t ss rrt rt t - - PDF document

❖♥ t❤❡ ❜❛s✐s ♣r♦♣❡rt② ♦❢ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠s ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦rs ✇✐t❤ r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❆❧❡①❛♥❞❡r ▼❛❦✐♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ s♣❡❝tr❛❧ t❤❡♦r② ♦❢ ♥♦♥✲s❡❧❢❛❞❥♦✐♥t ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ♦♥ ❛ ✜♥✐t❡ ✐♥t❡r✈❛❧ ❢♦r nt❤ ♦r❞❡r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ y(n) + p1(x)y(n−1) + . . . + pn(x)y = λy ✇❛s ❜❡❣✉♥ ❜② ❇✐r❦❤♦✛ ✐♥ ❤✐s t✇♦ ♣❛♣❡rs ❬✷✱ ✸❪ ♦❢ ✶✾✵✽ ✇❤❡r❡ ❤❡ ✐♥tr♦❞✉❝❡❞ r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✜rst t✐♠❡✳ ■t ✇❛s ❝♦♥t✐♥✉❡❞ ❜② ❚❛♠❛r❦✐♥ ❬✸✵✱ ✸✶❪ ❛♥❞ ❙t♦♥❡ ❬✷✽✱ ✷✾❪✳ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ✜rst ♦r❞❡r s②st❡♠s ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ 1 i B dy dx + Q(x)y = λy, y = col(y1, . . . , yn) (1), ✇❤❡r❡ B ✐s ❛ ♥♦♥s✐♥❣✉❧❛r ❞✐❛❣♦♥❛❧ n × n ♠❛tr✐①✱ B = diag(b−1

1 In1, . . . , b−1 r Inr) ∈ Cn×n,

n = n1 + . . . nr, ✇✐t❤ ❝♦♠♣❧❡① ❡♥tr✐❡s bj = bk✱ ❛♥❞ Q(x) ✐s ❛ ♣♦t❡♥t✐❛❧ ♠❛tr✐① t❛❦❡s ✐ts ♦r✐❣✐♥ ✐♥ t❤❡ ♣❛♣❡r ❜② ❇✐r❦❤♦✛ ❛♥❞ ▲❛♥❣❡r ❬✹❪✳ ❆❢t❡r✇❛r❞s t❤❡✐r ✐♥✈❡st✐❣❛t✐♦♥s ✇❡r❡ ❞❡✈❡❧♦♣❡❞ ✐♥ ♠❛♥② ❞✐r❡❝t✐♦♥s✳ ▼❛❧❛♠✉❞ ❛♥❞ ❖r✐❞♦r♦❣❛ ✐♥ ❬✷✵❪ ❡st❛❜❧✐s❤❡❞ ✜rst ❣❡♥❡r❛❧ r❡s✉❧ts ♦♥ ❝♦♠♣❧❡t❡♥❡ss ♦❢ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠s ♦❢ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ❞✐✛❡r❡♥t✐❛❧ s②st❡♠s ✭✶✮✳ ❆ ❧✐tt❧❡ ❜✐t ❧❛t❡r ▲✉♥②♦✈ ❛♥❞ ▼❛❧❛♠✉❞ ✐♥ ❬✶✼❪ ♦❜t❛✐♥❡❞ ✜rst ❣❡♥❡r❛❧ r❡s✉❧ts ♦♥ ❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ✭❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ✇✐t❤ ♣❛r❡♥t❤❡s❡s✮ ❢♦r ♠❡♥t✐♦♥❡❞ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ✇✐t❤ ❛ ♣♦t❡♥t✐❛❧ ♠❛tr✐① Q(x) ∈ L∞✳ ❚❤❡r❡ ✐s ❛♥ ❡♥♦r♠♦✉s ❧✐t❡r❛t✉r❡ r❡❧❛t❡❞ t♦ t❤❡ s♣❡❝tr❛❧ t❤❡♦r② ♦✉t❧✐♥❡❞ ❛❜♦✈❡✱ ❛♥❞ ✇❡ r❡❢❡r t♦ ❬✻✱ ✼✱ ✶✻✱ ✷✷✱ ✷✺❪ ❛♥❞ t❤❡✐r ❡①t❡♥s✐✈❡ r❡❢❡r❡♥❝❡ ❧✐sts ❢♦r t❤✐s ❛❝t✐✈✐t②✳ ■♥ t❤❡ ♣r❡s❡♥t ♣❛♣❡r✱ ✇❡ st✉❞② t❤❡ ❉✐r❛❝ s②st❡♠ By′ + V y = λy, (2) ✇❤❡r❡ y = col(y1(x), y2(x))✱ B = −i 0 i

• ,

V =

• P(x)

Q(x)

• ,

✶

t❤❡ ❢✉♥❝t✐♦♥s P(x), Q(x) ∈ L1(0, π)✱ ✇✐t❤ t✇♦✲♣♦✐♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s U(y) = Cy(0) + Dy(π) = 0, (3) ✇❤❡r❡ C = a11 a12 a21 a22

• ,

D = a13 a14 a23 a24

• ,

t❤❡ ❝♦❡✣❝✐❡♥ts aij ❛r❡ ❛r❜✐tr❛r② ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ❛♥❞ r♦✇s ♦❢ t❤❡ ♠❛tr✐① A = a11 a12 a13 a14 a21 a22 a23 a24

• ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦♣❡r❛t♦r Ly = By′ + V y ❛s ❛ ❧✐♥❡❛r

♦♣❡r❛t♦r ♦♥ t❤❡ s♣❛❝❡ H = L2(0, π) ⊕ L2(0, π)✱ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ D(L) = {y ∈ W 1

1 [0, π] : Ly ∈ H✱ Uj(y) = 0 (j = 1, 2)}✳

❙♣❡❝tr❛❧ ♣r♦❜❧❡♠s ❢♦r t❤❡ ♦♣❡r❛t♦r L ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭✸✮ ✇❡r❡ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❛ ❧♦t ♦❢ ♣❛♣❡rs✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ❬✷✷❪ t❤❛t t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ♦❢ ♣r♦❜❧❡♠ ✭✷✮✱ ✭✸✮ ✇✐t❤ r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐s ❝♦♠♣❧❡t❡ ✐♥ H✳ ❚r♦♦s❤✐♥ ❛♥❞ ❨❛♠❛♠♦t♦ ❬✸✷✱ ✸✸❪ ♣r♦✈❡❞ t❤❡ ❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ❢♦r t❤❡ ♦♣❡r❛t♦r L ✐♥ t❤❡ ❝❛s❡ ♦❢ s❡♣❛r❛t❡❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛♥❞ V ∈ L2(0, π)✳ ❉✐r❛❝ ♦♣❡r❛t♦rs ✇✐t❤ ♥♦♥s♠♦♦t❤ ♣♦t❡♥t✐❛❧s ✇❡r❡ ❝♦♥s✐❞❡r❡❞ ❜② ❇✉r❧✉ts❦❛②❛✱ ❑♦r♥❡✈ ❛♥❞ ❑❤r♦♠♦✈ ❬✺✱ ✶✺❪✳ ❘❡❣✉❧❛r ❉✐r❛❝ ♣r♦❜❧❡♠s ✇✐t❤ ♣♦t❡♥t✐❛❧s V ∈ L2(0, π) ✇❡r❡ st✉❞✐❡❞ ❜② ❉❥❛❦♦✈ ❛♥❞ ▼✐t②❛❣✐♥ ❬✽✲ ✶✹❪✱ ❛♥❞ ❛❧s♦ ❜② ❆rs❧❛♥ ❬✶❪✳ ■t ✇❛s ❡st❛❜❧✐s❤❡❞ ❜② ▲✉♥②♦✈ ❛♥❞ ▼❛❧❛♠✉❞ ✐♥ ❬✶✽✱ ✶✾❪ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❜② ❙❛✈❝❤✉❦✱ ❙❛❞♦✈♥✐❝❤❛②❛ ❛♥❞ ❙❤❦❛❧✐❦♦✈ ✐♥ ❬✷✻✱ ✷✼❪ t❤❛t t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ♦❢ ♣r♦❜❧❡♠ ✭✷✮✱ ✭✸✮ ✇✐t❤ str♦♥❣❧② r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r♠s ❛ ❘✐❡s③ ❜❛s✐s ✐♥ H ❛♥❞ ❛ ❘✐❡s③ ❜❛s✐s ✇✐t❤ ♣❛r❡♥t❤❡s❡s ✐♥ H ✐♥ t❤❡ ❝❛s❡ ♦❢ r❡❣✉❧❛r ❜✉t ♥♦t str♦♥❣❧② r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ◆♦t❡ t❤❛t ✐♥ ❬✶✽✱ ✶✾✱ ✷✻✱ ✷✼❪ t❤❡ ♣♦t❡♥t✐❛❧ V (x) ∈ L1(0, π)✳ ❍♦✇❡✈❡r✱ ❢♦r r❡❣✉❧❛r ❜✉t ♥♦t str♦♥❣❧② r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭❡①❝❡♣t t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ♣❡r✐♦❞✐❝ ❛♥❞ ❛♥t✐♣❡r✐♦❞✐❝ ♦♥❡s ✇❤✐❝❤ ✇❛s ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✶✱ ✽✲✶✹✱ ✷✸✱ ✷✹❪ ✐❢ t❤❡ ❢✉♥❝t✐♦♥s P(x), Q(x) ∈ L2(0, π)✮ ❛❧❧ t❤❡ ♠❡♥t✐♦♥❡❞ ♣❛♣❡rs r❡♠❛✐♥ ♦♣❡♥ t❤❡ q✉❡st✐♦♥ ✇❤❡t❤❡r t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ❢♦r♠s ❛ ✉s✉❛❧ ❘✐❡s③ ❜❛s✐s r❛t❤❡r t❤❛♥ ❛ ❘✐❡s③ ❜❛s✐s ✇✐t❤ ♣❛r❡♥t❤❡s❡s✳ ❚❤❡ ♠❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤❡ ♣r❡s❡♥t ❛rt✐❝❧❡ ✐s t♦ st✉❞② t❤✐s ♣r♦❜❧❡♠✳ ✷✳ ❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❞❡t❡r♠✐♥❛♥t ❛♥❞ t❤❡ s♣❡❝tr✉♠ ❉❡♥♦t❡ ❜② E(x, λ) = e11(x, λ) e12(x, λ) e21(x, λ) e22(x, λ)

• ✷

t❤❡ ♠❛tr✐① ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥ s②st❡♠ t♦ ❡q✉❛t✐♦♥ ✭✶✮ ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ E(0, λ) = I✱ ✇❤❡r❡ I ✐s t❤❡ ✉♥✐t ♠❛tr✐①✳ ❉❡♥♦t❡ ❛❧s♦ ❜② Aij t❤❡ ❞❡t❡r♠✐♥❛♥t ❝♦♠♣♦s❡❞ ♦❢ t❤❡ it❤ ❛♥❞ jt❤ ❝♦❧✉♠♥s ♦❢ t❤❡ ♠❛tr✐① A✳ ■t ✇❛s s❤♦✇♥ ✐♥ ❬✶✾❪ ❜② t❤❡ ♠❡t❤♦❞ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥ ♦♣❡r❛t♦rs t❤❛t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❞❡t❡r♠✐♥❛♥t ∆(λ) ♦❢ ♣r♦❜❧❡♠ ✭✷✮✱ ✭✸✮ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❢♦r♠ ∆(λ) = A12 + A34 + A32e11(π, λ) + A14e22(π, λ) + A13e12(π, λ) + A42e21(π, λ) = = ∆0(λ) + π

0 r1(t)e−iλtdt +

π

0 r2(t)eiλtdt,

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ∆0(λ) = A12 + A34 − A23eiπλ + A14e−iπλ ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❞❡t❡r♠✐♥❛♥t ♦❢ ♣r♦❜❧❡♠ By′ = λy, U(y) = 0, (4) ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s rj(t) ∈ L1(0, π)✱ j = 1, 2✳ ❉❡✜♥✐t✐♦♥ ✶✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭✸✮ ❛r❡ ❝❛❧❧❡❞ r❡❣✉❧❛r ✐❢ A14A23 = 0, (5) ❛♥❞ str♦♥❣❧② r❡❣✉❧❛r ✐❢ ❛❞❞✐t✐♦♥❛❧❧② (A12 + A34)2 + 4A14A23 = 0. (6) ❉❡✜♥✐t✐♦♥ ✷✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭✸✮ ❛r❡ ❝❛❧❧❡❞ r❡❣✉❧❛r ❜✉t ♥♦t str♦♥❣❧② r❡❣✉❧❛r ✐❢ ✭✺✮ ❤♦❧❞s ❜✉t ✭✻✮ ❢❛✐❧s✱ ✐✳❡✳✱ (A12 + A34)2 + 4A14A23 = 0. (7) ■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t t❤❡ ❡✐❣❡♥✈❛❧✉❡s λ0

k ♦❢ ♦♣❡r❛t♦r L0y = By′ ✇✐t❤ r❡❣✉❧❛r

❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r♠ t✇♦ s❡r✐❡s λ0

k = λ0 n,1 = − i π ln z1 + 2n ✐❢ k = 2n✱ ❛♥❞

λ0

k = λ0 n,2 = − i π ln z2 + 2n ✐❢ k = 2n + 1✱ n ∈ Z✱ ✇❤❡r❡ z1 ❛♥❞ z2 ❛r❡ t❤❡ r♦♦ts ♦❢

t❤❡ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ A23z2 − (A12 + A34)z − A14 = 0 ❛♥❞ ❛ ❜r❛♥❝❤ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠ ✇✐t❤ ✈❛❧✉❡s ✐♥ t❤❡ str✐♣ −π < Imz ≤ π ✐s ✜①❡❞✳ ❆❧s♦✱ ✐t ✇❛s s❤♦✇♥ ✐♥ ❬✶✽✱✶✾✱ ✷✻✱✷✼❪ t❤❛t t❤❡ ❡✐❣❡♥✈❛❧✉❡s λk ♦❢ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ✇✐t❤ r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s s❛t✐s❢② t❤❡ ❛s②♠♣t♦t✐❝ r❡❧❛t✐♦♥ λk = λ0

k + o(1), ✸

✇❤❡r❡ λ0

k ❛r❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥♣❡rt✉r❜❡❞ ♦♣❡r❛t♦r✳

❋✉rt❤❡r✱ ✇❡ ❝♦♥s✐❞❡r ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ♦♥❧② ✇✐t❤ r❡❣✉❧❛r ❜✉t ♥♦t str♦♥❣❧② r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❖♥❡ ❝❛♥ r❡❛❞✐❧② s❡❡ t❤❛t ✐♥ t❤✐s ❝❛s❡ z1 = z2 = A12 + A34 2A23 = z, ❛♥❞ t❤❡ s♣❡❝tr✉♠ ❝♦♥s✐sts ♦❢ ♣❛✐r✇✐s❡ ❝❧♦s❡ ❡✐❣❡♥✈❛❧✉❡s λk = λn,1 = − i

π ln z+2n+

εn,1 ✐❢ k = 2n✱ ❛♥❞ λk = λn,2 = − i

π ln z + 2n + εn,2 ✐❢ k = 2n + 1✱ k ∈ Z✱ εn,j → 0

✐❢ k → ±∞✳ ■❢ ❢♦r ❛❧❧ n s✉❝❤ t❤❛t |n| > n0✱ λn,1 = λn,2✱ t❤❡♥ t❤❡ s♣❡❝tr✉♠ ✐s ❝❛❧❧❡❞ ❛s②♠♣t♦t✐❝❛❧❧② ♠✉❧t✐♣❧❡✳ ■❢ ❢♦r ❛❧❧ n s✉❝❤ t❤❛t |n| > n0✱ λn,1 = λn,2✱ t❤❡♥ t❤❡ s♣❡❝tr✉♠ ✐s ❝❛❧❧❡❞ ❛s②♠♣t♦t✐❝❛❧❧② s✐♠♣❧❡✳ ❚♦ ❛♥❛❧②③❡ t❤✐s ❝❧❛ss ♦❢ ♣r♦❜❧❡♠s✱ ✐t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❞✐✈✐❞❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭✸✮ s❛t✐s❢②✐♥❣ ✭✺✮ ❛♥❞ ✭✼✮ ✐♥t♦ t✇♦ t②♣❡s✿ ✭❛✮ A13 = A24 = 0❀ ✭❜✮ |A13| + |A24| > 0✳ ❚❤❡s❡ t✇♦ ❝❛s❡s s❤♦✉❧❞ ❜❡ ❝♦♥s✐❞❡r❡❞ s❡♣❛r❛t❡❧②✳ ✸✳ ▼❛✐♥ r❡s✉❧ts ❈❛s❡ ✭❛✮✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭❛✮ ❛r❡ ❝❛❧❧❡❞ ♣❡r✐♦❞✐❝✲t②♣❡ ❝♦♥❞✐t✐♦♥s ❬✶✹❪✳ P❡r✐♦❞✐❝✲t②♣❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② t❤❡ ♠❛tr✐① 1 0 a 0 0 a 0 1

• ,

(8) ✇❤❡r❡ a = 0✳ ■❢ a = −1✱ t❤❡♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭✽✮ ❛r❡ ♣❡r✐♦❞✐❝✱ ❛♥❞ ✐❢ a = 1✱ ♦♥❡s ❛r❡ ❛♥t✐♣❡r✐♦❞✐❝✳ ■t ✇❛s s❤♦✇♥ ✐♥ ❬✶✹❪ t❤❛t ❢♦r ♣r♦❜❧❡♠ ✭✹✮ ✇✐t❤ ♣❡r✐♦❞✐❝✲t②♣❡ ❝♦♥❞✐t✐♦♥s ❛❧❧ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❞♦✉❜❧❡ ❛♥❞ ❛❧❧ ❝♦rr❡s♣♦♥❞✐♥❣ r♦♦t s✉❜s♣❛❝❡s ❝♦♥s✐st ♦❢ t✇♦ ❡✐❣❡♥❢✉♥❝t✐♦♥s✳ ❘❡✇r✐t❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ✇✐t❤ ♣❡r✐♦❞✐❝✲t②♣❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ ♠♦r❡ ✈✐s✉❛❧ ❢♦r♠ −iy′

1 + P(x)y2 = λy1,

iy′

2 + Q(x)y1 = λy2,

y1(0) + ay1(π) = 0, ay2(0) + y2(π) = 0. (9) ▲❡t a = reiϕ, −π < ϕ ≤ π✳ ❉❡♥♦t❡ τ0 = ϕ+π

π

+ i ln r

π ✳ ❚❤❡♥ ♣r♦❜❧❡♠ ✭✾✮ ❤❛s t✇♦

s❡r✐❡s ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s λn,j = τ0 + 2n + εn,j,

✹

✇❤❡r❡ n ∈ Z✱ εn,j → 0 ❛s |n| → ∞✱ j = 1, 2✳ ❚❤❡ ❝♦♥❥✉❣❛t❡ ♣r♦❜❧❡♠ ❤❛s t❤❡ ❢♦r♠ −iz′

1 + ¯

Q(x)z2 = λz1, iz′

2 + ¯

P(x)z1 = λz2, ¯ az1(0) + z1(π) = 0, z2(0) + ¯ az2(π) = 0. ❚❤❡♦r❡♠ ✶✳ ❙✉♣♣♦s❡ P(x), Q(x) ∈ W 1

2 (0, π)✳ ❚❤❡ s②st❡♠ ♦❢ ❡✐❣❡♥✲ ❛♥❞ ❛ss♦❝✐❛t❡❞

❢✉♥❝t✐♦♥s ♦❢ ♣r♦❜❧❡♠ ✭✾✮ ❢♦r♠s ❛ ❘✐❡s③ ❜❛s✐s ✐♥ H ✐❢ P(0) = a2P(π), Q(π) = a2Q(0). (10) ❘❡♠❛r❦ ✶✳ ■❢ ❝♦♥❞✐t✐♦♥ ✭✶✵✮ ❤♦❧❞s✱ t❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ ♣r♦❜❧❡♠ ✭✾✮ ✐s ❛s②♠♣t♦t✐❝❛❧❧② s✐♠♣❧❡✳ ❉❡♥♦t❡ ❜② Ψ t❤❡ s❡t ♦❢ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s (P(x), Q(x)) ∈ L1(0, π) ⊕ L1(0, π) s✉❝❤ t❤❛t t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ♦❢ ♣r♦❜❧❡♠ ✭✾✮ ❢♦r♠s ❛ ❘✐❡s③ ❜❛s✐s ✐♥ H✱ Ψ = (L1(0, π) ⊕ L1(0, π)) \ Ψ✳ ❚❤❡♦r❡♠ ✷✳ ❚❤❡ s❡ts Ψ ❛♥❞ Ψ ❛r❡ ❡✈❡r②✇❤❡r❡ ❞❡♥s❡ ✐♥ L1(0, π) ⊕ L1(0, π)✳ ❈❛s❡ ✭❜✮✳ ❚❤✐s ❝❛s❡ ❝♦♥t❛✐♥s ❛ ❧♦t ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❝♦♥❞✐t✐♦♥s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♠❛tr✐①❡s 1 b a 0 0 a 0 1

• ♦r

1 0 a 0 0 a b 1

• ,

✇❤❡r❡ a = 0, b = 0✳ ■t ✇❛s s❤♦✇♥ ✐♥ ❬✶✹❪ t❤❛t ❢♦r ♣r♦❜❧❡♠ ✭✹✮ ✇✐t❤ ♥♦♥♣❡r✐♦❞✐❝✲t②♣❡ ❝♦♥❞✐t✐♦♥s ❛❧❧ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❞♦✉❜❧❡ ❛♥❞ ❛❧❧ ❝♦rr❡s♣♦♥❞✐♥❣ r♦♦t s✉❜s♣❛❝❡s ❝♦♥s✐st ♦❢ ♦♥❡ ❡✐❣❡♥❢✉♥❝t✐♦♥ ❛♥❞ ♦♥❡ ❛ss♦❝✐❛t❡❞ ❢✉♥❝t✐♦♥✳ ❚❤❡♦r❡♠ ✸✳ ■❢ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛r❡ ♥♦t ♣❡r✐♦❞✐❝✲t②♣❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡♥ t❤❡ s②st❡♠ ♦❢ ❡✐❣❡♥✲ ❛♥❞ ❛ss♦❝✐❛t❡❞ ❢✉♥❝t✐♦♥s ♦❢ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❢♦r♠s ❛ ❘✐❡s③ ❜❛s✐s ✐♥ H ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s♣❡❝tr✉♠ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♠✉❧t✐♣❧❡✳ ❚❤❡♦r❡♠ ✹✳ ■❢ P(x), Q(x) ∈ W 1

2 (0, π) ❛♥❞

A13Q(π) + A24P(0) A23 = A13Q(0) + A24P(π) A14 , t❤❡♥ t❤❡ s♣❡❝tr✉♠ ✐s ❛s②♠♣t♦t✐❝❛❧❧② s✐♠♣❧❡✳ ❈♦r♦❧❧❛r② ✶✳ ❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✹ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ✐s ♥♦t ❛ ❜❛s✐s ✐♥ L2(0, π)✳

✺

❉❡♥♦t❡ ❜② Υ t❤❡ s❡t ♦❢ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s (P(x), Q(x)) ∈ L1(0, π) ⊕ L1(0, π) s✉❝❤ t❤❛t t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ♦❢ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❢♦r♠s ❛ ❘✐❡s③ ❜❛s✐s ✐♥ H✱ Υ = (L1(0, π) ⊕ L1(0, π)) \ Υ✱ ❚❤❡♦r❡♠ ✺✳ ❚❤❡ s❡t Υ ✐s ❡✈❡r②✇❤❡r❡ ❞❡♥s❡ ✐♥ L1(0, π) ⊕ L1(0, π)✳ ❘❡♠❛r❦ ✷✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♣♦t❡♥t✐❛❧s ♣r♦✈✐❞✐♥❣ t❤❡ ❛s②♠♣t♦t✐❝ ♠✉❧t✐♣❧✐❝✐t② ♦❢ t❤❡ s♣❡❝tr✉♠ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s✱ t❤❡ st✉❞② ♦❢ ✇❤✐❝❤ ✐s ❝❛rr✐❡❞ ♦✉t ❜② ♦t❤❡r ♠❡t❤♦❞s✳ ❘❡♠❛r❦ ✸✳ ■♥ t❤❡ ❝❛s❡ ♦❢ r❡❣✉❧❛r ❜✉t ♥♦t str♦♥❣❧② r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♣❛rt✐❝✉❧❛r ❢♦r♠ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡ ♣♦t❡♥t✐❛❧ V (x) t❤❡ s②st❡♠ ♦❢ r♦♦t ❢✉♥❝t✐♦♥s ♠❛② ❤❛✈❡ ♦r ♠❛② ♥♦t ❤❛✈❡ t❤❡ ❜❛s✐s ♣r♦♣❡rt②✱ ❛♥❞ ❡✈❡♥ ❢♦r ✜①❡❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ t❤✐s ♣r♦♣❡rt② ♠❛② ❛♣♣❡❛r ♦r ❞✐s❛♣♣❡❛r ✉♥❞❡r ❛r❜✐tr❛r② s♠❛❧❧ ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ ❝♦❡✣❝✐❡♥t V (x) ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡tr✐❝✳ ✸✳ ◆♦♥✲r❡❣✉❧❛r ❝❛s❡ ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭✷✮ ❛r❡ ❝❛❧❧❡❞ ✐rr❡❣✉❧❛r ✐❢ A14A23 = 0, (A12 + A34)(|A23| + |A14|) = 0 (11) ❛♥❞ ♦♥❡s ❛r❡ ❝❛❧❧❡❞ ❞❡❣❡♥❡r❛t❡ ✐❢ A14A23 = 0, (A12 + A34)(|A23| + |A14|) = 0. (12) ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✐♥ t❤❡ ❝❛s❡ ♦❢ ❞❡❣❡♥❡r❛t❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥ ∆0(λ) = 0 ❡✐t❤❡r ❤❛s ♥♦ r♦♦ts ♦r ∆0(λ) ≡ 0✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭✷✮ ❛r❡ ♥♦t r❡❣✉❧❛r t❤❡ ❝♦♠♣❧❡t❡♥❡ss ♣r♦♣❡rt② ❡ss❡♥t✐❛❧❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♣♦t❡♥t✐❛❧ V (x)✳ ■♥ t❤✐s ❝❛s❡ t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ♦❢ ♥♦♥♣❡rt✉r❜❡❞ ♦♣❡r❛t♦r ✭✹✮ (P(x) = Q(x) = 0) ✐s ♥♦t ❝♦♠♣❧❡t❡ ✐♥ H ❬✷✷❪✱ ❤♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ ❝♦♠♣❧❡t❡ ✐❢ V (x) ≡ 0✳ ❘❡❝❡♥t❧②✱ ▲✉♥②♦✈ ❛♥❞ ▼❛❧❛♠✉❞ ♣r♦✈❡❞ ✐♥ ❬✶✼❪ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥✿ ❚❤❡♦r❡♠ ✻✳ ▲❡t t❤❡ ❢✉♥❝t✐♦♥s P(x), Q(x) ❜❡ ❝♦♥t✐♥✉♦✉s ❛t t❤❡ ❡♥❞♣♦✐♥ts 0 ❛♥❞ π✳ ❚❤❡♥ t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ♦❢ ♣r♦❜❧❡♠ ✭✷✮✱ ✭✸✮ ✐s ❝♦♠♣❧❡t❡ ❛♥❞ ♠✐♥✐♠❛❧ ✐♥ H ✇❤❡♥❡✈❡r |A32| + | − A13P(0) + A42Q(π)| = 0 ❛♥❞ |A14| + | − A13P(π) + A42Q(0)| = 0.

✻

■❢ t❤❡ ❢✉♥❝t✐♦♥s P(x), Q(x) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ♦❢ ❝♦♠♣❧❡t❡♥❡ss ❛♥❞ ♠✐♥✐♠❛❧✐t② ✇❡r❡ ❡st❛❜❧✐s❤❡❞ ✐♥ ❬✷✶❪✳ ▼❛❧❛♠✉❞ ❛♥❞ ❖r✐❞♦r♦❣❛ ❛❧s♦ ♦❜t❛✐♥❡❞ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ♦❢ ❝♦♠♣❧❡t❡♥❡ss✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ t❤❡② ❡st❛❜❧✐s❤❡❞ ❬✷✵❪ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✳ ❚❤❡♦r❡♠ ✼✳ ■❢ A14 = A32 = 0 ❜✉t A13A42 = 0 ❛♥❞ 0 / ∈ supp R1(x) ∪ R2(x)✱ ✇❤❡r❡ R1(x) = A13P(x) − A42Q(x − π), R2(x) = A13P(π − x) − A42Q(x)✱ t❤❡♥ t❤❡ r♦♦t ❢✉♥❝t✐♦♥ s②st❡♠ ✐s ♥♦t ❝♦♠♣❧❡t❡ ✐♥ H✳ ◆♦t✐❝❡✱ t❤❛t ▼❛❧❛♠✉❞ ❛♥❞ ❤✐s ❝♦❛✉t❤♦rs ❆❣✐❜❛❧♦✈❛✱ ▲✉♥②♦✈✱ ❖r✐❞♦r♦❣❛ ❤❛✈❡ r❡❝❡✐✈❡❞ ❛ ❧♦t ♦❢ r❡s✉❧ts ♦♥ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ✭✐♥❝♦♠♣❧❡t❡♥❡ss✮ ♦❢ r♦♦t ✈❡❝t♦rs ❢♦r ♠✉❝❤ ♠♦r❡ ❣❡♥❡r❛❧ ✜rst✲♦r❞❡r s②st❡♠s✳ ✺✳ ❈♦♥❝❧✉s✐♦♥✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r❡s❡♥t ❛ t❛❜❧❡ s✉♠♠❛r✐③✐♥❣ t❤❡ s♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s✱ ♦✉t❧✐♥❡❞ ✐♥ t❤❡ ■♥tr♦❞✉❝t✐♦♥ ❢♦r ♦♣❡r❛t♦r ✭✷✮✱ ✭✸✮✳ ❚❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ✐♥❞✐❝❛t❡s ❝❧❛ss✐✜❝❛t✐♦♥ ❢♦r t❤❡ ❝❛s❡ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ t②♣❡ ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭❙❘❂str❡♥❣t❤❡♥❡❞ r❡❣✉❧❛r✱ ❲❘❂✇❡❛❦❧② r❡❣✉❧❛r❂r❡❣✉❧❛r ❜✉t ♥♦t str❡♥❣t❤❡♥❡❞ r❡❣✉❧❛r✱ ■❘❂✐rr❡❣✉❧❛r✱ ❉❊●❂❞❡❣❡♥❡r❛t❡✮✳ ❨❊❙✴◆❖ ♠❡❛♥s t❤❛t t❤❡ ✐♥❞✐❝❛t❡❞ ♣r♦♣❡rt② ♠❛② ❛♣♣❡❛r ♦r ❞✐s❛♣♣❡❛r ✉♥❞❡r ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥t q(x)❀ ❄✴◆❖ ♠❡❛♥s t❤❛t ✐t ❤❛s ❜❡❡♥ ♣r♦✈❡❞ t❤❛t ❢♦r ❛ s✉❜s❡t ♦❢ ♣♦t❡♥t✐❛❧s q(x) ∈ L1(0, π) t❤❡ ♣r♦♣❡rt② ❞♦❡s ♥♦t t❛❦❡ ♣❧❛❝❡✱ ❛♥❞ ❛♥ ❡①❛♠♣❧❡ ✇❤❡♥ t❤❡ ♣r♦♣❡rt② ❤♦❧❞s ✐s ✉♥❦♥♦✇♥✱ t❤✉s✱ t❤❡ ❞❡✜♥✐t✐✈❡ s♦❧✉t✐♦♥ ❤❛s ♥♦t ❜❡❡♥ r❡❝❡✐✈❡❞✳ ❈❛s❡ ❈❧❛ss ❈♦♥❞✐t✐♦♥s ♦♥ t❤❡ Aij ❈♦♠♣❧❡t❡♥❡ss ❇❛s✐s ♣r♦♣❡rt② ✶✳ ❙❘ A14A23 = 0, (A12 + A34)2 + 4A14A23 = 0 ❨❊❙ ❨❊❙ ✷✳❛✳ ❲❘ A14A23 = 0, (A12 + A34)2 + 4A14A23 = 0✱ A13 = A24 = 0 ❨❊❙ ❨❊❙✴◆❖ ✷✳❜✳ ❲❘ A14A23 = 0, (A12 + A34)2 + 4A14A23 = 0✱ |A13|+|A24| > ❨❊❙ ❨❊❙✴◆❖ ✸✳ ■❘ A14A23 = 0, (A12 + A34)(|A23| + |A14|) = 0 ❨❊❙✴◆❖ ❄✴◆❖ ✹✳ ❉❊● A14A23 = 0, (A12 + A34)(|A23| + |A14|) = 0 ❨❊❙✴◆❖ ❄✴◆❖

✼

❘❊❋❊❘❊◆❈❊❙ ❬✶❪ ■✳ ❆rs❧❛♥✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ s♠♦♦t❤♥❡ss ♦❢ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❉✐r❛❝ ♦♣❡r❛t♦r s✉❜❥❡❝t t♦ ❣❡♥❡r❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛♥❞ ✐ts ❜❛s✐s ♣r♦♣❡rt②✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✹✹✼ ✭✷✵✶✼✮ ✽✹✲✶✵✽✳ ❬✷❪ ●✳❉✳ ❇✐r❦❤♦✛✱ ❖♥ t❤❡ ❛s②♠♣t♦t✐❝ ❝❤❛r❛❝t❡r ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❝❡rt❛✐♥ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❝♦♥t❛✐♥✐♥❣ ❛ ♣❛r❛♠❡t❡r✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✾ ✭✶✾✵✽✮ ✷✶✾✲✷✸✶✳ ❬✸❪ ●✳❉✳ ❇✐r❦❤♦✛✱ ❇♦✉♥❞❛r② ✈❛❧✉❡ ❛♥❞ ❡①♣❛♥s✐♦♥s ♣r♦❜❧❡♠s ♦❢ ♦r❞✐♥❛r② ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✾ ✭✶✾✵✽✮ ✸✼✸✲✸✾✺✳ ❬✹❪ ●✳❉✳ ❇✐r❦❤♦✛✱ ❘✳❊✳ ▲❛♥❣❡r✱ ❚❤❡ ❜♦✉♥❞❛r② ♣r♦❜❧❡♠s ❛♥❞ ❞❡✈❡❧♦♣♠❡♥ts ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s②st❡♠ ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ✜rst ♦r❞❡r✱ Pr♦❝✳ ❆♠❡r✳ ❆❝❛❞✳ ❆rts ❙❝✐✳ ✺✽ ✭✶✾✷✸✮ ✹✾✲✶✷✽✳ ❬✺❪ ▼✳❙❤✳ ❇✉r❧✉ts❦❛②❛✱ ❱✳❱✳ ❑♦r♥❡✈✱ ❆✳P✳ ❑❤r♦♠♦✈✱ ❉✐r❛❝ s②st❡♠ ✇✐t❤ ♥♦♥❞✐✛❡r❡♥t✐❛❜❧❡ ♣♦t❡♥t✐❛❧ ❛♥❞ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ❩❤✳ ❱②❝❤✐s❧✳ ▼❛t✳ ▼❛t✳ ❋✐③✳ ✭✷✵✶✷✮ ✺✷ ✶✻✷✶✲✶✻✸✷ ✭✐♥ ❘✉ss✐❛♥✮✳ ❬✻❪ ❊✳❆✳ ❈♦❞❞✐♥❣t♦♥ ❛♥❞ ◆✳ ▲❡✈✐♥s♦♥✱ ❚❤❡♦r② ♦❢ ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭▼❝●r❛✇✲❍✐❧❧✱ ◆❡✇✲❨♦r❦✱ ✶✾✺✺✮✳ ❬✼❪ ◆✳ ❉❛♥❢♦r❞ ❛♥❞ ❏✳❚✳ ❙❝❤✇❛rt③✱ ▲✐♥❡❛r ❖♣❡r❛t♦rs✱ P❛rt ■■■✱ ❙♣❡❝tr❛❧ ❖♣❡r❛t♦rs ✭❲✐❧❡②✱ ◆❡✇✲❨♦r❦✱ ✶✾✼✶✮✳ ❬✽❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ■♥st❛❜✐❧✐t② ③♦♥❡s ♦❢ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♣❡r✐♦❞✐❝ ❙❝❤r♦❞✐♥❣❡r ❛♥❞ ❉✐r❛❝ ♦♣❡r❛t♦rs✱ ❘✉ss✐❛♥ ▼❛t❤✳ ❙✉r✈❡②s ✻✶ ✭✷✵✵✻✮ ✻✻✸✲✼✻✻✳ ❬✾❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ❇❛r✐✲▼❛r❦✉s ♣r♦♣❡rt② ❢♦r ❘✐❡s③ ♣r♦❥❡❝t✐♦♥s ♦❢ ✶❉ ♣❡r✐♦❞✐❝ ❉✐r❛❝ ♦♣❡r❛t♦rs✱▼❛t❤✳ ◆❛❝❤r✳ ✷✽✸ ✭✷✵✶✵✮ ✹✹✸✲✹✻✷✳ ❬✶✵❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ✶❉ ❉✐r❛❝ ♦♣❡r❛t♦rs ✇✐t❤ s♣❡❝✐❛❧ ♣❡r✐♦❞✐❝ ♣♦t❡♥t✐❛❧s✱ ❇✉❧❧✳ P♦❧✳ ❆❝❛❞✳ ❙❝✐✳ ▼❛t❤✳ ✻✵ ✭✷✵✶✷✮ ✺✾✲✼✺✳ ❬✶✶❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ❊q✉✐❝♦♥✈❡r❣❡♥❝❡ ♦❢ s♣❡❝tr❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ✶❉ ❉✐r❛❝ ♦♣❡r❛t♦rs ✇✐t❤ r❡❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ❏✳ ❆♣♣r♦①✳ ❚❤❡♦r② ✶✻✹ ✭✷✵✶✷✮ ✽✼✾✲✾✷✼✳ ❬✶✷❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ❈r✐t❡r✐❛ ❢♦r ❡①✐st❡♥❝❡ ♦❢ ❘✐❡s③ ❜❛s❡s ❝♦♥s✐st✐♥❣ ♦❢ r♦♦t ❢✉♥❝t✐♦♥s ♦❢ ❍✐❧❧ ❛♥❞ ✶❉ ❉✐r❛❝ ♦♣❡r❛t♦rs✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✷✻✸ ✭✷✵✶✷✮ ✷✸✵✵✲ ✷✸✸✷✳ ❬✶✸❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ❘✐❡s③ ❜❛s❡s ❝♦♥s✐st✐♥❣ ♦❢ r♦♦t ❢✉♥❝t✐♦♥s ♦❢ ✶❉ ❉✐r❛❝ ♦♣❡r❛t♦rs✱ Pr♦❝✳ ❆▼❙ ✶✹✶ ✭✷✵✶✸✮ ✶✸✻✶✲✶✸✼✺✳ ❬✶✹❪ P✳ ❉❥❛❦♦✈✱ ❇✳ ▼✐t②❛❣✐♥✱ ❯♥❝♦♥❞✐t✐♦♥❛❧ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙♣❡❝tr❛❧ ❉❡❝♦♠♣♦s✐t✐♦♥s

✽

♦❢ ✶❉ ❉✐r❛❝ ♦♣❡r❛t♦rs ✇✐t❤ ❘❡❣✉❧❛r ❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s✱ ■♥❞✐❛♥❛ ❯♥✐✈✳ ▼❛t❤✳ ❏✳ ✻✶ ✭✷✵✶✷✮ ✸✺✾✲✸✾✽✳ ❬✶✺❪ ❱✳❱✳ ❑♦r♥❡✈✱ ❆✳P✳ ❑❤r♦♠♦✈✱ ❉✐r❛❝ s②st❡♠ ✇✐t❤ ♥♦♥❞✐✛❡r❡♥t✐❛❜❧❡ ♣♦t❡♥t✐❛❧ ❛♥❞ ❛♥t✐♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ■③✈✳ ❙❛r❛t✳ ▼❛t✳ ▼❡❦①✳ ■♥❢♦r♠✳ ✭✷✵✶✸✮ ✶✸ ✷✽✲✸✺ ✭✐♥ ❘✉ss✐❛♥✮✳ ❬✶✻❪ ❏✳ ▲♦❝❦❡r✱ ❙♣❡❝tr❛❧ ❚❤❡♦r② ♦❢ ◆♦♥✲s❡❧❢✲❛❞❥♦✐♥t ❚✇♦✲♣♦✐♥t ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ ▼❛t❤✳ ❙✉r✈❡②s ▼♦♥♦❣r✳ ✭✶✾✷ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ❆♠st❡r❞❛♠✱ ✷✵✵✸✮✳ ❬✶✼❪ ❆✳ ▲✉♥②♦✈✱ ▼✳ ▼❛❧❛♠✉❞✱ ❖♥ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ❛♥❞ ❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ♦❢ r♦♦t s✉❜s♣❛❝❡s ♦❢ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ✜rst ♦r❞❡r s②st❡♠s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❏✳ ❙♣❡❝tr✳ ❚❤❡♦r② ✭✷✵✶✺✮ ✺ ✶✼✲✼✵✳ ❬✶✽❪ ❆✳ ▲✉♥②♦✈✱ ▼✳ ▼❛❧❛♠✉❞✱ ❖♥ t❤❡ ❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ♦❢ r♦♦t ✈❡❝t♦rs s②st❡♠ ❢♦r 2 × 2 ❉✐r❛❝ t②♣❡ s②st❡♠s✱ ❉♦❦❧✳ ▼❛t❤✳ ✭✷✵✶✹✮ ✾✵ ✺✺✻✲✺✻✶✳ ❬✶✾❪ ❆✳ ▲✉♥②♦✈✱ ▼✳ ▼❛❧❛♠✉❞✱ ❖♥ t❤❡ ❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ♦❢ r♦♦t ✈❡❝t♦rs s②st❡♠ ❢♦r 2 × 2 ❉✐r❛❝ t②♣❡ ♦♣❡r❛t♦rs✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✭✷✵✶✻✮ ✹✹✶ ✺✼✲✶✵✸✳ ❬✷✵❪ ▼✳ ▼❛❧❛♠✉❞✱ ▲✳ ❖r✐❞♦r♦❣❛✱ ❖♥ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ♦❢ r♦♦t s✉❜s♣❛❝❡s ♦❢ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ✜rst ♦r❞❡r s②st❡♠s ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✭✷✵✶✷✮ ✷✻✸ ✶✾✸✾✲✶✾✽✵✳ ❬✷✶❪ ❆✳❆✳ ▲✉♥②♦✈✱ ▼✳▼✳ ▼❛❧❛♠✉❞✳ ❖♥ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ♦❢ r♦♦t ✈❡❝t♦rs ❢♦r ✜rst ♦r❞❡r s②st❡♠s✿ ❆♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ❘❡❣❣❡ ♣r♦❜❧❡♠✳ ❉♦❦❧✳ ▼❛t❤✳ ❱✳ ✽✽ ◆♦✳ ✸ ✭✷✵✶✸✮ ✻✼✽✲✻✽✸✳ ❬✷✷❪ ❱✳❆✳ ▼❛r❝❤❡♥❦♦✱ ❙t✉r♠✲▲✐♦✉✈✐❧❧❡ ❖♣❡r❛t♦rs ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s ✭❇✐r❦❤☎ ❛✉s❡r✱ ❇❛s❡❧✱ ✶✾✽✻✱ ❑✐❡✈✱ ✶✾✼✼✮✳ ❬✷✸❪ ❇✳ ▼✐t②❛❣✐♥✱ ❙♣❡❝tr❛❧ ❊①♣❛♥s✐♦♥s ♦❢ ❖♥❡✲❞✐♠❡♥s✐♦♥❛❧ P❡r✐♦❞✐❝ ❉✐r❛❝ ❖♣❡r❛t♦rs✱ ❉②♥❛♠✐❝s ♦❢ P❉❊ ✶ ✭✷✵✵✹✮ ✶✷✺✲✶✾✶✳ ❬✷✹❪ ❇✳ ▼✐t②❛❣✐♥✱ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❡①♣❛♥s✐♦♥s ✐♥ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❉✐r❛❝ ♦♣❡r❛t♦r✱ ❉♦❦❧✳ ▼❛t❤✳ ✭✷✵✵✸✮ ✻✽ ✸✽✽✲✸✾✶✳ ❬✷✺❪ ▼✳❆✳ ◆❛✐♠❛r❦✱ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❖♣❡r❛t♦rs ✭❯♥❣❛r✱ ◆❡✇✲❨♦r❦✱ ✶✾✻✼✱ ◆❛✉❦❛✱ ▼♦s❝♦✇✱ ✶✾✻✾✮✳ ❬✷✻❪ ❆✳▼✳ ❙❛✈❝❤✉❦✱ ❆✳❆✳ ❙❤❦❛❧✐❦♦✈✱ ❚❤❡ ❉✐r❛❝ ❖♣❡r❛t♦r ✇✐t❤ ❈♦♠♣❧❡①✲❱❛❧✉❡❞ ❙✉♠♠❛❜❧❡ P♦t❡♥t✐❛❧✱ ▼❛t❤✳ ◆♦t❡s ✾✻ ✭✷✵✶✹✮ ✼✼✼✲✽✶✵✳ ❬✷✼❪ ❆✳▼✳ ❙❛✈❝❤✉❦✱ ■✳ ❱✳ ❙❛❞♦✈♥✐❝❤❛②❛✱ ❘✐❡s③ ❜❛s✐s ♣r♦♣❡rt② ✇✐t❤ ❜r❛❝❦❡ts ❢♦r t❤❡ ❉✐r❛❝ s②st❡♠ ✇✐t❤ s✉♠♠❛❜❧❡ ♣♦t❡♥t✐❛❧✱ ❏✳ ▼❛t❤✳ ❙❝✐✳ ✭✷✵✶✽✮ ✷✷✸ ✺✶✹✲✺✹✵✳ ❬✷✽❪ ▼✳❍✳ ❙t♦♥❡✱ ❆ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ s❡r✐❡s ♦❢ ❋♦✉r✐❡r ❛♥❞ ❇✐r❦❤♦✛✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✷✾ ✭✶✾✷✻✮ ✻✾✺✲✼✻✶✳ ❬✷✾❪ ▼✳❍✳ ❙t♦♥❡✱ ■rr❡❣✉❧❛r ❞✐✛❡r❡♥t✐❛❧ s②st❡♠s ♦❢ ♦r❞❡r t✇♦ ❛♥❞ t❤❡ r❡❧❛t❡❞

✾

❡①♣❛♥s✐♦♥s ♣r♦❜❧❡♠s✱ ❚r❛♥s✳ ❆♠❡r✳ ❙♦❝✳ ✷✾ ✭✶✾✷✼✮ ✷✸✲✺✸✳ ❬✸✵❪ ❏✳ ❚❛♠❛r❦✐♥✱ ❙✉r ◗✉❡❧q✉❡s P♦✐♥ts ❞❡ ❧❛ ❚❤❡♦r✐❡ ❞❡s ❊q✉❛t✐♦♥s ❉✐✛❡r❡♥t✐❡❧❧❡s ▲✐♥❡❛✐r❡s ❖r❞✐♥❛✐r❡s ❡t s✉r ❧❛ ●❡♥❡r❛❧✐s❛t✐♦♥ ❞❡ ❧❛ s❡r✐❡ ❞❡ ❋♦✉r✐❡r✱ ❘❡♥❞✳ ❈✐r❝✳ ▼❛t❡♠✳ P❛❧❡r♠♦ ✸✹ ✭✶✾✶✷✮ ✸✹✺✲✸✽✷✳ ❬✸✶❪ ❏✳ ❚❛♠❛r❦✐♥✱ ❙♦♠❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠ ♦❢ t❤❡ t❤❡♦r② ♦❢ ♦r❞✐♥❛r② ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ❡①♣❛♥s✐♦♥s ♦❢ ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ ✐♥ s❡r✐❡s ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ❢✉♥❝t✐♦♥s✱ ▼❛t❤✳ ❩✳ ✭✶✾✷✼✮✱ ✶✲✺✹✳ ❬✸✷❪ ■✳ ❚r♦♦s❤✐♥ ❛♥❞ ▼✳ ❨❛♠♦♠♦t♦✱ ❘✐❡s③ ❜❛s✐s ♦❢ r♦♦t ✈❡❝t♦rs ♦❢ ❛ ♥♦♥✲ s②♠♠❡tr✐❝ s②st❡♠ ♦❢ ✜rst✲♦r❞❡r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ ✐♥✈❡rs❡ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s✱ ❆♣♣❧✳ ❆♥❛❧✳ ✽✵ ✭✷✵✵✶✮ ✶✾✲✺✶✳ ❬✸✸❪ ■✳ ❚r♦♦s❤✐♥ ❛♥❞ ▼✳ ❨❛♠♦♠♦t♦✱ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❛♥ ✐♥✈❡rs❡ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ♥♦♥✲s②♠♠❡tr✐❝ s②st❡♠s ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ ❏✳ ■♥✈❡rs❡ ■❧❧✲P♦s❡❞ Pr♦❜❧✳ ✶✵ ✭✷✵✵✷✮ ✻✹✸✲✻✺✽✳

✶✵