r t❤❛t z✳ ❚❤✉s✱ ❢♦r λ ∈ ∂D \ {±1}✱ Jper,λ ❤❛s p ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡❛❝❤ s✐♠♣❧❡✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

❚❤❡ t❤✐r❞ t♦♦❧ ❝♦♥❝❡r♥s t❤❡ p✲st❡♣ tr❛♥s❢❡r ♠❛tr✐①✳ Tp(z)( u1

a0u0 ) = λ( u1 a0u0 ) ✐s ❡q✉✐✈❛❧❡♥t t♦ ( u1 a0u0 ) ❣❡♥❡r❛t✐♥❣ ❛

❋❧♦q✉❡t s♦❧✉t✐♦♥ ✦ ✭◆♦t❡✿ a0 ♠❛② ♥♦t ❜❡ ✶✳✮ ■♥ t❡r♠s ♦❢ t❤❡ ❖P✬s ❢♦r {an, bn}∞

n=1✱

Tp(z) =

• pp(z)

−qp(z) appp−1(z) −apqp−1(z)

• ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t✱ ∆(z)✱ ✐s ❞❡✜♥❡❞ ❜②

∆(z) = ❚r

• Tp(z)
• = pp(z) − apqp−1(z)

✐s ❛ ✭r❡❛❧✮ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ❡①❛❝t❧② p✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

❙✐♥❝❡ det

• Tp(z)
• = 1✱ ✐t ❤❛s ❛❧❣❡❜r❛✐❝ ❡✐❣❡♥✈❛❧✉❡s λ ❛♥❞

λ−1 ✇❤❡r❡ ∆(z) = λ + λ−1❀ ∆(z) = 2 cos θ ✐❢ λ = eiθ✳ ❋❧♦q✉❡t s♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞ t♦ ❣❡♦♠❡tr✐❝ ❡✐❣❡♥✈❛❧✉❡s ❢♦r Tp(z)✳ ■❢ λ = ±1✱ ✐t ❤❛s ♠✉❧t✐♣❧✐❝✐t② ♦♥❡✱ s♦ ✐s ❣❡♦♠tr✐❝✳ λ = ±1 ❤❛s ♠✉❧t✐♣❧✐❝✐t② ✷✱ s♦ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❡ ♦r t✇♦ ❋❧♦q✉❡t s♦❧✉t✐♦♥s✳ ❆♥ ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ∆(z) ∈ (−2, 2) ✐♠♣❧✐❡s ❛❧❧ z✬s ❛r❡ r❡❛❧ ✐s ∆−1 (−2, 2)

• ⊂ R✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

❆ ❜❛s✐❝ ❢❛❝t ♦❢ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s ✐s t❤❛t ✐❢ f(z) ✐s r❡❛❧ ✭✐✳❡✳✱ f(¯ z) = f(z)✮✱ x0 ∈ R ✇✐t❤ f′(x0) = 0✱ t❤❡r❡ ❛r❡ ♥♦♥✲r❡❛❧ z✬s ♥❡❛r x0 ✇✐t❤ f(z) r❡❛❧ ❛♥❞ ♥❡❛r f(x0)✳ ❚❤✉s✱ ∆−1 (−2, 2)

• ⊂ R ⇒ ∆′(x0) = 0 ✐❢ ∆(x0) ∈ (−2, 2)✳

❚❤✉s✱ ∆−1 (−2, 2)

• = (α1, β1) ∪ (α2, β2) ∪ . . . ∪ (αp, βp)

✇❤❡r❡ α1 < β1 ≤ α2 < β2 ≤ α3 < . . . < βp ✇✐t❤ ∆ ❛ s♠♦♦t❤ ❜✐❥❡❝t✐♦♥ ♦❢ (αj, βj) t♦ (−2, 2)✳ ❈♦✉❧❞ ❜❡ ♦r✐❡♥t❛t✐♦♥ r❡✈❡rs✐♥❣ ♦r ♥♦t✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

❙✐♥❝❡ ∆(x) → ∞ ❛s x → ∞✱ ✇❡ ♠✉st ❤❛✈❡ ∆(βp) = 2✳ ■t ❢♦❧❧♦✇s t❤❛t ∆(αp) = −2✱ ∆(βp−1) = −2✱ ∆(αp−1) = 2 . . . ✐✳❡✳✱ ∆(βj) = (−1)p−j2✱ ∆(αj) = (−1)p−j−12 ■❢ t❤❡ α✬s ❛♥❞ β✬s ❛r❡ ❛❧❧ ❞✐st✐♥❝t✱ ✇❡ ❤❛✈❡ p ♣♦✐♥ts ✇❤❡r❡ ∆(x) = 2 ❛♥❞ p ✇❤❡r❡ ∆(x) = −2✳ ❙✐♥❝❡ deg ∆ = p✱ t❤❡s❡ ❛r❡ ❛❧❧ t❤❡ ♣♦✐♥ts✳ ■❢ βj−1 = αj✱ t❤❡r❡ ✐s ♦♥❡ ❧❡ss ♣♦✐♥t ✇❤❡r❡ ∆(x) = (−1)p−j−12✱ ❜✉t ∆′(αj) = 0 s✐♥❝❡ ∆ − (−1)p−j−12 ❤❛s t❤❡ s❛♠❡ s✐❣♥ ♦♥ ❜♦t❤ s✐❞❡s ♦❢ αj✳ ■t ❢♦❧❧♦✇s t❤❛t

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❖♣❡♥s ❛♥❞ ❈❧♦s❡❞ ●❛♣s

❚❤❡♦r❡♠✳ ∆−1 [−2, 2]

• = ∪p

j=1[αj, βj] ❛♥❞

∆−1 {−2, 2}

• = {αj, βj}p

j=1 ❛♥❞

∆′(αj) = 0 ⇔ αj = βj−1, ∆′(βj) = 0 ⇔ βj = αj+1 ❛♥❞ ✐♥ t❤❛t ❝❛s❡✱ ∆

′′ ✐s ♥♦t ③❡r♦ ❛t t❤❛t ♣♦✐♥t✳

❚❤❡ [αj, βj] ❛r❡ ❝❛❧❧❡❞ t❤❡ ❜❛♥❞s ❛♥❞ (βj, αj+1) t❤❡ ❣❛♣s✳ ■❢ βj < αj+1✱ ✇❡ s❛② t❤❛t ❣❛♣ j ✐s ♦♣❡♥✳ ■❢ βj = αj+1✱ ✇❡ s❛② ❣❛♣ j ✐s ❝❧♦s❡❞✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❖♣❡♥s ❛♥❞ ❈❧♦s❡❞ ●❛♣s

❋✉rt❤✉r ❛♥❛❧②s✐s s❤♦✇s ❛t ❛ ❝❧♦s❡❞ ❣❛♣ ✭✇✐t❤ ∆(α) = 2 ❢♦r s✐♠♣❧✐❝✐t②✮ t❤❡r❡ ❛r❡ t✇♦ ♣❡r✐♦❞✐❝ ✭❋❧♦q✉❡t✮ s♦❧✉t✐♦♥s✱ ✇❤✐❧❡ ❛t ❡❛❝❤ ♦❢ t❤❡ ❡❞❣❡s ♦❢ ❛♥ ♦♣❡♥ ❣❛♣ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♣❡r✐♦❞✐❝ ✭❋❧♦q✉❡t✮ s♦❧✉t✐♦♥✳ ❚❤❡ tr❛♥s❢❡r ♠❛tr✐① ❤❛s ❛ ❏♦r❞❛♥ ❛♥♦♠❛❧②✱ ✐✳❡✳✱ det = 1✱ ❚r = 2✱ ❜✉t T = ( 1 0

0 1 )✳

❊❛❝❤ ♦❢ t❤❡ ❣❛♣s ✇❤❡r❡ ∆(x) ≥ 2 ❤❛s t✇♦ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥s✖❡✐t❤❡r t✇♦ ❛t βj = αj+1 ♦r ♦♥❡ ❡❛❝❤ ❛t βj ❛♥❞ αj+1 s♦ t❤❡r❡ ❛r❡ p ♣❡r✐♦❞✐❝ ❋❧♦q✉❡t s♦❧✉t✐♦♥s✱ ❛s t❤❡r❡ ♠✉st ❜❡ ❢r♦♠ t❤❡ Jper ❛♥❛❧②s✐s✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❙♣❡❝tr✉♠ ❛♥❞ ❙♣❡❝tr❛❧ ❚②♣❡s

■❢ z ✐s s✉❝❤ t❤❛t ∆(z) ∈ [−2, 2]✱ t❤❡♥ t❤❡ r♦♦ts ♦❢ λ + λ−1 = ∆(z) ❤❛✈❡ |λ| > 1✱ |λ−1| < 1✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t s♦❧✉t✐♦♥s u± ❞❡❝❛②✐♥❣ ❡①♣♦♥❡♥t✐❛❧❧② ❛t ±∞ s♦ t❤❡✐r ❲r♦♥s❦✐❛♥ ✐s ♥♦t ③❡r♦✳ ❇② t❤❡ ❡❛r❧✐❡r ❛♥❛❧②s✐s✱ Gnm(z) = u+

max(n,m)(z)u− min(m.n)(z)/W(z)

✐s t❤❡ ♠❛tr✐① ❢♦r

• J − z

−1✱ ✐✳❡✳✱ z / ∈ σ(J)✳ ■❢ ∆(z) ∈ [−2, 2]✱ t❤❡r❡ ✐s ❛ ❜♦✉♥❞❡❞ ❋❧♦q✉❡t s♦❧✉t✐♦♥ ✭s✐♥❝❡ |λ| = 1✮✳ ❚❤❡♥ (J − z)[uχ[−N,N]] ✐s ❜♦✉♥❞❡❞✱ ❜✉t s✐♥❝❡ p

j=1|um+j|2 ✐s ❝♦♥st❛♥t✱ uχ[−N,N] → ∞ s♦ z ∈ σ(J)✳

❚❤✉s ❚❤❡♦r❡♠✳ σ(J) = ∪p

j=1[αj, βj]✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❙♣❡❝tr✉♠ ❛♥❞ ❙♣❡❝tr❛❧ ❚②♣❡s

■❢ ∆(z) ∈ (−2, 2)✱ ✇❡ ❣❡t t❤❛t ❛❧❧ s♦❧✉t✐♦♥s ❛r❡ ❜♦✉♥❞❡❞ ❛t ±∞ ❛♥❞ t❤❡♥ ❜② ❛ ❲r♦♥s❦✐❛♥ ❛r❣✉♠❡♥t✱ |un|2 + |un+1|2 ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✳ ❙♦ ❜② ❛ ❈❛r♠♦♥❛✲t②♣❡ ❢♦r♠✉❧❛✱ ♦♥❡ s❤♦✉❧❞ ❡①♣❡❝t ♣✉r❡❧② ❛✳❝✳ s♣❡❝tr✉♠✳ ❇✉t t❤✐s ✐s ✇❤♦❧❡ ❧✐♥❡✱ ♥♦t ❤❛❧❢ ❧✐♥❡ ✦ ❍❡r❡ ✐s ❛ r❡♣❧❛❝❡♠❡♥t✿ ❆✇❛② ❢r♦♠ t❤❡ ❜❛♥❞s✱ Gnn = u+

n u− n /W ❛s ✇❡✬✈❡ s❡❡♥✳ ❇② ❝♦♥t✐♥✉✐t② ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢

tr❛♥s❢❡r ♠❛tr✐① ✐♥ z✱ u±

n ❤❛s ❛ ❧✐♠✐t ❛t z = x + iε ✇✐t❤ ε ↓ 0

✇❤✐❝❤ ❛r❡ ❋❧♦q✉❡t s♦❧✉t✐♦♥s✳ ❚❤✐s ✐s tr✉❡ ❛t ❧❡❛st ❛t ✐♥t❡r✐♦rs ♦❢ ❜❛♥❞s ✇❤❡r❡ t❤❡ tr❛♥s❢❡r ♠❛tr✐① ❤❛s ❞✐st✐♥❝t ❡✐❣❡♥✈❛❧✉❡s✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

❙♣❡❝tr✉♠ ❛♥❞ ❙♣❡❝tr❛❧ ❚②♣❡s

W ✐s ♥♦♥✲✈❛♥✐s❤✐♥❣ ♦♥ ❡❛❝❤ (αj, βj) s✐♥❝❡ u+ ❛♥❞ u− ❛r❡ ❞✐st✐♥❝t ❋❧♦q✉❡t s♦❧✉t✐♦♥s ✭e±iθ✮✳ ❚❤✉s✱ Gnn(z) ✐s ❝♦♥t✐♥✉♦✉s ❢r♦♠ C+ t♦ C+ ∪ R \ {αj, βj}p

j=1✳

❇✉t ✐❢ µ(n) ✐s t❤❡ s♣❡❝tr❛❧ ♠❡❛s✉r❡ ♦❢ δn✿ Gnn(z) = dµ(n)(x) x − z ❚❤❡ ❝♦♥t✐♥✉✐t② ✐♠♣❧✐❡s dµ(n) ✐s ♣✉r❡❧② ❛✳❝✳✱ s♦ ✇❡ ❤❛✈❡ ♣r♦✈❡♥ ❚❤❡♦r❡♠✳ ❆ ♣❡r✐♦❞✐❝ t✇♦✲s✐❞❡❞ ❏❛❝♦❜✐ ♠❛tr✐① ❤❛s ♣✉r❡❧② ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s s♣❡❝tr✉♠✳ ❖♥❡ ❝❛♥ ✇r✐t❡ ♦✉t ❛♥ ❡①♣❧✐❝✐t s♣❡❝tr❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ ❋❧♦q✉❡t s♦❧✉t✐♦♥s ✇✐t❤ z ∈ (αj, βj) ❛s ❝♦♥t✐♥✉✉♠ ❡✐❣❡♥❢✉♥❝t✐♦♥s✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P♦t❡♥t✐❛❧ ❚❤❡♦r②

❲❡ st❛rt ✇✐t❤ ❛ ♣✉③③❧❡✳ ∆ ❞❡t❡r♠✐♥❡s α1 < β1 ≤ α2 < β2 ≤ . . . ❛s t❤❡ r♦♦ts ♦❢ ∆2 − 4✳ ❈♦♥✈❡rs❡❧②✱ ❣✐✈❡♥ βp, αp−1, βp−2, . . .✱ ∆ − 2 ✐s ❞❡t❡r♠✐♥❡❞ ✉♣ t♦ ❛ ❝♦♥st❛♥t s✐♥❝❡ ✇❡ ❦♥♦✇ ✐ts ③❡r♦s✳ ❚❤❛t ❝♦♥st❛♥t ✐s ❞❡t❡r♠✐♥❡❞ ❜② αp ✇❤❡♥ ∆ ✐s −2✳ ❚❤✉s✱ βp, αp−1, βp−2 ♣❧✉s αp ❞❡t❡r♠✐♥❡ t❤❡ r❡♠❛✐♥✐♥❣ p − 1 α✬s ❛♥❞ β✬s✳ ❲❤② t❤✐s r✐❣✐❞✐t②❄ ❲❤② ❝❛♥✬t ✇❡ ❤❛✈❡ 2p ❛r❜✐tr❛r② α✬s ❛♥❞ β✬s❄ ❚❤❡ ❛♥s✇❡r ✇✐❧❧ ❧✐❡ ✐♥ ♣♦t❡♥t✐❛❧ t❤❡♦r②✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P♦t❡♥t✐❛❧ ❚❤❡♦r②

❋♦r ❛♥② z ∈ C✱ t❤❡r❡ ❛r❡ t✇♦ ❋❧♦q✉❡t ✐♥❞✐❝❡s✱ λ±✱ s♦❧✈✐♥❣ λ + λ−1 = ∆(z)✳ ■❢ |λ+| ≥ 1✱ ✇❡ s❡❡ t❤❛t γ(z) = lim

n→∞

1 n log Tn(λ) = 1 p log |λ+(z)| ❙♦❧✈✐♥❣ t❤❡ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ❢♦r λ γ(z) = 1 p

• log
• ∆(z)

2 +

• ∆(z)

2 2 − 1

• ❖♥ e = ∪p

j=1[αj, βj]✱ |. . .| = 1✱ s♦ γ(z) ≥ 0✱

γ(z) = 0 ♦♥ e✳ γ(z) ✐s ❤❛r♠♦♥✐❝ ♦♥ C \ e s✐♥❝❡ ∆

2 +

∆

2

• − 1 ✐s ❛♥❛❧②t✐❝ ❛♥❞ ♥♦♥✲✈❛♥✐s❤✐♥❣ t❤❡r❡

❛♥❞ γ(z) = log (|z|) + O(1) ❛t ∞✱ s✐♥❝❡ ∆(z) ✐s ❛ ❞❡❣r❡❡ p ♣♦❧②♥♦♠✐❛❧✳

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P♦t❡♥t✐❛❧ ❚❤❡♦r②

❚❤✉s γ(z) = Ge(z) ✐s t❤❡ ♣♦t❡♥t✐❛❧ t❤❡♦r✐sts✬ ●r❡❡♥✬s ❢✉♥❝t✐♦♥✳ ❚❤✉s✱ ❚❤❡♦r❡♠✳ γ(z) ❛s ❣✐✈❡♥ ❛❜♦✈❡ ✐s t❤❡ ♣♦t❡♥t✐❛❧ t❤❡♦r✐sts✬

• r❡❡♥✬s ❢✉♥❝t✐♦♥ ❛♥❞ ♣❡r✐♦❞✐❝ ❏❛❝♦❜✐ ♣❛r❛♠❡t❡rs ❛r❡

❛ss♦❝✐❛t❡❞ t♦ r❡❣✉❧❛r ♠❡❛s✉r❡s ✭✐♥ t❤❡ ❙t❛❤❧✕❚♦t✐❦ s❡♥s❡✮✳ ❈♦r♦❧❧❛r②✳ C(e) = (a1 · · · ap)1/p

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P♦t❡♥t✐❛❧ ❚❤❡♦r②

❇② ❣❡♥❡r❛❧ ♣r✐♥❝✐♣❧❡s✱ ✐❢ Ge ✐s s♠♦♦t❤ ✉♣ t♦ e ♦♥ eint✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♠❡❛s✉r❡ dρe(x) = fe(x)dx ✇❤❡r❡ fe(x) = 1 π ∂ ∂yGe(x + iy) |y=0 ❚❤✉s✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♠❡❛s✉r❡ ✐s fe(z) = 1 pπ |∆′(x)|

• 4 − ∆2(x)

= 1 pπ

• d

dx arccos ∆(x) 2

❋❧♦q✉❡t ❙♦❧✉t✐♦♥s P❡r✐♦❞✐❝ ❏❛❝♦❜✐ ▼❛tr✐❝❡s ❚❤❡ ❉✐s❝r✐♠✐♥❛♥t

• ❛♣s

❙♣❡❝tr✉♠ P♦t❡♥t✐❛❧ ❚❤❡♦r②

P♦t❡♥t✐❛❧ ❚❤❡♦r②

■♥ ❡❛❝❤✱ ❜❛♥❞ ∆(λ) ❣♦❡s ❢r♦♠ −2 t♦ ✷✱ s♦ arccos( ∆

2 ) ❢r♦♠

π t♦ 0✳ ❚❤✉s✱ ❚❤❡♦r❡♠✳ ρe

• [αj, βj]
• ❂ 1

p✳

❚❤✐s ❡①♣❧❛✐♥s t❤❡ ♣✉③③❧❡ ♠❡♥t✐♦♥❡❞ ❡❛r❧✐❡r✳ ❚❤✐s ✐s ❛❧s♦ ❛ ❞❡♥s✐t② ♦❢ ③❡r♦s ✇❛② ♦❢ ✉♥❞❡rst❛♥❞✐♥❣ ✇❤② t❤❡ ❛❜♦✈❡ fe ✐s t❤❡ ❉❖❙✳ ❋♦r t❤❡ ♣❡r✐♦❞✐❝ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤ ❛ ❜♦① ♦❢ s✐③❡ kp ❛r❡ t❤❡ ❋❧♦q✉❡t s♦❧✉t✐♦♥s ✇✐t❤ λ = e2πij/k✱ j = 0, 1, 2, . . . , k − 1✳