❖♥ 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 n t❤ ♦r❞❡r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ y ( n ) + p 1 ( x ) y ( n − 1) + . . . + p n ( 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 ( y 1 , . . . , y n ) (1) , ✇❤❡r❡ B ✐s ❛ ♥♦♥s✐♥❣✉❧❛r ❞✐❛❣♦♥❛❧ n × n ♠❛tr✐①✱ B = diag ( b − 1 1 I n 1 , . . . , b − 1 r I n r ) ∈ C n × n , n = n 1 + . . . n r , ✇✐t❤ ❝♦♠♣❧❡① ❡♥tr✐❡s b j � = b k ✱ ❛♥❞ 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❡♠ B y ′ + V y = λ y , (2) ✇❤❡r❡ y = col ( y 1 ( x ) , y 2 ( x )) ✱ � − i 0 � � � 0 P ( x ) B = , V = , 0 i Q ( x ) 0 ✶
t❤❡ ❢✉♥❝t✐♦♥s P ( x ) , Q ( x ) ∈ L 1 (0 , π ) ✱ ✇✐t❤ t✇♦✲♣♦✐♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s U ( y ) = C y (0) + D y ( π ) = 0 , (3) ✇❤❡r❡ � a 11 a 12 � � a 13 a 14 � C = , D = , a 21 a 22 a 23 a 24 t❤❡ ❝♦❡✣❝✐❡♥ts a ij ❛r❡ ❛r❜✐tr❛r② ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ❛♥❞ r♦✇s ♦❢ t❤❡ ♠❛tr✐① � a 11 a 12 a 13 a 14 � A = a 21 a 22 a 23 a 24 ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦♣❡r❛t♦r L y = B y ′ + V y ❛s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦♥ t❤❡ s♣❛❝❡ H = L 2 (0 , π ) ⊕ L 2 (0 , π ) ✱ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ D ( L ) = { y ∈ W 1 1 [0 , π ] : L y ∈ H ✱ U j ( 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 ∈ L 2 (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 ∈ L 2 (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 ) ∈ L 1 (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 ) ∈ L 2 (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 11 ( x, λ ) e 12 ( x, λ ) � E ( x, λ ) = e 21 ( x, λ ) e 22 ( x, λ ) ✷
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