t tr stts t r - - PowerPoint PPT Presentation

✶✴✷✵

▼♦❞❡❧✐♥❣ ✜♥✐t❡ ❡♥tr♦♣② st❛t❡s ✇✐t❤ ❢r❡❡ ❢❡r♠✐♦♥s

❖❧❡❦s❛♥❞r ●❛♠❛②✉♥

❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r❞❛♠

✇♦r❦✲✐♥✲♣r♦❣r❡ss t♦❣❡t❤❡r ✇✐t❤✿

◆✳ ■♦r❣♦✈✱ ❨✉✳ ❩❤✉r❛✈❧❡✈✱ ❉✳ ▼✳ ❈❤❡r♥♦✇✐t③ ❛♥❞ ❏✳❙✳ ❈❛✉①

❘❆◗■❙✱ ❆♥♥❡❝②✱ ✵✸✴✵✾✴✷✵✷✵

✷✴✷✵

❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ✶❉ s②st❡♠s

q|O(x, t)O(✵, ✵)|q =

• ❦

|q|O|❦|✷e−itE❦+ixP❦ ◮ ◆✉♠❡r✐❝s ✭❆❇❆❈❯❙✮ ◮ ❋✐❡❧❞ t❤❡♦r② ✭kFx ≫ ✶✱ k✷

Ft ≫ ✶✮

◮ ▲✐♥❡❛r ❙♣❡❝tr✉♠ ◮ O = P(∂ϕ, ∂✷ϕ, eiϕ) ◮ ❯♥✐✈❡rs❛❧✐t② ❢r♦♠ ♠✐❝r♦s❝♦♣✐❝

▲❡❢t✿ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ❆❇❆❈❯❙ ❛♥❞ ✐♥❡❧❛st✐❝ ♥❡✉tr♦♥ s❝❛tt❡r✐♥❣ ❢♦r ❑❈✉❋✸ ✳ ❬P❘▲ ✶✶✶ ✶✸✼✷✵✺❪✳ ❘✐❣❤t✿ ❚❤❡ t❤r❡s❤♦❧❞ s✐♥❣✉❧❛r✐t✐❡s ✐♥ t❤❡ ◆♦♥✲▲✐♥❡❛r ▲✉tt✐♥❣❡r ▲✐q✉✐❞✳ ❇♦t❤ ❛♣♣r♦❛❝❤❡s ❢❛✐❧ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✦✦✦

✷✴✷✵

❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ✶❉ s②st❡♠s

q|O(x, t)O(✵, ✵)|q =

• ❦

|q|O|❦|✷e−itE❦+ixP❦ ◮ ◆✉♠❡r✐❝s ✭❆❇❆❈❯❙✮ ◮ ❋✐❡❧❞ t❤❡♦r② ✭kFx ≫ ✶✱ k✷

Ft ≫ ✶✮

◮ ▲✐♥❡❛r ❙♣❡❝tr✉♠ ◮ O = P(∂ϕ, ∂✷ϕ, eiϕ) ◮ ❯♥✐✈❡rs❛❧✐t② ❢r♦♠ ♠✐❝r♦s❝♦♣✐❝

▲❡❢t✿ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ❆❇❆❈❯❙ ❛♥❞ ✐♥❡❧❛st✐❝ ♥❡✉tr♦♥ s❝❛tt❡r✐♥❣ ❢♦r ❑❈✉❋✸ ✳ ❬P❘▲ ✶✶✶ ✶✸✼✷✵✺❪✳ ❘✐❣❤t✿ ❚❤❡ t❤r❡s❤♦❧❞ s✐♥❣✉❧❛r✐t✐❡s ✐♥ t❤❡ ◆♦♥✲▲✐♥❡❛r ▲✉tt✐♥❣❡r ▲✐q✉✐❞✳ ❇♦t❤ ❛♣♣r♦❛❝❤❡s ❢❛✐❧ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✦✦✦

✸✴✷✵

❖✉t❧✐♥❡

◮ ❲❛r♠✲✉♣✿ s✐♥❡✲❦❡r♥❡❧

◮ ▼✐❝r♦s❝♦♣✐❝ ❜♦s♦♥✐③❛t✐♦♥ T = ✵ ◮ P❤❛s❡ ❞r❡ss✐♥❣ ◮ ❙t❛t✐❝ ❝♦rr❡❧❛t✐♦♥s

◮ ❳❨✲♠♦❞❡❧

◮ ❋❡rr♦♠❛❣♥❡t✐❝ ◮ P❛r❛♠❛❣♥❡t✐❝

◮ ❉②♥❛♠✐❝s

✹✴✷✵

❙✐♥❡ ❑❡r♥❡❧

τ(x, t = ✵) = det

[−kF ,kF ]

• ✶ + e✷πiν − ✶

π sin x(q − p)/✷ q − p

• ❋♦r♠✲❢❛❝t♦r ♣r❡s❡♥t❛t✐♦♥

τ(x, t) = O(x, t)O(✵, ✵) =

• k✶<k✷···<kN

|q|O|❦|✷e−itE❦+iP❦x |q✿ ❢r❡❡ ❢❡r♠✐♦♥s✿ qi = ✷πni

L ❀

|❦✿ s❤✐❢t❡❞ ❢r❡❡ ❢❡r♠✐♦♥s✿ ki = ✷π(ni −ν)

L

P❦ =

• i

ki, E❦ =

• i

k✷

i /✷

❚❤❡ ❢♦r♠✲❢❛❝t♦r ✭♦✈❡r❧❛♣✮✿ |q|O|❦|✷ = ✷ L sin πν ✷N det

N×N

✶ ki − qj ✷ .

✺✴✷✵

❋✐❡❧❞ t❤❡♦r② tr❡❛t♠❡♥t ❂ ▼✐❝r♦s❝♦♣✐❝ ❜♦s♦♥✐③❛t✐♦♥ ✭T = ✵ ✮

◮ ❋♦r♠✲❋❛❝t♦r s✉♠♠❛t✐♦♥ τ(x, t) =

• ❦

|q|O|❦|✷e−ixP❦+itE❦ = det(✶ + ˆ V ) ◮ ❖rt❤♦❣♦♥❛❧✐t② ❈❛t❛str♦♣❤❡✿ |q|O|❦✈❛❝|✷ = A/N✷α ◮ ❙♦❢t✲♠♦❞❡ s✉♠♠❛t✐♦♥ τ(x, t) ∼

• ■❘

|q|O|❦|✷e−ixP❦+itE❦ = e

√αϕ(x,t)e−√αϕ(✵,✵) =

A (x − kF t)α(x + kF t)α ◮ ◆♦♥❧✐♥❡❛r ❝♦♥tr✐❜✉t✐♦♥s τ(x, t) ∼

• Q+■❘

|q|O|❦|✷e−ixP❦+itE❦+ix(Q−kF )+it(Q✷−k✷

F )/✷ =

B √t(x − kF t) ˜

α(x + kF t)α ❙❧❛✈♥♦✈ ✭✶✾✽✾✮❀ ❙❧❛✈♥♦✈ ❛♥❞ ❑♦r❡♣✐♥ ✭✶✾✾✶✮❀ ❆✳ ❙❤❛s❤✐✱ ▲✳ ■✳ ●❧❛③♠❛♥✱ ❏✳✲❙✳ ❈❛✉①✱ ❛♥❞ ❆✳ ■♠❛♠❜❡❦♦✈ ✭✷✵✶✶✮❀ ◆✳ ❑✐t❛♥✐♥❡✱ ❑✳❑✳ ❑♦③❧♦✇s❦✐✱ ❏✳✲▼✳ ▼❛✐❧❧❡t✱ ◆✳❆✳ ❙❧❛✈♥♦✈✱ ❛♥❞ ❱✳ ❚❡rr❛s ✭✷✵✵✾✲✷✵✶✷✮❀ ❑✳❑✳ ❑♦③❧♦✇s❦✐✱ ❏✳✲▼✳ ▼❛✐❧❧❡t ✭✷✵✶✺✮❀

✻✴✷✵

❈♦♠❜✐♥❛t♦r✐❝s ♦❢ ♦rt❤♦❣♦♥❛❧✐t② ❝❛t❛str♦♣❤❡

• ❡♥❡r✐❝ ♦✈❡r❧❛♣

|❦✈❛❝|q|✷ = ✷ L sin πν ✷N

• i>j

(ki − kj)✷

i>j

(qi − qj)✷

• i,j

(ki − qj)✷ . ❋❡r♠✐ s❡❛ ✐♥t❡❣❡rs kj = ✷π L (nj − ν), qj = ✷π L nj, nj = − N − ✶ ✷ + j − ✶, j = ✶, ✷ . . . N |❦✈❛❝|q|✷ = sin πν πν ✷N

i=j

• ✶ −

ν i − j −✷ = G ✷(✶ − ν)G ✷(✶ + ν)G ✹(N + ✶) G ✷(N − ν + ✶)G ✷(N + ν + ✶) . |❦✈❛❝|q|✷ = G ✷(✶ − ν)G ✷(✶ + ν) N✷ν✷ ❋♦r ν = ν(k)✱ ✭ν± = ν(±kF ), kF = πL/N✮ |✈❛❝|❋❙|✷ = G ✷(✶ − ν−)G ✷(✶ + ν+)(✷π)ν−−ν+ Nν✷

−+ν✷ +

exp

• [−kF ,kF ]✷

ν(λ)ν(µ) (λ − µ + i✵)✷ dλdµ

✼✴✷✵

❙t❛t✐❝ ✰ ③❡r♦ t❡♠♣❡r❛t✉r❡

τ(x, t = ✵) = G ✷(✶ − ν)G ✷(✶ + ν) (−✷ix)ν✷(✷ix)ν✷ e−✷iνx + (ν → ν + Z)

✽✴✷✵

❋✐♥✐t❡ t❡♠♣❡r❛t✉r❡

τ(x) = det

• ✶ + nF(q)

π (e✷πiν − ✶)sin(x(p − q)) p − q

• ◮ ■t ✐s ❝❤❛❧❧❡♥❣✐♥❣ t♦ ❞♦ ♠✐❝r♦s❝♦♣✐❝

◮ ❖✈❡r❧❛♣s ❛r❡ t♦ s♠❛❧❧ ∼ e−cN ◮ ❚♦♦ ♠❛♥② s♦❢t ♠♦❞❡s ∼ ecN

◮ ❍❡✉r✐st✐❝ ❛♣♣r♦❛❝❤ ✐♥st❡❛❞✦ ❉r❡ss✐♥❣✿ ✐♥❤♦♠♦❣❡♥❡♦✉s ❛♥❞ ❝♦♠♣❧❡① ✈❛❧✉❡❞✦✦✦

nF (q) π (e✷πiν−✶) = e✷πiνT (q) − ✶ π , ν → νT (q) = ✶ ✷πi log(✶+(e✷πiν−✶)nF (q))

◮ τ(x, t) ≈ exp

• −i

∞

−∞

(x − qt)νT(q)dq +

• R✷

νT(q)νT(q′) (q′ − q + i✵)✷ dqdq′

✾✴✷✵

5 10 15 20 25 x

• 0.2

0.2 0.4 0.6 0.8 1.0 Re[τ[x]]

T=0.1

5 10 15 20 25 x 0.2 0.4 0.6 0.8 1.0 Re[τ[x]]

T=1.0

❚r(e−βHO(x, t) . . . )/❚r❡−β❍ = O(z = x+it) . . . S✶×R✶

z→z′=e✷πz/β

= ∼ O(z′) . . . R✷ ❈❋❚ ♣r❡❞✐❝t✐♦♥ ❢♦r ❝♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤✿ τ(x)

• T=✵ = A

xν✷ = ⇒ τ(x) = A (sinh(xT)/T)ν✷ = e−x/ξ = ⇒ ✶/ξ ∼ T???

✶✵✴✷✵

❊①❛♠♣❧❡s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s

◮ ▼♦❜✐❧❡ ✐♠♣✉r✐t② ❬❙❝✐P♦st P❤②s✳ ✽✱ ✵✺✸ ✭✷✵✷✵✮✱ ◆❡✇ ❏✳ P❤②s✳ ✶✽ ✭✷✵✶✻✮✱ ✵✹✺✵✵✺❪ ρ(y) = det

[−✶,✶](✶ + ˆ

K + δ ˆ K) − det

[−✶,✶](✶ + ˆ

K) K(p, q) = sin[πF(p)]ei(p−q)y/✷(cot[πF(p)] + i) − ei(q−p)y/✷(cot[πF(q)] + i) π(p − q) sin[πF(q)] δK(p, q) = ✶ π sin[πF(p)]e−i(p+q)y/✷ sin[πF(q)] ◮ ❘❡t✉r♥ ♣r♦❜❛❜✐❧✐t② ❢r♦♠ t❤❡ ❞♦♠❛✐♥ ✇❛❧❧ ✐♥✐t✐❛❧ st❛t❡ |❉❲ = | ↑↑ . . . ↑↓↓ . . . ↓ ❬❏✳▼✳ ❙t❡♣❤❛♥✱ ❏✳ ❙t❛t ✭✷✵✶✼✮❪ ❉❲|eτHXXX |❉❲ = det R+

• ✶ − e−p✷/✹ sin √τ(p − q)

π(p − q) e−q✷/✹

• ◮ P❡rs✐st❡♥❝❡ ♦❢ s♣✐♥ ❝♦♥✜❣✉r❛t✐♦♥s ❬■✳ ❉♦r♥✐❝ ✭✷✵✶✽✮❪ nF (q) = ✶/ cosh(q)

◮ ❈❧❛ss✐❝❛❧ ✐♥t❡❣r❛❜❧❡ s②st❡♠s nF (q) = r(q)

✶✶✴✷✵

❍XY = −✶ ✷

L

• j=✶

✶ + γ ✷ σx

j σx j+✶ + ✶ − γ

✷ σy

j σy j+✶ + hσz j

• ❙♣❡❝tr✉♠ ♦❢ ❢❡r♠✐♦♥✐❝ ✭▼❛❥♦r❛♥❛✮ ❡①❝✐t❛t✐♦♥s

E(q) =

• (h − cos q)✷ + γ✷ sin✷ q

❇♦❣♦❧②✉❜♦✈ r♦t❛t✐♦♥ ❛♥❣❧❡ eiθ(q) = h − cos q − iγ sin q

• (h − cos q)✷ + γ✷ sin✷ q

◗✉❛♥t✉♠ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❢♦r T = ✵ ❛t h = ✶✳ ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ❚rσx

x+✶σx ✶e−β❍XY

❚re−β❍XY = A(T, h, γ)e−x/ξ(T,h,γ)

✶✷✴✷✵

❋r❡❞❤♦❧♠ ❞❡t❡r♠✐♥❛♥t ♣r❡s❡♥t❛t✐♦♥ ❬❆✳●✳ ■③❡r❣✐♥✱ ❱✳❙✳ ❑❛♣✐t♦♥♦✈✱ ◆✳❆✳ ❑✐t❛♥✐♥❡✱ s♦❧✈✲✐♥t✴✾✼✶✵✵✷✽❪ τ(x) ≡ ❚rσx

x+✶σx ✶e−β❍XY

❚re−β❍XY = det

[−π,π] (✶ + V + δV ) − det [−π,π] (✶ + V )

V = −ωF(q) π e

i(p−q) ✷

sin x(p−q)

✷

sin p−q

✷

, δV = −ωF(q) π e−i(p+q)x/✷ ωF(q) = ✶ ✷

• ✶ − eiθ(q) tanh βE(q)

✷

✶✸✴✷✵

❋♦r♠✲❢❛❝t♦rs τ(x) =

• q

|❦|q|✷e

−ix

• N+✶
• i=✶

ki−

N

• i=✶

qi

• ✇✐t❤

eikL = e−✷πiν(k), eiqL = ✶, L = N + ✶. |❦|q|✷ = A N+✶

• i=✶

ωi sin πνi L ✷

N+✶

• i>j

sin✷ ki−kj

✷ N

• i>j

sin✷ qj−qi

✷ N+✶

• i=✶

N

• j=✶

sin✷ ki−qj

✷

◆❖❚■❈❊✿ ✭✐✮ ❇r✐❧❧♦✉✐♥ ③♦♥❡❀ ✭✐✐✮ ▲❛r❣❡ ①✦ e✷πiν(k) = ✶ − ✷ωF(k) = eiθ(k) tanh βE(k) ✷ .

✶✹✴✷✵

|❦|q(✵)|✷ = O(✶) |❦|q(✵)|✷ = ✶/L ❊①❝✐t❛t✐♦♥s |❦|q(✵)

a→N+✶|✷ ∼

✶ (a + ν−)✷ |❦|q(✵)

a→N+✶|✷ ∼ f (a/L)

L

✶✺✴✷✵

❋❡rr♦♠❛❣♥❡t✐❝ h ≤ ✶

τ(x) =

• q

|❦|q|✷e

−ix

• N+✶
• i=✶

ki −

N

• i=✶

qi

• τ(x) ≈ eixδPA✵

L/✷

• a=−L/✷

✶ (a + ν−)✷ = A(T, h, γ)e−x/ξ(T,h,γ) ξ−✶ = −i

π

• −π

ν(q)dq = − ✶ ✷π

π

• −π

dk log tanh βE(k) ✷ log A = log ✷π e +

π

• −π

dq

q

• −π

dk

• ✷(ν(q) − ν(k)) − ✶

(✷π − q + k)✷ − (ν(q) − ν(k))✷ ✹ sin✷ q−k

✷

• −

π

• −π

dqν(q)

• ✹q

π✷ − q✷ − tan q ✷

✶✻✴✷✵

P❛r❛♠❛❣♥❡t✐❝ h > ✶

τ(x) =

• a

|❦|q(✵)

a→N+✶|✷eix∆P−ixqa = R(x)

✷π exp  ix∆P − ✶ ✹

π

• −π

dq

q

• −π

dk (ν(q) − ν(k))✷ sin✷ q−k

✷

  R(x) ❝♦rr❡s♣♦♥❞s t♦ ❤♦❧❡s✬ ❝♦♥tr✐❜✉t✐♦♥ R(x) =

π

• −π

dk e−ikx(✶ − e−✷πiν(k)) exp  −

π

• −π

ν(q) cot q − k + i✵ ✷ dq   ξ−✶ = − ✶ ✷π

π

• −π

dk log tanh βE(k) ✷ + log y+ y± = h +

• h✷ + γ✷ − ✶

✶ ± γ log A = log ✷ β

• h✷ + γ✷ − ✶

− i

π

• −π

dq ν(q) eiq + y+ eiq − y+ − ✶ ✹

π

• −π

dq

q

• −π

dk (ν(q) − ν(k))✷ sin✷ q−k

✷

✶✼✴✷✵

β = ✶.✶✱ γ = ✵.✷✺

✶✽✴✷✵

❊②t❛♥ ❇❛r♦✉❝❤ ❛♥❞ ❇❛rr② ▼✳ ▼❝❈♦② P❤②s✳ ❘❡✈✳ ❆ ✸✱ ✼✽✻ ✭✶✾✼✶✮ A = XY , ✇❤❡r❡✿ X =

∞

• l=✶
• ✶ − λ−✶

✶ f✷l−✶

✶ − λ−✶

✶ g✷l−✶

✶ − λ−✶

✷ f✷l−✶

✶ − λ−✶

✷ g✷l−✶

• ✶ − λ−✶

✶ f✷l

✶ − λ−✶

✶ g✷l

✶ − λ−✶

✷ f✷l

✶ − λ−✶

✷ g✷l

• Y =

∞

• i,j=✶

(✶ − f✷jf✷i−✶)(✶ − f✷if✷j−✶)(✶ − g✷jg✷i−✶)(✶ − g✷ig✷j−✶) (✶ − f✷jf✷i)(✶ − f✷j−✶f✷i−✶)(✶ − g✷jg✷i)(✶ − g✷j−✶g✷i−✶) × × (✶ − f✷jg✷i−✶)(✶ − f✷ig✷j−✶)(✶ − g✷jf✷i−✶)(✶ − g✷if✷j−✶) (✶ − f✷jg✷i)(✶ − g✷jf✷i)(✶ − g✷j−✶f✷i−✶)(✶ − f✷j−✶g✷i−✶) ❛♥❞ λ✶, λ✷, f , g ❛r❡ ❞❡✜♥❡❞ ❛s λ✶ =

• h +
• h✷ −
• ✶ − γ✷✶/✷

/(✶−γ), λ✷ =

• h −
• h✷ −
• ✶ − γ✷✶/✷

/(✶−γ) fk = h + Wk ✶ − γ✷ − h + Wk ✶ − γ✷ ✷ − ✶ ✶/✷ , gk = h − Wk ✶ − γ✷ − h − Wk ✶ − γ✷ ✷ − ✶ ✶/✷ ✇✐t❤ Wk =

• γ✷h✷ −
• ✶ − γ✷

γ✷ + (kπ)✷β−✷✶/✷

✶✾✴✷✵

❉②♥❛♠✐❝s ✭Pr❡❧✐♠✐♥❛r②✮

νT (q) = ✶ ✷πi log

• ✶ + (e✷πiν − ✶)nF (q)
• θ(x−qt)− ✶

✷πi log

• ✶ + (e−✷πiν − ✶)nF (q)
• θ(qt−x).

❊r❢ (x − qt)(✶ + i) ✷√t

• → ❙✐❣♥(x − qt)
• 100
• 50

50 100 x

• 0.03
• 0.02
• 0.01

0.01 0.02 0.03

T=0.1, τ(x,50)

• 40
• 20

20 40 x

• 1.5×10-9
• 1.×10-9
• 5.×10-10

5.×10-10 1.×10-9 1.5×10-9

T=1.0, τ(x,50)

❋✐❣✉r❡✿ ❚❤❡ ❡①❛❝t ❡①♣r❡ss✐♦♥s ✭s♦❧✐❞ ❧✐♥❡s✮ ✈s✳ t❤❡ ❛s②♠♣t♦t✐❝ ✭❞♦tt❡❞ ❧✐♥❡s✮ ❢♦r

ν = ✵.✹✳ ✭❘❡❛❧ ♣❛rt ✐s ❜❧❛❝❦ ❛♥❞ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s r❡❞✮✳

✷✵✴✷✵

❙✉♠♠❛r② ❛♥❞ ♦✉t❧♦♦❦

◮ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ❢♦r ❋r❡❞❤♦❧♠ ❞❡t❡r♠✐♥❛♥t ✇✐t❤♦✉t s♦❧✈✐♥❣ ❘❍P ◮ P❤❛s❡ s❤✐❢t ❞r❡ss✐♥❣ ◮ ❉✐✛❡r❡♥t t②♣❡s ♦❢ s♦❢t ♠♦❞❡ ❝♦♥tr✐❜✉t✐♦♥s ◮ ❯♥✐✈❡rs❛❧✐t② ◮ ❆s②♠♣t♦t✐❝ ❢♦r ❝❧❛ss✐❝❛❧ ✐♥t❡❣r❛❜❧❡ ♠♦❞❡❧s❄ ◮ ❘❡❧❛t✐♦♥ ✇✐t❤ ◗❚▼❄ ❚❤❡r♠❛❧ ❢♦r♠✲❢❛❝t♦rs❄

✷✶✴✷✵

❊①tr❛ s❧✐❞❡s

✷✷✴✷✵

▲✐❡❜✲▲✐♥✐❣❡r ♠♦❞❡❧

H = −

N

• i=✶

∂✷ ∂x✷

i

+ c

• i<j

δ(xi − xj) ◮ ❉❡♥s✐t② ♦❢ r♦♦ts ρ(λi) = ✶ L(λi+✶ − λi) ⇔ ρ(λ) − ✶ ✷π

q

• −q

K(λ − µ)ρ(µ)dµ = ✶ ✷π ◮ ❉r❡ss❡❞ ♠♦♠❡♥t✉♠ ✭❝♦✉♥t✐♥❣ ❢✉♥❝t✐♦♥✮ eiLP(µ) − ✶ = ✵ P(µ) = µ +

q

• −q

θ(µ − λ)ρ(λ)dλ ◮ ❙❤✐❢t ❢✉♥❝t✐♦♥ F(λi) = µi − λi λi+✶ − λi ⇔ F(λ) − ✶ ✷π q

−q

K(λ − µ)F(µ)dµ = π + θ(λ − q) ✷π θ(λ) = ✷ arctan λ c K(λ) = ∂λθ(λ) = ✷c λ✷ + c✷

✷✸✴✷✵

❙♦❢t ♠♦❞❡ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ♦♥❡✲♣❛rt✐❝❧❡ ❞❡♥s✐t② ♠❛tr✐①

ρ(x) =

• |µ

e−ix(Pλ−Pµ)|µ|ˆ Ψ|λ|✷ = R

• det(✶ + ˆ

K + δ ˆ K) − det(✶ + ˆ K)

• K(λ, λ′) = sin[πF(λ)] sin[πF(λ′)]×

× ei(P(λ)−P(λ′))y/✷(cot[πF(λ)] + i) − ei(P(λ′)−P(λ))y/✷(cot[πF(λ′)] + i) π(λ − λ′) δK(λ, λ′) = ✶ π sin[πF(λ)]e−i(P(λ)+P(λ′))y/✷ sin[πF(λ′)]

✷✹✴✷✵

❆❇❆❈❯❙ ✈❡rs✉s s♦❢t ♠♦❞❡s

Q = ✵; c = ✹; N/L = ✶

✷✺✴✷✵

❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❛t ③❡r♦ t❡♠♣❡r❛t✉r❡

τ(x, t) = G✷(✶ − F)G✷(✶ + F) (✷i(t − x))F✷ (✷i(x + t))F✷ e−✷iFx + G✷(✶ − F)G✷(F) F ✷(✷i(t − x))(✶+F)✷ (✷i(x + t))F✷ x − t x + t ✷F e−i(t−x)✷/(✷t)−✷iFx (x/t − ✶)✷

• ✷π

−it θ(x✷ > t✷)+ + G✷(−F)G✷(✶ + F) F ✷(✷i(t − x))(✶−F)✷ (✷i(x + t))F✷ x + t x − t ✷F ei(t−x)✷/(✷t)−✷iFx (x/t − ✶)✷

• ✷π

it θ(x✷ < t✷) + (F → F + Z)