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❈❛♥♦♥✐❝❛❧ s②st❡♠s ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ✐♥❞❡✜♥✐t❡ ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ A ∈ U κ ( J p ) ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧

❈❛♥♦♥✐❝❛❧ s②st❡♠s ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ✐♥❞❡✜♥✐t❡ ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ A ∈ U κ ( J p ) ■♥ ♠❡♠♦r✐❛♠ ❨✉✳▼✳❇❡r❡③❛♥s❦② ❈♦♥❢❡r❡♥❝❡✿ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s ❑②✐✈✱ ❯❦r❛✐♥❡✱ ✼✕✶✶✱ ❙❡♣t❡♠❜❡r✱ ✷✵✷✵✳ ❚❤❡ t♦♣✐❝s ♦❢ t❤❡ ❝♦♥❢❡r❡♥❝❡✿ ❖♣❡r❛t♦r t❤❡♦r② ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ▼♦♠❡♥t ❛♥❞ ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❜❧❡♠s ■♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧②s✐s ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❛♥❛❧②s✐s ❛♥❞ ♦♣❡r❛t♦r ❛❧❣❡❜r❛s ◆♦♥❧✐♥❡❛r ❛♥❛❧②s✐s ❛♥❞ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ▼❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ♦❢ ❝♦♠♣❧❡① s②st❡♠s ❛♥❞ ❢r❛❝t❛❧ ❝❛❧❝✉❧✉s ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧

❈❛♥♦♥✐❝❛❧ s②st❡♠s ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ✐♥❞❡✜♥✐t❡ ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ A ∈ U κ ( J p ) ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧ ❱✐❡♥♥❛✱ ❉❡❝❡♠❜❡r ✷✷✱ ✷✵✶✾ ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧

❈❛♥♦♥✐❝❛❧ s②st❡♠s ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ✐♥❞❡✜♥✐t❡ ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ A ∈ U κ ( J p ) ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳✱ ✷✼✼ ✭✷✵✶✾✮✱ ♥♦✳ ✶✱ ✸✶✲✶✶✵✳ ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧

❈❧❛ss U E ( J ) ❈❛♥♦♥✐❝❛❧ s②st❡♠s ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ✐♥❞❡✜♥✐t❡ ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❘❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡❧ ❍✐❧❜❡rt s♣❛❝❡s ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ A ∈ U κ ( J p ) ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❈❛♥♦♥✐❝❛❧ s②st❡♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ ❍❛♠✐❧t♦♥✐❛♥ H ( x ) d dx u ( x , λ ) = i λ u ( x , λ ) H ( x ) J , ❛✳❡✳ [ 0 , d ] ✭✶✮ d > 0 , ✇❤❡r❡ J ✐s ❛ m × m ✲s✐❣♥❛t✉r❡ ♠❛tr✐①✱ ✐✳❡✳ J = J ∗ = J − 1 ✱ ❛♥❞ H ∈ L 1 ([ 0 , d ]) , H ( x ) ≥ 0 , tr❛❝❡ H ( x ) = 1 ❛✳❡✳ [ 0 , d ] ✭✷✮ ❚❤❡ ♠❛tr✐① s♦❧✉t✐♦♥ U ( x , λ ) ♦❢ ✭✷✮ s✉❝❤ t❤❛t U ( 0 , λ ) = I ✐s ❝❛❧❧❡❞ t❤❡ ♠❛tr✐③❛♥t ♦❢ t❤❡ s②st❡♠ ✭✷✮✳ ❚❤❡ ♠❛tr✐① ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✭♠✈❢✮ U ( x , λ ) ✐s ❡♥t✐r❡ ✐♥ λ ❛♥❞ s❛t✐s✜❡s d dx ( U ( x , λ ) JU ( x , ω ) ∗ ) = i ( λ − ω ) U ( x , λ ) H ( x ) U ( x , ω ) ∗ . ✭✸✮ ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧

❈❧❛ss U E ( J ) ❈❛♥♦♥✐❝❛❧ s②st❡♠s ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ✐♥❞❡✜♥✐t❡ ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s ❘❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡❧ ❍✐❧❜❡rt s♣❛❝❡s ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ A ∈ U κ ( J p ) ■♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r ❞❡ ❇r❛♥❣❡s ♠❛tr✐❝❡s A ( λ ) := U ( d , λ ) ✐s ❝❛❧❧❡❞ t❤❡ ♠♦♥♦❞r♦♠② ♠❛tr✐① ♦❢ t❤❡ s②st❡♠ ✭✷✮✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ✭✸✮ t❤❛t t❤❡ ❦❡r♥❡❧ ω ( λ ) = J − A ( λ ) JA ( ω ) ∗ K A ( ρ ω ( λ ) := − 2 π i ( λ − ω )) , ρ ω ( λ ) ✐s ♥♦♥♥❡❣❛t✐✈❡ ♦♥ C + × C + ✱ ✐✳❡✳ ❢♦r ❡✈❡r② ω j ∈ C ✱ u j ∈ C m n � u ∗ j K A ω j ( ω i ) u i ≥ 0 . i , j = 1 ❉❡✜♥✐t✐♦♥ ✶✱ ▼✳❙✳ ▲✐✈✞ s✐❝✬✺✹ ▼✈❢ A ( λ ) ✐s s❛✐❞ t♦ ❜❡ ✐♥ t❤❡ ❝❧❛ss U ( J ) ✱ ✐❢✿ ✭✐✮ K A ω ( λ ) ✐s ♥♦♥♥❡❣❛t✐✈❡ ✐♥ C + × C + ❀ ✭✐✐✮ J − A ( µ ) JA ( µ ) ∗ = 0 ♦♥ t❤❡ r❡❛❧ ❧✐♥❡ R ✳ ❘✐❣❣❡❞ ❞❡ ❇r❛♥❣❡s✲P♦♥tr②❛❣✐♥ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❡♠❜❡❞❞✐♥❣ ❱♦❧♦❞②♠②r ❉❡r❦❛❝❤ ❱❛s②❧ ❙t✉s ❉♦♥❡ts❦ ❯♥✐✈❡rs✐t②✱ ❱✐♥♥✐ts②❛✱ ❯❦r❛✐♥❡ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❍❛rr② ❉②♠ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ❘❡❤♦✈♦t✱ ■sr❛❡❧