❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❈♦♦r❞❡♥❛çã♦ ❞♦s ❈✉rs♦s ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛
▼♦❞❡❧♦s ●ê♠❡♦s ❡♠ ❚❡♦r✐❛s ❞❡
❈❛♠♣♦s ❊s❝❛❧❛r❡s
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦
❏♦s❡❝❧é❝✐♦ ❉✉tr❛ ❉❛♥t❛s
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❈♦♦r❞❡♥❛çã♦ ❞♦s ❈✉rs♦s ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛
▼♦❞❡❧♦s ●ê♠❡♦s ❡♠ ❚❡♦r✐❛s ❞❡
❈❛♠♣♦s ❊s❝❛❧❛r❡s
❚❡s❡ r❡❛❧✐③❛❞❛ s♦❜ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢✳ ❉r✳ ❉✐♦♥✐s✐♦ ❇❛③❡✐❛ ❋✐❧❤♦✱ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛✱ ❡♠ ❝♦♠♣❧❡♠❡♥t❛çã♦ ❛♦s r❡✲ q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳
▼♦❞❡❧♦s ●ê♠❡♦s ❡♠ ❚❡♦r✐❛s ❞❡
❈❛♠♣♦s ❊s❝❛❧❛r❡s
❏♦s❡❝❧é❝✐♦ ❉✉tr❛ ❉❛♥t❛s
❆♣r♦✈❛❞❛ ❡♠
❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛
Pr♦❢✳ ❉r✳ ❉✐♦♥✐s✐♦ ❇❛③❡✐❛ ❋✐❧❤♦ ❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❈❧❛✉❞✐♦ ❇❡♥❡❞✐t♦ ❙✐❧✈❛ ❋✉rt❛❞♦ ❊①❛♠✐♥❛❞♦r ■♥t❡r♥♦
Pr♦❢✳ ❉r✳ ❋❡r♥❛♥❞♦ ❏♦r❣❡ ❙❛♠♣❛✐♦ ▼♦r❛❡s ❊①❛♠✐♥❛❞♦r ■♥t❡r♥♦
Pr♦❢✳ ❉r✳ ❊❞✉❛r❞♦ ▼❛r❝♦s ❘♦❞r✐❣✉❡s ❞♦s P❛ss♦s ❊①❛♠✐♥❛❞♦r ❊①t❡r♥♦ ✭❯❋❈●✮
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧❛ ✈✐❞❛ ❡ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ♥❛s❝❡r ♥✉♠❛ ❢❛♠í❧✐❛ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥♦✉ ❝r❡s❝❡r ❝♦♠ ❞✐❣♥✐❞❛❞❡✳ ❈♦♠ s❛t✐s❢❛çã♦✱ ♠❛♥✐❢❡st♦ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❛♦s ♠❡✉s
q✉❡r✐❞♦s ♣❛✐s✱ ❊❞✇✐r❣❡♥s ❉✉tr❛ ❡ ❏♦ã♦ P❛✉❧✐♥♦ ✭❏♦ã♦ ❞❛ ❇❛rr❛❝❛✮✱ ♣❡❧❛ sá❜✐❛ ❡❞✉❝❛çã♦ ❡ ♣❡❧♦ ③❡❧♦ ♣❛r❛ ❝♦♠✐❣♦ ❞✉r❛♥t❡ t♦❞❛ ❛ ♠✐♥❤❛ ✈✐❞❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ✈✐❞❛ ❡s❝♦❧❛r❀ ❛ ❡❧❡s ❛ ♠✐♥❤❛ ❛❞♠✐r❛çã♦✱ ❛ ♠✐♥❤❛ ❣r❛t✐❞ã♦✳ ❙♦✉ ❣r❛t♦ à ♠✐♥❤❛ ❡s♣♦s❛ ❆❝á❝✐❛ ❉✉tr❛✱ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛✱
♣❛❝✐ê♥❝✐❛ ❡ ♣♦r t♦❞♦ ♦ ❝❛r✐♥❤♦ ❞❡❞✐❝❛❞♦❀ à ♠✐♥❤❛ ✐r♠ã ❑❛❧✐♥❛ ❈❧é❝✐❛✱ à ♠✐♥❤❛ s♦❜r✐♥❤❛ ❑❛r❡♥ ❲❡♠✐❧❧②✱ ❛♦ ♠❡✉ ✐r♠ã♦ ▲❡♦♥❛r❞♦ ❈♦st❛ ✭❖ ●♦r❞♦✮✱ ❛♦s ♠❡✉s t✐♦s ❋r❛♥❝✐s❝♦ ❉✉tr❛ ❡ ❊❧③❛ ❉✉tr❛✱ ❛♦s ♠❡✉s ❛✈ós ❙❡❜❛st✐ã♦ ✭❇❛st♦ ❞❛ ●♦♠❛✮ ❡ ❏♦❛♥❛✱ ❡ ❛♦s ❞❡♠❛✐s ❢❛♠✐❧✐❛r❡s ❝♦♠
♦s q✉❛✐s ❝♦♥✈✐✈♦ ❢♦r❛ ❞♦ ♠✉♥❞♦ ❞♦ tr❛❜❛❧❤♦ ❡ ❞♦s ❡st✉❞♦s✳
▼❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛♦s ♠❡✉s ✐r♠ã♦s ❏❛♠✐❧t♦♥ ❘♦❞r✐❣✉❡s ❡ ▼❛♥♦❡❧ ❉❛♥t❛s ✭❖ ❈❛❧✐①t♦✮✱
♣❡❧❛ ❛♠✐③❛❞❡ q✉❡ ❝♦♠❡ç❛♠♦s ❛ ❝♦♥str✉✐r ❞✉r❛♥t❡ ♥♦ss❛ ✈✐❞❛ ❞❡ ❡st✉❞❛♥t❡s✳ ❊①t❡♥❞♦ ♦s ❛❣r❛❞❡❝✐♠❡♥t♦s às s✉❛s r❡s♣❡❝t✐✈❛s ❡s♣♦s❛s✱ ❋❛❜r✐❝✐❛♥❡ ❡ ▼❛r✐❛ ❏♦sé✱ ❡ ❛♦s ❞❡♠❛✐s ❢❛♠✐❧✐❛r❡s ♣❡❧❛ ❝❛❧♦r♦s❛ ❛❝♦❧❤✐❞❛ q✉❛♥❞♦ ♣r❡❝✐s❡✐✳
➚s ❞❡♠❛✐s ♣❡ss♦❛s q✉❡ ♠❡ ❛❝♦❧❤❡r❛♠ ❡♠ ❏♦ã♦ P❡ss♦❛✿ ♦s ♠❡✉s t✐♦s ❇♦s❝♦ ❡ ●♦r❡t❡✱ ♠❡✉ ♣r✐♠♦ ❏❛✐❧s♦♥✱ ♠❡✉s ✈✐③✐♥❤♦s ▲✉✐③❛✱ ❏❡ssé✱ ▲✐♥❞❛❝✐ ❡ ❆♥t♦♥②✳ ❆ ❘❛q✉❡❧ ▲✐♠❛ ♣❡❧❛ ❛t❡♥çã♦
❞✐s♣❡♥s❛❞❛ ❡♠ ♠♦♠❡♥t♦s r❡❛❧♠❡♥t❡ ❞✐❢í❝❡✐s✳
❆♦ ♣r♦❢❡ss♦r ❋á❜✐♦ ▼❡❞❡✐r♦s✱ ❛t✉❛❧ ❝♦♦r❞❡♥❛❞♦r ❞♦ ❈✉rs♦ ❞❡ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ❋ís✐❝❛ ❞♦
❈❡♥tr♦ ❞❡ ❊❞✉❝❛çã♦ ❡ ❙❛ú❞❡ ✭❈❊❙✮ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❈❛♠♣✐♥❛ ●r❛♥❞❡ ✭❯❋❈●✮✱ ♣♦r ❝♦❧❛❜♦r❛r ❝♦♠ ❛ ♠✐♥❤❛ t❛r❡❢❛ ❞❡ ❝♦♥❝✐❧✐❛çã♦ ❡♥tr❡ ♦ tr❛❜❛❧❤♦ ❡ ♦ ❝✉rs♦ ❞❡ ❞♦✉t♦r❛❞♦✳
❆♦ ♣r♦❢❡ss♦r ❉✐♦♥✐s✐♦ ❇❛③❡✐❛✱ ♦s ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣❡❧❛ s❛❜❡❞♦r✐❛ ❝♦♠♣❛rt✐❧❤❛❞❛ ❡ ♣❡❧❛ ✈❛❧♦r♦s❛ ♦r✐❡♥t❛çã♦ ❞✉r❛♥t❡ ❛s ❡t❛♣❛s ❞❡ ♠❡str❛❞♦ ❡ ❞♦✉t♦r❛❞♦✳ ❊♠ s❡✉ ♥♦♠❡ ❛❣r❛❞❡ç♦ ❛♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ✭❉❋✮✳ ▼✐♥❤❛ ❣r❛t✐❞ã♦ ❛✐♥❞❛
✭❯❋P❇✮✱ ♣❡❧❛s ❝♦♥❞✐çõ❡s ♣r♦♣♦r❝✐♦♥❛❞❛s✱ ❡ à ❈❛♣❡s✱ ♣❡❧♦ t❡♠♣♦ ❡♠ q✉❡ ❢♦✐ ♣♦ssí✈❡❧ ♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢❛③❡♠♦s ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ ♥♦✈❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❝❤❛♠❛❞♦s ❦✲ ❞❡❢❡✐t♦s✱ q✉❡ sã♦ ❞❡❢❡✐t♦s t♦♣♦❧ó❣✐❝♦s ❝♦♠ t❡r♠♦ ❝✐♥ét✐❝♦ ♥ã♦✲❝❛♥ô♥✐❝♦✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱
❡st✉❞❛♠♦s ✉♠❛ ❝❧❛ss❡ ❞❡ ❦✲❞❡❢❡✐t♦s ❡♠ ♠♦❞❡❧♦s ❞❡ t❡♦r✐❛s ❞❡ ❝❛♠♣♦s ❡s❝❛❧❛r❡s ❞✐st✐♥t♦s ❞❛ t❡♦r✐❛ ♣❛❞rã♦ ♠❛s q✉❡ ❞❡s❝r❡✈❡♠✱ ❝❛s♦ ❛ ❝❛s♦✱ ♦ ♠❡s♠♦ ❞❡❢❡✐t♦ ❝♦♠ ❛ ♠❡s♠❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❛q✉❡❧❡ ❞❡s❝r✐t♦ ♣❡❧❛ t❡♦r✐❛ ❣♦✈❡r♥❛❞❛ ♣❡❧❛ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛ ♣❛❞rã♦✳ ❊♠ t❡♦r✐❛s
q✉❡ ❛♣r❡s❡♥t❛♠ t❛✐s r❡❧❛çõ❡s✱ ♠♦❞❡❧♦s ❞✐st✐♥t♦s s✉♣♦rt❛♠ ❛ ♠❡s♠❛ ❡str✉t✉r❛ t♦♣♦❧ó❣✐❝❛❀ ❞❛í ❝❤❛♠á✲❧♦s ❞❡ ♠♦❞❡❧♦s ❣ê♠❡♦s✳ ❈♦♥str✉í♠♦s✱ ❡♥tã♦✱ ✉♠ ♠♦❞❡❧♦ ❞❡ t❡♦r✐❛ ❣ê♠❡❛✱ q✉❡ ❞❡♥♦♠✐♥❛♠♦s ♠♦❞❡❧♦ ❆▲❚❲✱ ❡ ❡♥❝♦♥tr❛♠♦s ❛s r❡❧❛çõ❡s ❡①✐st❡♥t❡s ❡♥tr❡ ❡❧❡s✱ ✐♥❝❧✉✐♥❞♦ ❛s
r❡❧❛çõ❡s ❡♥tr❡ ♦s ♣♦t❡♥❝✐❛✐s ❞❡ ❛♠❜♦s✱ q✉❡✱ ❡♠❜♦r❛ ❞✐st✐♥t♦s✱ ❛♣r❡s❡♥t❛♠ ♠í♥✐♠♦s ❝♦♥❡❝t❛❞♦s ♣❡❧♦ ♠❡s♠♦ ❝❛♠♣♦ s♦❧✉çã♦✱ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❡stát✐❝❛s ❡ ❡stá✈❡✐s✳ ❖s r❡s✉❧t❛❞♦s
sã♦ ✐❧✉str❛❞♦s ❝♦♠ ✈ár✐♦s ❡①❡♠♣❧♦s✳ ❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❞✐st✐♥❣✉✐r ❛s t❡♦r✐❛s✱ ❛♥❛❧✐s❛♠♦s ❛ s✐t✉❛çã♦ ❡♠ q✉❡ ❛ ❝♦♠♣♦♥❡♥t❡T11 ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ é ♥ã♦✲♥✉❧❛✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡
❛ q✉❡❜r❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ♣r❡ssã♦ ♥✉❧❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s
❡stát✐❝❛s✳ ❈♦♠ ♦ ♠❡s♠♦ ♦❜❥❡t✐✈♦ ❞❡ ❞✐st✐♥çã♦✱ ✜③❡♠♦s ✉♠ ❡st✉❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❧✐♥❡❛r ❞♦s ❞❡❢❡✐t♦s ❡ ♦❜t✐✈❡♠♦s q✉❡✱ ❡♠❜♦r❛ r❡♣r❡s❡♥t❡♠ ♦ ♠❡s♠♦ ❞❡❢❡✐t♦✱ ❝❛s♦ ❛ ❝❛s♦✱ ✉♠❛ t❡♦r✐❛ ♥ã♦ é ✉♠❛ s✐♠♣❧❡s r❡♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ♦✉tr❛✳ ❋✐③❡♠♦s ❛✐♥❞❛ ✉♠❛ ❡①t❡♥sã♦ ❞❛ ♥❛t✉r❡③❛ ❣ê♠❡❛
❡♥tr❡ ♠♦❞❡❧♦s ♠❛✐s ❣❡r❛✐s ❞❡ t❡♦r✐❛s ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r r❡❛❧ ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❛♦ ❝❡♥ár✐♦ ❞❡ ❜r❛♥❛✳ ■♥✈❡st✐❣❛♠♦s t❛♠❜é♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❣ê♠❡♦ ❡♥tr❡ ♦s ♠♦❞❡❧♦s ♣❛❞rã♦ ❡ t❛q✉✐ô♥✐❝♦ ❡♠ ❝♦s♠♦❧♦❣✐❛ ❋❘❲✱ ♦♥❞❡ ♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❡✈♦❧✉✐ ❝♦♠ ♦ t❡♠♣♦✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ ❞♦ ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ♥❡✇ ❢❡❛t✉r❡s ♦❢ s♦✲❝❛❧❧❡❞ ❦✲❞❡❢❡❝ts✱ ✇❤✐❝❤ ❛r❡ t♦♣♦❧♦❣✐❝❛❧ ❞❡❢❡❝ts ✇✐t❤ ♥♦♥✲❝❛♥♦♥✐❝❛❧ ❦✐♥❡t✐❝ t❡r♠✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ st✉❞② ❛ ❝❧❛ss ♦❢ ❦✲❞❡❢❡❝ts ✐♥ ♠♦❞❡❧s ♦❢ s❝❛❧❛r ✜❡❧❞ t❤❡♦r✐❡s ❞✐st✐♥❝t ❢r♦♠ st❛♥❞❛r❞ t❤❡♦r② ❜✉t ❞✐s❝r✐❜✐♥❣✱ ❝❛s❡ t♦ ❝❛s❡✱ t❤❡ ✈❡r②
s❛♠❡ ❞❡❢❡❝t str✉❝t✉r❡ ✇✐t❤ t❤❡ ✈❡r② s❛♠❡ ❡♥❡r❣② ❞❡♥s✐t② ❛s t❤❛t ❞❡s❝r✐❜❡❞ ❜② t❤❡ t❤❡♦r② ❣♦✈❡r♥❡❞ ❜② st❛♥❞❛r❞ ▲❛❣r❛♥❣❡ ❞❡♥s✐t②✳ ■♥ t❡♦r✐❡s ✇❤✐❝❤ ♣r❡s❡♥ts s✉❝❤ r❡❧❛t✐♦♥s❤✐♣s✱ ❞✐st✐♥❝t
♠♦❞❡❧s s✉♣♣♦rt t❤❡ s❛♠❡ t♦♣♦❧♦❣✐❝❛❧ str✉❝t✉r❡❀ ✇❤② ❝❛❧❧ t❤❡♠ ♦❢ t✇✐♥❧✐❦❡ ♠♦❞❡❧s✳ ❲❡ t❤❡♥ ❜✉✐❧❞ ❛ ♠♦❞❡❧ ♦❢ t✇✐♥ t❤❡♦r②✱ ✇❤✐❝❤ ✇❡ ❝❛❧❧ ❆▲❚❲ ♠♦❞❡❧✱ ❛♥❞ ✜♥❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ t❤❡♠✱ ✐♥❝❧✉❞✐♥❣ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ♣♦t❡♥t✐❛❧s ♦❢ ❜♦t❤✱ ✇❤✐❝❤✱ ❛❧t❤♦✉❣❤ ❞✐st✐♥❝t✱
t❤❡② ♣r❡s❡♥t ♠✐♥✐♠❛ t❤❛t ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② t❤❡ s❛♠❡ ✜❡❧❞ s♦❧✉t✐♦♥✱ ❢♦r t❤❡ ❝❛s❡ ♦❢ st❛t✐❝ ❛♥❞ st❛❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥s✳ ❚❤❡ r❡s✉❧ts ❛r❡ ✐❧✉str❛t❡❞ ✇✐t❤ s❡✈❡r❛❧ ❡①❛♠♣❧❡s✳ ■♥ ♦r❞❡r t♦ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ t❤❡♦r✐❡s✱ ✇❡ ❛♥❛❧②③❡ t❤❡ s✐t✉❛t✐♦♥ ✐♥ ✇❤✐❝❤ t❤❡ ❝♦♠♣♦♥❡♥t T11 ♦❢ t❤❡
❡♥❡r❣②✲♠♦♠❡♥t✉♠ t❡♥s♦r ✐s ♥♦♥③❡r♦✱ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❜r❡❛❦✐♥❣ t❤❡ ♣r❡ss✉r❡❧❡ss ❝♦♥❞✐t✐♦♥ r❡q✉✐r❡❞ t♦ ❡♥s✉r❡ st❛❜✐❧✐t② ♦❢ st❛t✐❝ s♦❧✉t✐♦♥s✳ ❲✐t❤ t❤❡ s❛♠❡ ♣✉r♣♦s❡ ♦❢ ❞✐st✐♥❝t✐♦♥✱ ✇❡ ❞✐❞ ❛ st✉❞② ♦❢ ❧✐♥❡❛r st❛❜✐❧✐t② ♦❢ ❞❡❢❡❝ts ❛♥❞ ✇❡ ❢♦✉♥❞ t❤❛t✱ ❛❧t❤♦✉❣❤ r❡♣r❡s❡♥t✐♥❣ t❤❡ s❛♠❡ ❞❡❢❡❝t
str✉❝t✉r❡✱ ❝❛s❡ t♦ ❝❛s❡✱ ❛ t❤❡♦r② ✐s ♥♦t ❛ s✐♠♣❧❡ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ t❤❡ ♦t❤❡r✳ ❲❡ ❛❧s♦ ♠❛❞❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ t✇✐♥ ♥❛t✉r❡ ❜❡t✇❡❡♥ ♠♦r❡ ❣❡♥❡r❛❧ ♠♦❞❡❧s ♦❢ r❡❛❧ s❝❛❧❛r ✜❡❧❞ t❤❡r✐❡s ❛♥❞ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❜r❛♥❡✇♦r❧❞ s❝❡♥❛r✐♦✳ ❲❡ ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞ t❤❡ ❜❡❤❛✈✐♦r t✇✐♥ ❜❡t✇❡❡♥ st❛♥❞❛r❞
❛♥❞ t❛❝❤②♦♥✐❝ ♠♦❞❡❧s ✐♥ ❋❘❲ ❝♦s♠♦❧♦❣②✱ ✇❤❡r❡ t❤❡ s❝❛❧❛r ✜❡❧❞ ❡✈♦❧✈❡s ♦✈❡r t✐♠❡✳
❈♦♥t❡ú❞♦
■♥tr♦❞✉çã♦ ✶
✶ ❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ ✸
✶✳✶ ❖ ❛r❣✉♠❡♥t♦ ❞❡ ❉❡rr✐❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷ ❖ ♠♦❞❡❧♦ ♣❛❞rã♦ ✶✷
✷✳✶ ❉❡❢❡✐t♦s ❞♦ t✐♣♦ ❦✐♥❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✷ ❉❡❢❡✐t♦s ❞♦ t✐♣♦ ❧✉♠♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸ ❖ ♠♦❞❡❧♦ ♠♦❞✐✜❝❛❞♦ ✷✹
✹ ❆ ♥❛t✉r❡③❛ ❣ê♠❡❛ ❞♦s ♠♦❞❡❧♦s ✸✵
✹✳✶ Pr♦♣r✐❡❞❛❞❡s ❡s♣❡❝í✜❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✹✳✷ ❋♦r♠❛❧✐s♠♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹✳✸ ❉✐st✐♥❣✉✐♥❞♦ ❛s t❡♦r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
❈♦♥t❡ú❞♦
✹✳✸✳✶ T11 ♥ã♦✲♥✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✹✳✸✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✺ ❖✉tr♦s ♠♦❞❡❧♦s ❣ê♠❡♦s ✹✹
✺✳✶ ❯♠ ♠♦❞❡❧♦ ♠❛✐s ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✺✳✷ ❖ ❝❡♥ár✐♦ ❞❡ ❜r❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✻ Pr❡s❡♥ç❛ ❞❡ ♠♦❞❡❧♦s ❣ê♠❡♦s ❡♠ ❝♦s♠♦❧♦❣✐❛ ✺✺
✻✳✶ ❉✐♥â♠✐❝❛ ♣❛❞rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✻✳✷ ❉✐♥â♠✐❝❛ t❛q✉✐ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✻✳✸ ❋♦r♠❛❧✐s♠♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✻✳✹ Pr❡s❡♥ç❛ ❞❡ ❝✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✹
❇✐❜❧✐♦❣r❛✜❛ ✻✼
■♥tr♦❞✉çã♦
❊str✉t✉r❛s t♦♣♦❧ó❣✐❝❛s ❡♠ t❡♦r✐❛s ❞❡ ❝❛♠♣♦s ❝❧áss✐❝♦s ❡stã♦ ♣r❡s❡♥t❡s ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ❢ís✐❝❛✳ ❊str✉t✉r❛s ❝♦♠♦ ❦✐♥❦s✱ ✈órt✐❝❡s ❡ ♠♦♥♦♣♦❧♦s sã♦ ✐♠♣♦rt❛♥t❡s ❡♠ ❢ís✐❝❛ ❞❡ ❛❧t❛s
❡♥❡r❣✐❛s✱ s✉r❣✐♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❞✐✈❡rs♦s ❝♦♥t❡①t♦s ♥❛ ❈♦s♠♦❧♦❣✐❛ ❬✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻❪✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠ t❡r ❡①✐st✐❞♦ ❡♠ ❞✐❢❡r❡♥t❡s ❢❛s❡s ❞❛ ❡✈♦❧✉çã♦ ❞♦ ✉♥✐✈❡rs♦ ❬✼✱ ✽✱ ✾✱ ✶✵❪✳ ◆♦ ✉♥✐✈❡r♦ ♣r✐♠♦r❞✐❛❧✱ t❛✐s ❞❡❢❡✐t♦s ♣♦❞❡♠ t❡r s✐❞♦ ❢♦r♠❛❞♦s à ♠❡❞✐❞❛ ❡♠ q✉❡ ♦ ✉♥✐✈❡rs♦
r❡s❢r✐♦✉ ❡ s✐♠❡tr✐❛s ❢♦r❛♠ q✉❡❜r❛❞❛s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✐♥✈❡st✐❣❛♠♦s ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❞❡❢❡✐t♦s t♦♣♦❧ó❣✐❝♦s ❡♠ t❡♦r✐❛s ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r ❝♦♠ t❡r♠♦s ❝✐♥ét✐❝♦s ♥ã♦✲❝❛♥ô♥✐❝♦s✱ s✐♠✐❧❛r❡s
àq✉❡❧❛s ❡♠♣r❡❣❛❞❛s ❡♠ ♠♦❞❡❧♦s ❞❡ ❦✲❡ssê♥❝✐❛ ❬✶✶✱ ✶✷✱ ✶✸❪✳ ❖s ❞❡❢❡✐t♦s t♦♣♦❧ó❣✐❝♦s ❡st✉❞❛❞♦s ❡♠ t❡♦r✐❛s ❣❡♥❡r❛❧✐③❛❞❛s ❞❡ss❛ ♥❛t✉r❡③❛ sã♦ ❞❡♥♦♠✐♥❛❞♦s ❦✲❞❡❢❡✐t♦s ❡ ❛❧❣✉♥s ❛s♣❡❝t♦s ❞❡st❡s ♦❜❥❡t♦s ❥á ❢♦r❛♠ ❡st✉❞❛❞♦s ❡♠ ❬✶✹✱ ✶✺✱ ✶✻✱ ✶✼✱ ✶✽✱ ✶✾✱ ✷✵✱ ✷✶❪✳ ■♥s♣✐r❛❞♦s ❡♠ ❬✷✷❪✱ ❡st✉❞❛♠♦s
♠♦❞❡❧♦s ❞❡ t❡♦r✐❛s ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r r❡❛❧ ❝♦♠ t❡r♠♦ ❝✐♥ét✐❝♦ ♥ã♦✲❝❛♥ô♥✐❝♦✱ ♠❛s q✉❡ s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❞❡ t❡r♠♦ ❝✐♥ét✐❝♦ ❝❛♥ô♥✐❝♦ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ q✉❡ ♥♦s ♣❡r♠✐t❡ ❝❧❛ss✐✜❝á✲❧♦s ❝♦♠♦ ♠♦❞❡❧♦s ❣ê♠❡♦s✱ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡✱ ❡♠❜♦r❛ s❡❥❛♠ ❞✐st✐♥t♦s✱ ♦s ❞♦✐s
♠♦❞❡❧♦s ❞❡ t❡♦r✐❛s ❞❡s❝r❡✈❡♠ ✉♠❛ ♠❡s♠❛ ❡str✉t✉r❛ t♦♣♦❧ó❣✐❝❛✱ ❝❛s♦ ❛ ❝❛s♦✱ t❡♥❞♦ ♠❡s♠❛ s♦❧✉çã♦✱ ♣❛r❛ ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❝❛♠♣♦ ❡stát✐❝♦✱ ❡ ♠❡s♠❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛✳
■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s ❛s ❡q✉❛çõ❡s ♥❡❝❡ssár✐❛s ❛♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❞✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r r❡❛❧ ❡♠ ✭1 + 1✮ ❞✐♠❡♥sõ❡s ❡s♣❛ç♦✲
t❡♠♣♦r❛✐s✱ ♣❛rt✐♥❞♦ ❞❛s ❝♦♥s✐❞❡r❛çõ❡s ❢❡✐t❛s ❛❝❡r❝❛ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ♠í♠✐♥❛ ❛çã♦✱ ♣❛ss❛♥❞♦
♣❡❧❛s ❝♦♥❞✐çõ❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ s♦❧✉çã♦✱ ❝♦♠♦ ❞✐s❝✉t✐❞♦ ♥♦ ❛r❣✉♠❡♥t♦ ❞❡ ❉❡rr✐❝❦✱ ❡
■♥tr♦❞✉çã♦
❝✉❧♠✐♥❛♥❞♦ ♥♦ ❡st✉❞♦ ❣❡♥❡r❛❧✐③❛❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❧✐♥❡❛r ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❞❡❢❡✐t♦ ❛ ♣❛rt✐r ❞❡ ✢✉t✉❛çõ❡s ❡♠ t♦r♥♦ ❞♦ ❝❛♠♣♦ s♦❧✉çã♦✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ é ❢❡✐t❛ ✉♠❛ ❞❡s❝r✐çã♦ ❞❛
❞✐♥â♠✐❝❛ ♣❛❞rã♦✱ ❝♦♠ t❡r♠♦ ❝✐♥ét✐❝♦ ❝❛♥ô♥✐❝♦✱ ❡ ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ ❡str✉t✉r❛s t♦♣♦❧ó❣✐❝❛s✱ ♦✉ ❞♦ t✐♣♦ ❦✐♥❦✱ ❡ ❡str✉t✉r❛s ♥ã♦✲t♦♣♦❧ó❣✐❝❛s✱ ♦✉ ❞♦ t✐♣♦ ❧✉♠♣✱ s❡❣✉✐❞❛ ❞❡ ✈ár✐♦s ❡①❡♠♣❧♦s✳ ❖ ♠❡s♠♦ é ❢❡✐t♦ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❞❡st❛ ✈❡③ ♣❛r❛ ✉♠ ♠♦❞❡❧♦ ♠♦❞✐✜❝❛❞♦ q✉❡ ✐♥t✐t✉❧❛♠♦s
♠♦❞❡❧♦ ❆▲❚❲✳ ◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❝♦♠❡ç❛♠♦s ❛ tr❛ç❛r r❡❧❛çõ❡s ❡♥tr❡ ♦s ❞♦✐s ♠♦❞❡❧♦s tr❛t❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✳ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s s❡♠❡❧❤❛♥t❡s ❧❡✈❛r❛♠✲♥♦s ❛ ❥✉st✐✜❝❛r ❛ ❞❡♥♦♠✐♥❛çã♦ ❞❡ ♠♦❞❡❧♦s ❣ê♠❡♦s✱ ♦✉ t❡♦r✐❛s ❣ê♠❡❛s✳ ❯♠ ❢♦r♠❛❧✐s♠♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠
♣❛r❛ r❡s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s é ❞❡s❡♥✈♦❧✈✐❞♦✱ ❝♦♠♦ ❡♠ ❬✷✸❪✱ ❡ ❛ ♥❛t✉r❡③❛ ❣ê♠❡❛ é ✐♥❝♦r♣♦r❛❞❛ às ❡q✉❛çõ❡s r❡❡s❝r✐t❛s s❡❣✉♥❞♦ ❡ss❡ ❢♦r♠❛❧✐s♠♦✳ ❆✐♥❞❛ ♥❡st❡ ❝❛♣ít✉❧♦✱ ❡st✉❞❛♠♦s ❢♦r♠❛s
❞❡ ❞✐st✐♥❣✉✐r ❛s ❞✉❛s t❡♦r✐❛s✱ ❝♦♥s✐❞❡r❛♥❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♦ ❝♦♠♣♦♥❡♥t❡ T11 ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲
♠♦♠❡♥t♦ ♥ã♦✲♥✉❧♦ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❛ ♣ró♣r✐❛ ❡st❛❜✐❧✐❞❛❞❡ ❧✐♥❡❛r ❞❛s s♦❧✉çõ❡s✳ ▼♦❞❡❧♦s ♠❛✐s ❣❡r❛✐s ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❛♦ ❝❡♥ár✐♦ ❞❡ ❜r❛♥❛ ♣♦❞❡♠ s❡r ✈✐st♦s ♥♦ ❝❛♣ít✉❧♦ q✉✐♥t♦✳
❖ ❝❛♣ít✉❧♦ q✉❡ ❛♥t❡❝❡❞❡ ❛s ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ♠♦❞❡❧♦s ❣ê♠❡♦s ♥♦ ❝❡♥ár✐♦ ❞❡ ❝♦s♠♦❧♦❣✐❛✳ ❘❡❧❛❝✐♦♥❛♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ❡✈♦❧✉✐♥❞♦ ❝♦♠ ♦ t❡♠♣♦ s♦❜ ❞✐♥â♠✐❝❛ ♣❛❞rã♦ ❡ s♦❜ ❞✐♥â♠✐❝❛ t❛q✉✐ô♥✐❝❛✱ ❡♠ ✉♠ s✐t❡♠❛ ❡s♣❛ç♦✲t❡♠♣♦ ❝✉❥❛ ♠étr✐❝❛ é
❛ ❞❡ ❋r✐❡❞♠❛♥♥✲❘♦❜❡rts♦♥✲❲❛❧❦❡r✳
❈❛♣ít✉❧♦ ✶
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
❊st❡ ❝❛♣ít✉❧♦ é ❞❡✈♦t❛❞♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ❡q✉❛çõ❡s ❣❡r❛✐s q✉❡ ❢♦r♠❛♠ ♦ ❢❡rr❛♠❡♥t❛❧ ♠❛t❡♠át✐❝♦ ♥❡❝❡ssár✐♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ s❡ ♣♦ss❛
❡✈♦❧✉✐r ♥❛ ❧❡✐t✉r❛ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♣❛✉s❛s ♣❛r❛ ❞❡♠♦♥str❛çõ❡s ♠❛t❡♠át✐❝❛s ♠❛✐s ❞❡♠♦r❛❞❛s✳ ■♥s♣✐r❛❞♦s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛❞♦t❛❞♦ ❡♠ ❬✶✾✱ ✷✹❪✱ ❞❡s❝r❡✈❡♠♦s ✉♠❛ ❞✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❡str✉t✉r❛s ❞❡ ❞❡❢❡✐t♦s ❡♠ ♠♦❞❡❧♦s ❣♦✈❡r♥❛❞♦s ♣♦r ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r r❡❛❧
❡♠ (1,1) ❞✐♠❡♥sõ❡s ❡s♣❛ç♦✲t❡♠♣♦r❛✐s✳ ❆té ❡♥tã♦✱ ♥ã♦ ❢❛③❡♠♦s ✐❧✉str❛çõ❡s ❝♦♠ ❡①❡♠♣❧♦s
❡s♣❡❝í✜❝♦s✳ ❆ ❞❡s❝r✐çã♦ t❡♠ ❝❛rát❡r ❣❡r❛❧ ❡✱ ❢❡✐t❛ ❞❡ss❛ ❢♦r♠❛✱ t❡♠ ❛ ✈❛♥t❛❣❡♠ ❞❡ ♣❡r♠✐t✐r ♦ ❡st✉❞♦ ❞❡ ✉♠❛ ❞✐✈❡rs✐❞❛❞❡ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❆❧❣✉♠❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦
♣r✐♥❝✐♣❛❧♠❡♥t❡ à ❢♦r♠❛ ❞♦ t❡r♠♦ ❝✐♥ét✐❝♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛✱ s❡rã♦ tr❛t❛❞❛s ❡♠ ❞❡t❛❧❤❡s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳
❈♦♥s✐❞❡r❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ ❝❛❞❛ ♣♦♥t♦ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❡stá ❛ss♦❝✐❛❞❛ ❛ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r r❡❛❧ φ(x, t)✿ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ♣♦s✐çã♦ ❡ t❡♠♣♦✱ q✉❡
❝♦♥st✐t✉✐ ✉♠ s✐st❡♠❛ ❝♦♠ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ✭P❛r❛ ✉♠❛ r❡✈✐sã♦ ♠❛✐s
❞❡t❛❧❤❛❞❛✱ ❝♦♥s✉❧t❛r ❬✷✺✱ ✷✻❪✮✳ ❆ ❞✐♥â♠✐❝❛ ❞❡ φ é ❞❡s❝r✐t❛ ♣❡❧❛ ❛çã♦
S =
Z
d2xL(φ, X), ✭✶✳✶✮
♦♥❞❡L(φ, X)é ✉♠❛ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛ ❣❡♥❡r❛❧✐③❛❞❛✳ ❊st❛ é s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✉♠❛ ❢✉♥çã♦
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
❞♦ ❝❛♠♣♦ ❡ ❞❡ s✉❛s ♣r✐♠❡✐r❛s ❞❡r✐✈❛❞❛s✱ ❡ ❛ ❣r❛♥❞❡③❛
X = 1 2∂µφ∂
µφ, ✭✶✳✷✮
♦♥❞❡ µ, ν = 0,1✱ r❡♣r❡s❡♥t❛ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❝✐♥ét✐❝❛ ❡ ❣r❛❞✐❡♥t❡ ❞❛ ❞✐♥â♠✐❝❛✳ ❊♠ ♣r✐♥❝í♣✐♦✱
❛ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛L♣♦❞❡r✐❛ t❛♠❜é♠ ❞❡♣❡♥❞❡r ❞❡ ❞❡r✐✈❛❞❛s ❞❡ φ❡♠ ♦r❞❡♥s ♠❛✐s ❛❧t❛s✳ ❆ ❝♦♥s❡q✉ê♥❝✐❛ ✐♥❞❡s❡❥❛❞❛ é q✉❡ ❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ s❡r✐❛♠ ♥ã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱
♠❛s ❞❡ ♦r❞❡♥s s✉♣❡r✐♦r❡s✳
❆ ❛ss✐♥❛t✉r❛ ❞❛ ♠étr✐❝❛ é (+,−)✱ ♦♥❞❡ x0 é ❛ ❞✐♠❡♥sã♦ t❡♠♣♦r❛❧ ❡ x1 é ❛ ❞✐♠❡♥sã♦
❡s♣❛❝✐❛❧✳ ❊st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❣r❛♥❞❡③❛ ❛❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s ❡ t❡♠♣♦r❛✐s t❛♠❜é♠ ❛❞✐♠❡♥s✐♦♥❛✐s✳
❙❡ ✜③❡r♠♦s ✉♠❛ ♣❡q✉❡♥❛ ✈❛r✐❛çã♦ ❛r❜✐trár✐❛ δφ ♥♦ ❝❛♠♣♦ φ✱ ❛ ✈❛r✐❛çã♦ ❝♦♥s❡q✉❡♥t❡ ♥❛
❛çã♦ é ❞❛ ❢♦r♠❛
δS =
Z
d2x δL(φ, X). ✭✶✳✸✮
❆♣❧✐❝❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ♠í♥✐♠❛ ❛çã♦✱ ♦♥❞❡ s❡ r❡q✉❡r q✉❡Ss❡❥❛ ❡st❛❝✐♦♥ár✐❛ s♦❜ ✉♠❛ ✈❛r✐❛çã♦
❛r❜✐trár✐❛ δφq✉❡ s❡ ❛♥✉❧❛ ♥♦s ❧✐♠✐t❡s ❞❡ ✐♥t❡❣r❛çã♦ ❬✷✼✱ ✷✽❪✱ ♦❜t❡♠♦s ❛s ❡q✉❛çõ❡s ❝❧áss✐❝❛s ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ♦ s✐st❡♠❛ ❞❡s❝r✐t♦ ♣♦r S✿
∂µ(LX∂µφ) = Lφ. ✭✶✳✹✮
❊st❛s sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✱ ♣♦❞❡♥❞♦ s❡r r❡❡s❝r✐t❛s ♠❛✐s ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
LφX∂µφ∂µφ+LXX∂µφ∂αφ∂µ∂αφ+LX✷φ=Lφ, ✭✶✳✺✮
♦♥❞❡ ❡st❛♠♦s ✉s❛♥❞♦ ❛ ♥♦t❛çã♦ LX = ∂L/∂X ❡ Lφ = ∂L/∂φ✳ ❈♦♥❤❡❝✐❞❛ ❛ ❞❡♥s✐❞❛❞❡
❧❛❣r❛♥❣✐❛♥❛ q✉❡ ❣♦✈❡r♥❛ ❞❡t❡r♠✐♥❛❞❛ t❡♦r✐❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ❝♦♠♦ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ♥❛s ✈❛r✐á✈❡✐s x ❡t✳
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
❆ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ❛çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s ♥❛ ♠étr✐❝❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❝♦♥s❡r✈❛❞❛ ♣❛r❛ ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❝❛♠♣♦ q✉❡ ♦❜❡❞❡❝❡♠ à
❡q✉❛çã♦ ✭✶✳✺✮✱ ❡ q✉❡ ✐❞❡♥t✐✜❝❛♠♦s ❝♦♠♦ s❡♥❞♦ ♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦✱ ❝✉❥❛ ❢♦r♠❛ é
Tµν =L
X∂µφ∂νφ−gµνL. ✭✶✳✻✮
❆s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ sã♦
T00 =ρ=LXφ˙2− L, ✭✶✳✼✮
T01=T10=LXφφ˙ ′, ✭✶✳✽✮
T11=p=LXφ′2+L. ✭✶✳✾✮
❖s t❡r♠♦s T00=ρ ❡T11=p✱ ♥❛s ❡q✉❛çõ❡s ✭✶✳✼✮ ❡ ✭✶✳✾✮✱ r❡♣r❡s❡♥t❛♠ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛
❡ ❛ ♣r❡ssã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s sí♠❜♦❧♦s ✭˙✮ ❡ ✭′ ✮ r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡r✐✈❛❞❛s
❝♦♠ r❡s♣❡✐t♦ ❛♦ t❡♠♣♦ ❡ à ❝♦♦r❞❡♥❛❞❛ ❡s♣❛❝✐❛❧✳
❉❡s❞❡ q✉❡ ❝♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❝❛♠♣♦ ❡stát✐❝♦✱ ♦✉ s❡❥❛✱ φ = φ(x)✱ ❞❛ ❊q✳
✭✶✳✷✮✱ t❡♠♦s q✉❡
X =−1 2φ
′2, ✭✶✳✶✵✮
♦♥❞❡ ✉s❛♠♦s ❛ ♥♦t❛çã♦ φ′ =dφ/dx✳ ❆ ❊q✳ ✭✶✳✺✮ é ❡♥tã♦ r❡❞✉③✐❞❛ ❛
(2LXXX+LX)φ′′ = 2LφXX− Lφ. ✭✶✳✶✶✮
❈♦♠ ✉♠ ♣♦✉❝♦ ❞❡ á❧❣❡❜r❛✱ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✿
(L −2LXX)′ = 0. ✭✶✳✶✷✮
■♥t❡❣r❛♥❞♦ ❡st❛ ❡q✉❛çã♦✱ ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ ❆ss✐♠✱
L −2LXX =C, ✭✶✳✶✸✮
♦♥❞❡ C é ❛ ❝♦♥st❛♥t❡ ❞❡ ✐♥t❡❣r❛çã♦ ❛ q✉❛❧ ♥♦s r❡❢❡r✐♠♦s✳
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
❈♦♠♦ ❡st❛♠♦s tr❛t❛♥❞♦ ♦ ❝❛s♦ ❡♠ q✉❡ ❛s s♦❧✉çõ❡s sã♦ ❡stát✐❝❛s✱ ✉♠ ♦❧❤❛r s♦❜r❡ ❛ ❊q✳ ✭✶✳✾✮ ♣❡r♠✐t❡✲♥♦s ✐❞❡♥t✐✜❝❛r ❛ ❝♦♥st❛♥t❡ C ❝♦♠♦ s❡♥❞♦ ❛ ❝♦♠♣♦♥❡♥t❡ T11 ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲
♠♦♠❡♥t♦✱ ♦✉ s❡❥❛✱ ❛ ♣r❡ssã♦✿
C=L −2LX
−12φ′2
=LXφ′2+L=T11.
❆ ❡♥❡r❣✐❛ t♦t❛❧ ❞❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❝❛♠♣♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ✐♥t❡❣r❛♥❞♦ ❛ ❝♦♠♣♦♥❡♥t❡ T00
s♦❜r❡ t♦❞♦ ♦ ❡s♣❛ç♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛✳ P❛r❛ ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❝❛♠♣♦ ❡stát✐❝♦✱ t❡♠♦s
T00=−L ❡ ❛ ❡♥❡r❣✐❛ t♦t❛❧ é
E =−
Z ∞ −∞
dxL(φ, X). ✭✶✳✶✹✮
❊st❛ ❡♥❡r❣✐❛ ✐❞❡♥t✐✜❝❛ ❛ ♠❛ss❛ ❞❡ r❡♣♦✉s♦ ❞❛ ❡str✉t✉r❛ ❞♦ ❞❡❢❡✐t♦✳
✶✳✶ ❖ ❛r❣✉♠❡♥t♦ ❞❡ ❉❡rr✐❝❦
❖ t❡♦r❡♠❛ ❞❡ ❉❡rr✐❝❦ é ✐♠♣♦rt❛♥t❡ ♣♦rq✉❡ tr❛t❛ ❞❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ s♦❧✉çã♦ ❡stát✐❝❛✳ ❙❡❣✉♥❞♦ ❡ss❡ t❡♦r❡♠❛✱ ♥ã♦ ♣♦❞❡ ❤❛✈❡r s♦❧✉çõ❡s ❡stát✐❝❛s✱ ❞❡ ❡♥❡r❣✐❛ ✜♥✐t❛✱
❡♠ t❡♦r✐❛s ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r ❡♠ ♠❛✐s ❞♦ q✉❡ ✉♠❛ ❞✐♠❡♥sã♦ ❡s♣❛❝✐❛❧✳ ❉❡rr✐❝❦ ♣❡r❝❡❜❡✉ q✉❡✱ ♣❛r❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❝❛♠♣♦ q✉❡ é ✉♠ ♣♦♥t♦ ❡st❛❝✐♦♥ár✐♦ ❞❛ ❡♥❡r❣✐❛✱ ❡st❛ é ❡st❛❝✐♦♥ár✐❛ ♣❛r❛ t♦❞❛s ❛s ✈❛r✐❛çõ❡s✱ ✐♥❝❧✉✐♥❞♦ ♠✉❞❛♥ç❛s ❞❡ ❡s❝❛❧❛ ❬✷✽✱ ✷✾✱ ✸✵✱ ✸✶✱ ✸✷✱ ✸✸❪✳
P❛r❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r ❡stát✐❝♦φ(x)✱ ❞❡✜♥✐♠♦s ✉♠❛ s♦❧✉çã♦ r❡❡s❝❛❧❛❞❛
❞❛ ❢♦r♠❛
φ(λ)(x) = φ(λx), ✭✶✳✶✺✮
♦♥❞❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ❡s❝❛❧❛x→λx✱ s❡♥❞♦λ✉♠ ♣❛râ♠❡tr♦ ✜♥✐t♦✳ ❚❡♠♦s q✉❡
X(λ) =−1 2
dφ(λ)
dx
2
=−1 2
d
dx(φ(λx))
2
=−λ
2
2
dφ dx(λx)
2
=λ2X(λx).
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
❆ ❡♥❡r❣✐❛ ❛♥t❡s ❞❛ ♠✉❞❛♥ç❛ ❞❡ ❡s❝❛❧❛ é ❛ s❡❣✉✐♥t❡✿
E(φ) =−
Z ∞ −∞
dDxL(φ, X) =E0+E2. ✭✶✳✶✻✮
❆q✉✐ t❡♠♦s ❞❡❝♦♠♣♦st♦ ❛ ❡♥❡r❣✐❛ ❡♠ s✉❛s ♣❛rt❡s ♣♦t❡♥❝✐❛❧ ❡ ❝✐♥ét✐❝❛✳ ❆ ❡①♣r❡ssã♦ ❞❛ ❡♥❡r❣✐❛
❡s❝r✐t❛ ❛ss✐♠✱ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ s♦♠❛ ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s✱ r❡♣r❡s❡♥t❛ ✉♠❛ t❡♦r✐❛ ❣♦✈❡r♥❛❞❛ ♣❡❧❛ ❞✐♥â♠✐❝❛ ♣❛❞rã♦ ♦✉ ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡❧❛✳ ❖s s✉❜s❝r✐t♦s ✐♥❞✐❝❛♠ ❛s ♣♦tê♥❝✐❛s ❡①♣❧í❝✐t❛s ❞❡ λ q✉❡ ❛♣❛r❡❝❡♠ q✉❛♥❞♦ ❛ ✐♥t❡❣r❛❧ é r❡❡s❝❛❧❛❞❛✳ ❈♦♠ ❛ ♠✉❞❛♥ç❛ ❞❡ ❡s❝❛❧❛✱ t❡♠♦s
E(λ) =E(φ(λ)) =−
Z ∞ −∞
dDxL φ(λ), X(λ)
=−
Z ∞ −∞
dDxL φ(λx), λ2X(λx)
. ✭✶✳✶✼✮
❋❛③❡♥❞♦
y=λx ⇒ dDx=λ−DdDy, ✭✶✳✶✽✮
♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
E(λ) =−
Z
dDyλ−DL(φ, λ2X) ✭✶✳✶✾✮
❉❡❝♦♠♣♦♥❞♦ ❡st❛ ❡♥❡r❣✐❛ ❡♠ s✉❛s ♣❛rt❡s ♣♦t❡♥❝✐❛❧ ❡ ❝✐♥ét✐❝❛ ❝♦♠♦ ❢❡✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡
❝♦♠♣❛r❛♥❞♦✲❛ ❝♦♠ ❛ ❊q✳ ✭✶✳✶✻✮✱ ♦❜t❡♠♦s
E(λ)=λ−DE
0+λ2−DE2. ✭✶✳✷✵✮
❈♦♥s✐❞❡r❛♥❞♦ E0 ❡ E2 ❝♦♠♦ ❣r❛♥❞❡③❛s ♣♦s✐t✐✈❛s✱ ❛♥❛❧✐s❡♠♦s ♦ s❡❣✉✐♥t❡✿
E(λ) =
1
λE0+λE2, D= 1
1
λ2E0+E2, D= 2
1
λ3E0+
1
λE2, D= 3
✭✶✳✷✶✮
P♦❞❡♠♦s ♥♦t❛r q✉❡✱ ♣❛r❛ D = 1✱ t❡♠♦s ✉♠ ♣♦♥t♦ ❡st❛❝✐♦♥ár✐♦ ❡♠ λ =E0/E2✳ ❙❡ D > 1✱ ❛
❡♥❡r❣✐❛E(λ) ❞❡❝r❡s❝❡ ♠♦♥♦t♦♥✐❝❛♠❡♥t❡ à ♠❡❞✐❞❛ q✉❡D ❝r❡s❝❡✳ ❆ss✐♠✱ s♦❧✉çõ❡s ❡stát✐❝❛s ❡♠
t❡♦r✐❛s ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r ❝♦♠ ✉♠❛ ❡♥❡r❣✐❛ ❞♦ t✐♣♦ ✭✶✳✶✻✮ só sã♦ ♣♦ssí✈❡✐s ❡♠ ✉♠❛ ❞✐♠❡♥sã♦ ❡s♣❛❝✐❛❧✳
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
P♦❞❡♠♦s ❛✐♥❞❛ ✈❡r q✉❡E(λ)λ=1=E❀ ❡ ✐ss♦ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡♥❡r❣✐❛ t❡r ❛ ❢♦r♠❛ ❞❡s❝r✐t❛ ❡♠
✭✶✳✶✻✮✱ ♣♦❞❡♥❞♦ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ♣r♦❞✉t♦ ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠♦
é ♦ ❝❛s♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡s❝r✐t❛ ♣❡❧❛ ❞✐♥â♠✐❝❛ t❛q✉✐ô♥✐❝❛ ❬✸✹✱ ✸✺❪✳ ❆ss✐♠ ∂E(λ)/∂λ ❞❡✈❡ s❡r
♠✐♥✐♠✐③❛❞❛ ❡♠ λ= 1✳ P♦rt❛♥♦✱ ✉♠❛ s♦❧✉çã♦ ❡stát✐❝❛✱φ =φ(x)✱ ❞❡✈❡ s❛t✐s❢❛③❡r
dE(λ)
dλ
λ=1
=
Z ∞ −∞
dDx(DL −2LXX) = 0. ✭✶✳✷✷✮
P❛r❛ D= 1✱ t❡♠♦s
L −2LXX = 0, ✭✶✳✷✸✮
♦ q✉❡ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ❛♣❡♥❛s ❝♦♥✜❣✉r❛çõ❡s ❡stát✐❝❛s ❡ s❡♠ ♣r❡ssã♦ sã♦ ❡stá✈❡✐s✳ ❖❜s❡r✈❛♠♦s q✉❡ ❡st❛ ❡q✉❛çã♦ ❞❡♣❡♥❞❡ ❞♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❡ ❞❡ s✉❛s ♣r✐♠❡✐r❛s ❞❡r✐✈❛❞❛s✳ P♦rt❛♥t♦✱ é ✉♠❛
❡q✉❛çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ➱ t❛♠❜é♠ ✉♠❛ ❡q✉❛çã♦ ❞❡ ✈í♥❝✉❧♦✿ ❡❧❛ ✐♠♣õ❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ♣r❡ssã♦ ♥✉❧❛ ♣❛r❛ ❡①✐stê♥❝✐❛ ❞❛ ❡st❛❜✐❧✐❞❛❞❡✳
✶✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❧✐♥❡❛r
❙♦❧✉çõ❡s t♦♣♦❧ó❣✐❝❛s t❡♥❞❡♠ ❛ ♠❛♥t❡r ❛ t♦♣♦❧♦❣✐❛ ♠❡s♠♦ q✉❛♥❞♦ s♦❢r❡♠ ♣❡rt✉r❜❛çõ❡s✳
❖ ❡st✉❞♦ ❞❡st❛ ❡st❛❜✐❧✐❞❛❞❡ ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çõ❡s ❛❝❡r❝❛ ❞❛ ❡♥❡r❣✐❛ ❡ t♦♣♦❧♦❣✐❛ ❞❛s s♦❧✉çõ❡s✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ♣♦ss✐❜✐❧✐t❛ ❛ss♦❝✐❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥í✈❡✐s q✉â♥t✐❝♦s ❛ q✉❛❧q✉❡r s♦❧✉çã♦ ❝❧áss✐❝❛ ❡stát✐❝❛ ❡stá✈❡❧✱ ❢✉♥❞❛♠❡♥t❛❞♦ ♥❛ ♦❜s❡r✈❛çã♦ ❞❡ q✉❡ só❧✐t♦♥s ✲ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❝❧áss✐❝❛s
❞❡ ❝❛♠♣♦s ✲✱ s❡♠ q✉❛♥t✐③❛çã♦✱ sã♦ s✐♠✐❧❛r❡s ❛ ♣❛rtí❝✉❧❛s✳ ❈❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❡st❛❞♦s q✉â♥t✐❝♦s ♣♦❞❡♠ s❡r ❡①♣❛♥❞✐❞♦s ❡♠ ✉♠❛ sér✐❡ s❡♠✐❝❧áss✐❝❛❀ ♦s t❡r♠♦s ❞♦♠✐♥❛♥t❡s ❞❡st❛ sér✐❡
❡stã♦ r❡❧❛❝✐♦♥❛❞♦s às s♦❧✉çõ❡s ❝❧áss✐❝❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❬✸✶✱ ✸✻✱ ✸✼❪✳
❈♦♥s✐❞❡r❛♠♦s ♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❞❛ ❢♦r♠❛
φ(x, t) =φ(x) +η(x, t), ✭✶✳✷✹✮
♦♥❞❡ φ(x)r❡♣r❡s❡♥t❛ ♦ ❝❛♠♣♦ ❡stát✐❝♦ ❡ η(x, t)❞❡s❝r❡✈❡ ✉♠❛ ♣❡q✉❡♥❛ ✢✉t✉❛çã♦ ❡♠ t♦r♥♦ ❞❛
s♦❧✉çã♦ ❡stát✐❝❛✳ ▲♦❣♦✱ ♦ X = 12∂µφ∂µφ t❛♠❜é♠ s♦❢r❡ ✉♠❛ ♣❡q✉❡♥❛ ✈❛r✐❛çã♦✱ ♣♦❞❡♥❞♦ s❡r
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
r❡❡s❝r✐t♦ ❝♦♠♦
X(x, t) =X(x) + ¯X(x, t), ✭✶✳✷✺✮
♦♥❞❡
¯
X(x, t) =∂µφ∂µη+ 1 2∂µη∂
µη. ✭✶✳✷✻✮
P♦❞❡♠♦s ❡①♣❛♥❞✐r ❛ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❛té s❡❣✉♥❞❛ ♦r❞❡♠ ❡♠ η ❡ ❡♠ X✳ ❖ r❡s✉❧t❛❞♦ ❞❛ ❡①♣❛♥sã♦ é ♦ s❡❣✉✐♥t❡✿¯
L(φ+η, X+ ¯X) =L(φ, X) +ηLφ+ ¯XLX + η2
2 Lφφ+ ¯
X2
2 LXX+ηX¯LφX. ✭✶✳✷✼✮
❙✉❜st✐t✉✐♥❞♦ ✭✶✳✷✻✮ ❡♠ ✭✶✳✷✼✮ ❡ r❡✉♥✐♥❞♦ ❛♣❡♥❛s ♦s t❡r♠♦s q✉❛❞rát✐❝♦s✱ ❛ ♣❛rt❡ ❝♦rr❡s♣♦♥❞❡♥t❡
❞❛ ❛çã♦ é
S(2) = 1 2
Z
d2x
LX∂µη∂µη+LXX(∂µφ∂µη)2 + [Lφφ−∂µ(LφX∂µφ)]η2 . ✭✶✳✷✽✮
❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛η t❡♠ ❛ ❢♦r♠❛
∂µ(LX∂µη+LXX∂µφ∂αφ∂αη) = [Lφφ−∂µ(LφX∂µφ)]η, ✭✶✳✷✾✮
♦♥❞❡ t❡♠♦s ❞❡s♣r❡③❛❞♦ ❥✉st❛♠❡♥t❡ ♦s t❡r♠♦s q✉❛❞rát✐❝♦s ❡♠ η✱ ❡ ❝♦♥s✐❞❡r❛❞♦ ❛♣❡♥❛s ♦s t❡r♠♦s ❧✐♥❡❛r❡s✳ ❙❡♥❞♦ φ ✉♠❛ s♦❧✉çã♦ ❡stát✐❝❛✱ ❛ ❊q✳ ✭✶✳✷✾✮ t♦♠❛ ❛ ❢♦r♠❛
LXSη¨−[(2LXSXSXS +LXS)η
′]′
= [Lφφ+ (LφXSφ
′)′]η. ✭✶✳✸✵✮
❙✉♣♦♥❞♦
η(x, t) = η(x) cos(ωt), ✭✶✳✸✶✮
♥❛ ❊q✳ ✭✶✳✸✵✮ ♦❜t❡♠♦s
−[(2LXSXSXS+LXS)η
′]′ =
Lφφ+ (LφXSφ
′)′+ω2L
XS
η. ✭✶✳✸✷✮
❊st❛ ♣♦❞❡ ❛✐♥❞❛ s❡r r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛
−[a(x)η′]′
=b(x)η, ✭✶✳✸✸✮
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
♦♥❞❡
a(x) = 2LXSXSXS+LXS, ✭✶✳✸✹✮
b(x) = Lφφ+ (LφXSφ
′)′+ω2
LXS. ✭✶✳✸✺✮
❆❣♦r❛ ❝❤❛♠❡♠♦s
A2 ≡ 2LXSXSXS+LXS
LXS
✭✶✳✸✻✮
❡✱ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ❢❛ç❛♠♦s ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ ✈❛r✐á✈❡✐s
dx =Adz e η= √ u LXA
. ✭✶✳✸✼✮
❈♦♠ ✐ss♦✱ ♦❜t❡♠♦s
−uzz+U(z)u=ω2u, ✭✶✳✸✽✮
♦♥❞❡
U(z) = (
√
ALX)zz
√
ALX −
1
LX
Lφφ+
1 A LφX φz A z . ✭✶✳✸✾✮
❆ ❊q✳ ✭✶✳✸✽✮ ♦❜t✐❞❛ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ω2 q✉❡✱ ❢♦r♠❛❧♠❡♥t❡✱ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛
❡q✉❛çã♦ ❡st❛❝✐♦♥ár✐❛ ❞❡ ❙❝❤rö❞✐♥❣❡r ♥♦ ♣♦t❡♥❝✐❛❧ U(z)✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ s♦❧✉çã♦ ❡stát✐❝❛
φ(x) ❞❡♣❡♥❞❡ ❞❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ❞♦ ♦♣❡r❛❞♦r
− d
2
dz2 +U(z). ✭✶✳✹✵✮
P♦❞❡♠♦s ♥♦t❛r q✉❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ♥❡❣❛t✐✈♦s tr❛♥s❢♦r♠❛ ❛ ❢✉♥çã♦cos❞❛ ❊q✳ ✭✶✳✸✶✮
❡♠ cosh✳ ■st♦ ❝♦♥tr❛r✐❛ ❛ ♥♦ss❛ s✉♣♦s✐çã♦ ❞❡ ♣❡q✉❡♥❛ ✢✉t✉❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❝❛♠♣♦ ❡stát✐❝♦✳
◆❡st❡ ❝❛s♦✱ ❛s s♦❧✉çõ❡s sã♦ ✐♥stá✈❡✐s✱ ❞❡s❞❡ q✉❡ ❛ ♣❡rt✉r❜❛çã♦ ❝r❡s❝❡ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡✳ ❙❡ ♥ã♦ ❤á ❛✉t♦✈❛❧♦r❡s ♥❡❣❛t✐✈♦s✱ ❡♥tã♦ t♦❞♦s ♦s ω sã♦ r❡❛✐s ❡ ❛ ♣❡q✉❡♥❛ ♣❡rt✉r❜❛çã♦ η(x, t) ♥ã♦
❝r❡s❝❡ ❝♦♠ ♦ t❡♠♣♦✳ P♦❞❡♠♦s ♥♦t❛r q✉❡ ♦✉ ❡❧❛ ♦s❝✐❧❛✱ ♥♦ ❝❛s♦ ω2 >0✱ ♦✉ ❡♥tã♦ ♥ã♦ ❞❡♣❡♥❞❡
❞♦ t❡♠♣♦✳ ❚❡♠♦s✱ ♥❡st❡s ❝❛s♦s✱ ✉♠❛ s♦❧✉çã♦ ❡stá✈❡❧✳
◆♦ss♦ ✐♥t❡r❡ss❡ ❡stá ✈♦❧t❛❞♦ ♣❛r❛ s♦❧✉çõ❡s ❡stá✈❡✐s ❡ ♦ ♥ã♦✲❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛ ♣❡rt✉r❜❛çã♦ é ❛ss❡❣✉r❛❞♦ ♣❡❧❛ ❝♦♥❞✐çã♦ ❞❡ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡ ❬✶✹❪✳ ❚❛❧ ❝♦♥❞✐çã♦ é ❛ ✐♠♣♦s✐çã♦
❉✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛
❞❡ q✉❡ ❛s r❛í③❡s ❞❛ ❊q✳ ✭✶✳✸✵✮ s❡❥❛♠ r❡❛✐s ❡ ❞✐st✐♥t❛s✱ ♦ q✉❡ r❡s✉❧t❛ ❡♠
A2 ≡ 2LXSXSXS+LXS
LXS
>0. ✭✶✳✹✶✮
❯♠❛ ♠❡❧❤♦r ❞✐s❝✉ssã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✽✱ ✸✾✱ ✹✵✱ ✹✶❪✳
❈❛♣ít✉❧♦ ✷
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
◆❡st❡ ❡ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❡st❛r❡♠♦s ❡♠♣r❡❣❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡s❡♥✈♦❧✈✐❞❛s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ❛♦s ♠♦❞❡❧♦s ❞❡ t❡♦r✐❛s ❞❡ ❝❛♠♣♦ ❡s❝❛❧❛r ❝✉❥♦s ❞❡t❛❧❤❡s ♣r♦♣♦♠♦✲♥♦s ❛ ❡st✉❞❛r✳ ❖
♠♦❞❡❧♦ q✉❡ ❝♦♥s✐❞❡r❛♠♦s ♥♦ ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦ é ♦ ♠❛✐s s✐♠♣❧❡s✿ ♦ ❞❡ ✉♠ ú♥✐❝♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❝✉❥❛ ❞✐♥â♠✐❝❛ é ❣♦✈❡r♥❛❞❛ ♣❡❧❛ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛ ♣❛❞rã♦ ❞❛❞❛ ♣♦r
L=X−V(φ), ✭✷✳✶✮
♦♥❞❡ X = 12∂µφ∂µφ r❡♣r❡s❡♥t❛ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❝✐♥ét✐❝❛ ❡ ❣r❛❞✐❡♥t❡ ♣❛r❛ ❛ ❞✐♥â♠✐❝❛ ❡ V(φ)
❞❡s❝r❡✈❡ ♦ ♣♦t❡♥❝✐❛❧✿ ✉♠❛ ❢✉♥çã♦ ❞❡φ t♦♠❛❞❛ ❢r❡q✉❡♥t❡♠❡♥t❡ ❝♦♠♦ s❡♥❞♦ ♣♦❧✐♥♦♠✐❛❧✱ t❡♥❞♦ ❢♦r♠❛s ❡s♣❡❝í✜❝❛s ♣❛r❛ ❝❛❞❛ s✐t✉❛çã♦ ❝♦♥s✐❞❡r❛❞❛✳ ❆ ❞❡♥s✐❞❛❞❡ ❧❛❣r❛♥❣✐❛♥❛ ✭✷✳✶✮ é ✐♥✈❛r✐❛♥t❡
s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ❡ s❡ ❛♣r❡s❡♥t❛ ♥❛t✉r❛❧♠❡♥t❡ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❝✐♥ét✐❝❛ ❡ ♣♦t❡♥❝✐❛❧ ❞❛ ❡♥❡r❣✐❛✳ ❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ✭✶✳✹✮ ♣❛r❛ ❡st❡ ❝❛s♦ t♦♠❛ ❛ ❢♦r♠❛
∂µ∂µφ+Vφ= 0, ✭✷✳✷✮
♦♥❞❡ Vφ =dV /dφ✳ ❖ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ é
Tµν =∂µφ∂νφ−ηµνL. ✭✷✳✸✮
❉❡s❞❡ q✉❡ ❝♦♥s✐❞❡r❡♠♦s s♦❧✉çõ❡s ❡stát✐❝❛s✱ ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ t♦r♥❛✲s❡
φ′′=Vφ ✭✷✳✹✮
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ sã♦
T00=ρ(x) = 1 2φ
′2+V(φ), ✭✷✳✺✮
q✉❡ r❡♣r❡s❡♥t❛ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛✱ ❡
T11= 1 2φ
′2
−V(φ). ✭✷✳✻✮
❆ ❡♥❡r❣✐❛ ❞♦ ❝❛♠♣♦ ❡stát✐❝♦ é
E =
Z
dx
1 2φ
′2+V(φ)
. ✭✷✳✼✮
❆ ❝♦♥❞✐çã♦ ❞❡ ♣r❡ssã♦ ♥✉❧❛✱ ❊q✳ ✭✶✳✷✸✮✱ ❧❡✈❛ à s❡❣✉✐♥t❡ ❡q✉❛çã♦✿
1 2φ
′2 =V(φ). ✭✷✳✽✮
❈♦♠❜✐♥❛♥❞♦ ✭✷✳✽✮ ❝♦♠ ✭✷✳✺✮✱ ♦❜t❡♠♦s ❛ ✐♠♣♦rt❛♥t❡ r❡❧❛çã♦
ρ(x) =φ′2 = 2V(φ). ✭✷✳✾✮
❯♠❛ r❡❧❛çã♦ ❞❡st❡ t✐♣♦ é ✈❡r❞❛❞❡✐r❛ ✉♥✐❝❛♠❡♥t❡ ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❝♦♥s❡r✈❛❞❛
♣❛r❛ ❝♦♥✜❣✉r❛çõ❡s ❡stát✐❝❛s q✉❡ r❡❧❛❝✐♦♥❛ ♦ ✈❛❧♦r ❞♦ ❝❛♠♣♦ à s✉❛ ❞❡r✐✈❛❞❛ ❡s♣❛❝✐❛❧✳ ❊♠ ❣❡r❛❧✱ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♥ã♦ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ❛♣❡♥❛s✳
✷✳✶ ❉❡❢❡✐t♦s ❞♦ t✐♣♦ ❦✐♥❦
❆s ❡str✉t✉r❛s ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ só❧✐t♦♥s ❢♦r❛♠ ❞❡s❝♦❜❡rt❛s ♣❡❧♦ ❡♥❡❣❡♥❤❡✐r♦ ❡s❝♦❝ês ❏✳ ❙❝♦tt ❘✉ss❡❧❧✱ ❡♠ ✶✽✸✹✱ ❡♥q✉❛♥t♦ ♠♦♥t❛✈❛ s❡✉ ❝❛✈❛❧♦ ♣♦r ✉♠ ❝❛♥❛❧ ❞❡ á❣✉❛✱ q✉❛♥❞♦ ✈✐✉ ✉♠
❜❛r❝♦ ♣❛r❛r r❡♣❡♥t✐♥❛♠❡♥t❡✳ ❊❧❡ ♦❜s❡r✈♦✉ q✉❡ ✉♠ ✏♠♦rr♦✑ ❞❡ á❣✉❛ ❝♦♠❡ç♦✉ ❛ s❡ ♠♦✈❡r ❞❡s❞❡ ❛ ♣r♦❛ ❞♦ ❜❛r❝♦ ♣♦r ✉♠❛ ❣r❛♥❞❡ ❞✐stâ♥❝✐❛ ❝❛♥❛❧ ❛❜❛✐①♦✱ ♣r❡s❡r✈❛♥❞♦ ❛ ❢♦r♠❛ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡✳ ❙♦❧✉çõ❡s q✉❡ s❡ ♣r♦♣❛❣❛♠ s❡♠ ❞✐ss✐♣❛çã♦ ❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✉♥✐❢♦r♠❡ sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦
só❧✐t♦♥s ❡ ✉♠❛ ❞❡ s✉❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❛♣❧✐❝❛çõ❡s ❡stá✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ót✐❝❛✱ ♦♥❞❡ ❡❧❡s ♣♦❞❡♠ s❡ ♣r♦♣❛❣❛r ❡♠ ✜❜r❛s ót✐❝❛s s❡♠ ❞✐st♦rçã♦✳
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
❉❡❢❡✐t♦s ❝♦♠♦ ❦✐♥❦s ❡ ♣❛r❡❞❡s ❞❡ ❞♦♠í♥✐♦ ✭❛ s✉❛ ❡①t❡♥sã♦ ♣❛r❛ ♠❛✐s ❞♦ q✉❡ ✉♠❛ ❞✐♠❡♥sã♦ ❡s♣❛❝✐❛❧✮ sã♦ ♦s t✐♣♦s ♠❛✐s s✐♠♣❧❡s ❞❡ só❧✐t♦♥s✳ ❙ã♦ ♦❜❥❡t♦ ❞❡ ♣❡sq✉✐s❛ ❡♠ ✈ár✐❛s ár❡❛s ❞❛
❢ís✐❝❛✱ ❞❡s❞❡ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛ ❛ ❝♦s♠♦❧♦❣✐❛❀ ♣♦❞❡♠ t❡r ❡①✐st✐❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ✉♥✐✈❡rs♦ ♣r✐♠♦r❞✐❛❧✳ ❯♠❛ ❞✐❢❡r❡♥ç❛ ❝♦♥s✐st❡ ♥♦ ❢❛t♦ ❞❡ q✉❡ ♦s só❧✐t♦♥s ❞❡ á❣✉❛ ♣r❡s❡r✈❛♠ s✉❛ ✐❞❡♥t✐❞❛❞❡ ❞❡♣♦✐s ❞❡ ✉♠ ❡s♣❛❧❤❛♠❡♥t♦ ❡♥q✉❛♥t♦ q✉❡ ❦✐♥❦s ❡ ♣❛r❡❞❡s ❞❡ ❞♦♠í♥✐♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡
tê♠ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❬✸✷✱ ✸✻❪✳
❋✐❣✉r❛ ✷✳✶✿ P♦t❡♥❝✐❛❧ ✭✷✳✶✶✮✳ ❆s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠ ♦ ♣♦t❡♥❝✐❛❧ ♣❛r❛ a= 1 ❡ λ= 0.5, 1, 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❈♦♠♦ ✉♠ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♠♦❞❡❧♦ ❞❡s❝r✐t♦ ♣❡❧♦ ♣♦t❡♥❝✐❛❧ ❝♦♠ ❛✉t♦✲ ✐♥t❡r❛çã♦ q✉árt✐❝❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ φ4✱ q✉❡ s♦❢r❡ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❡ s✐♠❡tr✐❛❀
t❛❧✈❡③ ♦ ❡①❡♠♣❧♦ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ❞❡ ❡str✉t✉r❛s ❞❡ ❞❡❢❡✐t♦s s✉♣♦rt❛♥❞♦ s♦❧✉çõ❡s t♦♣♦❧ó❣✐❝❛s✳ ❆ ❞✐♥â♠✐❝❛ é ❞❡s❝r✐t❛ ♣♦r
L= 1 2(∂µφ)
2
− λ
2
2 φ
2
−a22
, ✭✷✳✶✵✮
♦♥❞❡ µ = 0,1✱ ❡ λ ❡ a sã♦ ♣❛râ♠❡tr♦s✳ ❊st❛ ❧❛❣r❛♥❣✐❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜ ❛ tr❛♥s❢♦r♠❛çã♦
φ → −φ✳ ❖ ♣♦t❡♥❝✐❛❧
V(φ) = 1 2λ
2 φ2
−a22
✭✷✳✶✶✮
t❡♠ ❞♦✐s ♠í♥✐♠♦s ❣❧♦❜❛✐s ❞❡❣❡♥❡r❛❞♦s ❡♠ φ = ±a ❡ ✉♠ ♠á①✐♠♦ ❡♠ φ = 0 ✭❱❡r ❋✐❣✳ ✷✳✶✮✳
❖s ♠í♥✐♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ sã♦ ❡q✉✐♣r♦✈á✈❡✐s ♥❛ s✐t✉❛çã♦ s✐♠étr✐❝❛✳ ❆ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❡
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
s✐♠❡tr✐❛ ♦❝♦rr❡ q✉❛♥❞♦ ♦ ❝❛♠♣♦ φ ❛ss✉♠❡ ✉♠ ❞❡ss❡s ♠í♥✐♠♦s✳
❋✐❣✉r❛ ✷✳✷✿ ❙♦❧✉çã♦ ✭✷✳✶✹✮✳ ❆s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠ ❛s s♦❧✉çõ❡s ❦✐♥❦ ❡ ❛♥t✐✲❦✐♥❦ ♣❛r❛ a= 1 ❡ λ= 0.5, 1, 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ é
¨
φ−φ′′+ 2λ2 φ2
−a2
φ= 0. ✭✷✳✶✷✮
❉✉❛s s♦❧✉çõ❡s tr✐✈✐❛✐s ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ sã♦φ=±a✳ ❊st❛s✱ ♣♦ré♠✱ tê♠ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♥✉❧❛ ❡✱ ❝♦♠♦ ✈✐♠♦s✱ ✐❞❡♥t✐✜❝❛♠ ♦s ✈á❝✉♦s ❝❧áss✐❝♦s ❞♦ ♠♦❞❡❧♦✳ ❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛
❝❛♠♣♦ ❡stát✐❝♦ t❡♠ ❛ ❢♦r♠❛
φ′′−2λ2φ(φ2−a2) = 0. ✭✷✳✶✸✮
❆s s♦❧✉çõ❡s ❞❡st❛ ❡q✉❛çã♦ sã♦ ❝❤❛♠❛❞❛s t♦♣♦❧ó❣✐❝❛s ♦✉ ❞♦ t✐♣♦ ❦✐♥❦✱ ❡ sã♦ ❞❛❞❛s ♣♦r
φ(x) =±atanh (λa(x−x0)), ✭✷✳✶✹✮
♦♥❞❡ x0 ❧♦❝❛❧✐③❛ ♦ ❝❡♥tr♦ ❞♦ ❦✐♥❦✳ ❆ ❧♦❝❛❧✐③❛çã♦ ❞♦ ❝❡♥tr♦ ❞❡ ✉♠❛ s♦❧✉çã♦ ❡stát✐❝❛ é
❛r❜✐trár✐❛✱ ♣♦✐s ♦ ♠♦❞❡❧♦ t❡♠ s✐♠❡tr✐❛ tr❛♥s❧❛❝✐♦♥❛❧✳ ❖ ❦✐♥❦ t❡♠ ❛♠♣❧✐t✉❞❡ ✐❣✉❛❧ ❛a❡ ❧❛r❣✉r❛ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ λa✳ ➱ ❝♦st✉♠❡ ❞❡♥♦♠✐♥❛r♠♦s ❛s s♦❧✉çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦
s✐♥❛❧ ♣♦s✐t✐✈♦ ❞❡ ❦✐♥❦s ❡ ❛q✉❡❧❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦ s✐♥❛❧ ♥❡❣❛t✐✈♦✱ ❞❡ ❛♥t✐✲❦✐♥❦s✳ ❆ ❋✐❣✳ ✷✳✷ ♠♦str❛ ❛s s♦❧✉çõ❡s ✭✷✳✶✹✮✱ ❝♦♥s✐❞❡r❛❞❛s ❛s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ s✐♥❛❧✱ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
❞❡ λ✱ ♦♥❞❡ ✜③❡♠♦s a = 1 ❡ x0 = 0✳ P♦❞❡♠♦s ❡♥tã♦ ♥♦t❛r q✉❡✱ q✉❛♥❞♦ x > 0✱ φ ❡✈♦❧✉✐ ❡♠
❞✐r❡çã♦ ❛ +a✳ P❛r❛ x < 0✱ φ ❡✈♦❧✉✐ ❡♠ ❞✐r❡çã♦ ❛ −a✳ ❯♠❛ s♦❧✉çã♦ t♦♣♦❧ó❣✐❝❛✱ ♣♦rt❛♥t♦✱
❝♦♥❡❝t❛ ♠í♥✐♠♦s ❞✐st✐♥t♦s ❡ ❞❡s❝r❡✈❡ ✉♠ s❡t♦r t♦♣♦❧ó❣✐❝♦✳
❙♦❧✉çõ❡s ❞♦ t✐♣♦ ❦✐♥❦ sã♦ ❝❛s♦s ❡s♣❡❝✐❛✐s ❞❡ s♦❧✉çõ❡s ♥ã♦✲❞✐ss✐♣❛t✐✈❛s✱ ♣❛r❛ ❛s q✉❛✐s ❛
❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡♠ ✉♠ ❞❛❞♦ ♣♦♥t♦ ♥ã♦ s❡ ❛♥✉❧❛ ♥♦ ❧✐♠✐t❡ ❞❡ t❡♠♣♦ ❣r❛♥❞❡✳ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♣❛r❛ ❡st❡ ❝❛s♦ é ❞❛❞❛ ♣♦r
ρ(x) = λ2a4sech4(λa(x−x0)), ✭✷✳✶✺✮
❡ t❡♠ ✉♠ ♠á①✐♠♦ ❡♠ x=x0✱ρmax =λ2a4✳ ●rá✜❝♦s ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡stã♦ ♣r❡s❡♥t❡s
♥❛ ❋✐❣✳ ✷✳✸✱ ♦♥❞❡ ♥♦✈❛♠❡♥t❡ ✜③❡♠♦s x0 = 0✳ ❆ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ ❦✐♥❦ é ♦❜t✐❞❛ ✐♥t❡❣r❛♥❞♦✲s❡
❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡♠ t♦❞♦ ♦ ❡s♣❛ç♦✳ ❱❡❥❛♠♦s✿
E =
Z +∞ −∞
dxρ(x) = 4λa
3
3 . ✭✷✳✶✻✮
➱ ❢á❝✐❧ ♦❜s❡r✈❛r q✉❡ ❡st❡ ✈❛❧♦r é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❧♦❝❛❧✐③❛çã♦ ❞♦ ❝❡♥tr♦ ❞♦ ❦✐♥❦✱ ♦ q✉❡ s✐❣♥✐✜❝❛
q✉❡ tr❛♥s❧❛çõ❡s ♥ã♦ ♠✉❞❛♠ ❛ s✉❛ ❡♥❡r❣✐❛✳
❋✐❣✉r❛ ✷✳✸✿ ❉❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ✭✷✳✶✺✮✳ ❆s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♣❛r❛ a= 1 ❡ λ= 0.5, 1, 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆ss♦❝✐❛❞❛ à s♦❧✉çã♦ ❞♦ t✐♣♦ ❦✐♥❦ ❡①✐st❡ ✉♠❛ ❝♦rr❡♥t❡ ❝♦♥s❡r✈❛❞❛✱
jµ = 1 2aǫ
µν∂νφ, ✭✷✳✶✼✮
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
♦♥❞❡µ, ν = 0,1❡ǫµν é ♦ sí♠❜♦❧♦ ❛♥t✐✲s✐♠étr✐❝♦ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✱ s❡♥❞♦ǫ01= 1✳ ❯t✐❧✐③❛♥❞♦
❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❛♥t✐✲s✐♠❡tr✐❛✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ jµ é ✉♠❛ ❣r❛♥❞❡③❛ ❝♦♥s❡r✈❛❞❛✱ ♦✉
s❡❥❛✱
∂µjµ = 0. ✭✷✳✶✽✮
❑✐♥❦s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ✉♠❛ ❣r❛♥❞❡③❛ ❞❡♥♦♠✐♥❛❞❛ ❝❛r❣❛ t♦♣♦❧ó❣✐❝❛✱ ❞❛❞❛ ♣♦r
Q=
Z
dxj0 = 1
2a[φ(x= +∞)−φ(x=−∞)]. ✭✷✳✶✾✮
❙✉❛ ❝♦♥s❡r✈❛çã♦ ❞á✲♥♦s ✐♥❢♦r♠❛çã♦ ❛❝❡r❝❛ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ❞❡❢❡✐t♦✳ ❈♦♥✜❣✉r❛çõ❡s ❞❡ ❞❡❢❡✐t♦ ❡stá✈❡✐s sã♦ t❛✐s q✉❡ dQ/dt= 0✳
❋✐❣✉r❛ ✷✳✹✿ P♦t❡♥❝✐❛❧ ✭✷✳✷✵✮✳ ❆s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠ ♦ ♣♦t❡♥❝✐❛❧ ♣❛r❛ a= 1 ❡ λ= 0.5, 1, 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❖✉tr♦ ❡①❡♠♣❧♦ ❞❡ ❞❡❢❡✐t♦ t♦♣♦❧ó❣✐❝♦ é ♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ s✐♥❡✲●♦r❞♦♥✱ q✉❡ t❡♠
s✐❞♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡st✉❞❛r ✉♠❛ ❣❛♠❛ ❞❡ ❢❡♥ô♠❡♥♦s ❡♠ ❞✐❢❡r❡♥t❡s ❝♦♥t❡①t♦s ❬✹✷✱ ✹✸✱ ✹✹✱ ✹✺❪✱ ❝✉❥♦ ♣♦t❡♥❝✐❛❧ t❡♠ ❛ ❢♦r♠❛
V(φ) = 1 2λ
2cos2(aφ), ✭✷✳✷✵✮
❡ ❡stá ♣❧♦t❛❞♦ ♥❛ ❋✐❣✳ ✷✳✹✳ ❊st❡ ♣♦t❡♥❝✐❛❧ t❡♠ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ♠í♥✐♠♦s q✉❡
❝♦rr❡s♣♦♥❞❡♠ ❛ φ = kπ/a✱ ♦♥❞❡ k é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ❯♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ♠í♥✐♠♦s s✐❣♥✐✜❝❛ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ❝♦rr❡♥t❡s ❧♦❝❛✐s ❝♦♥s❡r✈❛❞❛s ❡✱ ♣♦r s✉❛ ✈❡③✱ ✉♠ ♥ú♠❡r♦ t❛♠❜é♠
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
✐♥✜♥✐t♦ ❞❡ s❡t♦r❡s t♦♣♦❧ó❣✐❝♦s ❡q✉✐✈❛❧❡♥t❡s q✉❡ sã♦ ❝♦♥❡❝t❛❞♦s ♣❡❧❛s s♦❧✉çõ❡s
φ(x) =±1
aarcsin[tanh(λax)] +kπ; k = 0,±1,±2, . . . ✭✷✳✷✶✮
❯♠ ❣rá✜❝♦ ❞❡st❛s s♦❧✉çõ❡s é ✈✐st♦ ♥❛ ❋✐❣✳ ✷✳✺✱ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡λ✱ ❝♦♥t❡♠♣❧❛♥❞♦ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❦✐♥❦ ❡ ❛♥t✐✲❦✐♥❦✳
❋✐❣✉r❛ ✷✳✺✿ ❙♦❧✉çõ❡s ❞♦ ♠♦❞❡❧♦ s✐♥❡✲●♦r❞♦♥✳ ❙❡♥❞♦a =λ= 1✱ ❛s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛
❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠ ❛s s♦❧✉çõ❡s ❦✐♥❦ ❡ ❛♥t✐✲❦✐♥❦ ♣❛r❛ k = 0±1,±2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞❛ ❛ ❝❛❞❛ s♦❧✉çã♦ é ❞♦ t✐♣♦
ρ(x) =λ2sech2(λax). ✭✷✳✷✷✮
❯♠ ❣rá✜❝♦ ❞❡st❛ ❡①♣r❡ssã♦ ❡stá ♥❛ ❋✐❣✳ ✷✳✻✱ ♦♥❞❡ é ♣❡r❝❡♣tí✈❡❧ ❛ s❡♠❡❧❤❛♥ç❛ ❝♦♠ ❛ ❋✐❣✳ ✷✳✸✱ q✉❡ r❡♣r❡s❡♥t❛ ♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞❛ ❛♦s ❦✐♥❦s ❡stá ❝♦♥❝❡♥tr❛❞❛ ♥♦ ❝❡♥tr♦ ❞♦ ❞❡❢❡✐t♦✳ ■♥t❡❣r❛♥❞♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡♠ t♦❞♦ ♦ ❡s♣❛ç♦✱
♦❜t❡♠♦s ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞❡ ❝❛❞❛ s♦❧✉çã♦✿
E = 2
λ a
✭✷✳✷✸✮
▼❛✐s ✉♠❛ ✈❡③✱ ♦❜s❡r✈❛♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❧♦❝❛❧✐③❛çã♦ ❞♦ ❝❡♥tr♦ ❞♦ ❦✐♥❦✳
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
❋✐❣✉r❛ ✷✳✻✿ ❉❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞♦ ♠♦❞❡❧♦ s✐♥❡✲●♦r❞♦♥✳ ❆s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♣❛r❛ a = 1 ❡ λ= 0.5, 1, 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
✷✳✷ ❉❡❢❡✐t♦s ❞♦ t✐♣♦ ❧✉♠♣
❊str✉t✉r❛s ♥ã♦✲t♦♣♦❧ó❣✐❝❛s✱ ♦✉ ❞♦ t✐♣♦ ❧✉♠♣✱ ❛♣❛r❡❝❡♠ ❡♠ ✈ár✐♦s ❝♦♥t❡①t♦s ❞❡♥tr♦ ❞❛
❢ís✐❝❛✳ ❊♠ ❢ís✐❝❛ ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛✱ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❞❡s❝r❡✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ tr❛♥s♣♦rt❡ ❞❡ ❝❛r❣❛ ❡♠ ❝♦rr❡♥t❡s ❞✐❛tô♠✐❝❛s ❬✹✻✱ ✹✼✱ ✹✽✱ ✹✾✱ ✺✵❪✱ ❡♠ ❝♦♠✉♥✐❝❛çã♦ ó♣t✐❝❛✳ ❊♠ ❢ís✐❝❛ ❞❡ ❛❧t❛s ❡♥❡r❣✐❛s✱ ❡stã♦ ♣r❡s❡♥t❡s ♥♦s ❡st✉❞♦s s♦❜r❡ ❢♦r♠❛çã♦ ❞❡ ❡str✉t✉r❛
❬✺✶✱ ✺✷✱ ✺✸✱ ✺✹✱ ✺✺❪ ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠❛tér✐❛ ❡s❝✉r❛ ♣r❡s❡♥t❡ ♥❛s ❣❛❧á①✐❛s ❬✺✻✱ ✺✼❪✳
❯♠ ❡①❡♠♣❧♦ ❞❡ ❞❡❢❡✐t♦ ♥ã♦✲t♦♣♦❧ó❣✐❝♦ ♦✉ ❞♦ t✐♣♦ ❧✉♠♣ é ♦ ♠♦❞❡❧♦ φ3 ❞❡s❝r✐t♦ ♣❡❧♦
♣♦t❡♥❝✐❛❧
V3(φ) =
λ
2φ
2
1− φ
a
. ✭✷✳✷✹✮
❊st❡ t❡♠ ✉♠ ú♥✐❝♦ ♠í♥✐♠♦ ❧♦❝❛❧ ❡♠ φ= 0 ❡ ✉♠ ♠á①✐♠♦ ❡♠ φ= 2a/3❡ ❡stá ♣❧♦t❛❞♦ ♥❛ ❋✐❣✳
✷✳✼✳
❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❝❛♠♣♦ ❡stát✐❝♦ t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
φ′′−λφ
1− 3φ 2a
= 0. ✭✷✳✷✺✮
❖ ♠♦❞❡❧♦ ♣❛❞rã♦
❋✐❣✉r❛ ✷✳✼✿ P♦t❡♥❝✐❛❧ ❞♦ ♠♦❞❡❧♦ φ3✳ ❆s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛♠
♦ ♣♦t❡♥❝✐❛❧ ♣❛r❛ a = 1 ❡ λ= 0.5, 1, 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❋✐❣✉r❛ ✷✳✽✿ ❙♦❧✉çõ❡s ❞♦ ♠♦❞❡❧♦ φ3✳ ❙❡♥❞♦ a= 1✱ ❛s ❧✐♥❤❛s ❝♦♥tí♥✉❛✱ ♣♦♥t✐❧❤❛❞❛ ❡ tr❛❝❡❥❛❞❛
r❡♣r❡s❡♥t❛♠ ❛s s♦❧✉çõ❡s ❞♦ t✐♣♦ ❧✉♠♣ ♣❛r❛ λ= 0,5; 1; 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆ s♦❧✉çã♦ ❞❡ss❛ ❡q✉❛çã♦ é
φ(x) =asech2
√
λ
2 (x−x0)
!
. ✭✷✳✷✻✮
❙❡✉s ❣rá✜❝♦s sã♦ ♠♦str❛❞♦s ♥❛ ❋✐❣✳ ✷✳✽✱ ♦♥❞❡ ✜③❡♠♦s ♦ ❝❡♥tr♦ ❞❛ s♦❧✉çã♦ s❡r x0 = 0 ❡✱
❝♦♠ a ✜①♦✱ ❡s❝♦❧❤❡♠♦s ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ♣❛r❛ λ✳ ❆ ❞❡♥♦♠✐♥❛çã♦ ♥ã♦✲t♦♣♦❧ó❣✐❝❛ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞❡st❡ t✐♣♦ ❞✐③ r❡s♣❡✐t♦ à ❛✉sê♥❝✐❛ ❞❡ ❝❛r❣❛ t♦♣♦❧ó❣✐❝❛✳ P♦❞❡♠♦s ♥♦t❛r q✉❡ ♦s ✈❛❧♦r❡s
❛ss✐♥tót✐❝♦s ❞❡ ❝❛♠♣♦ φ(+∞) ❡φ(−∞) ❝♦✐♥❝✐❞❡♠✳ ❖ ❝❛♠♣♦φ✱ ♥❡st❡ ❝❛s♦✱ ♥ã♦ ❞❡s❝r❡✈❡ ✉♠ s❡t♦r t♦♣♦❧ó❣✐❝♦✱ ❝♦♠♦ ❛❝♦♥t❡❝❡ ♣❛r❛ s♦❧✉çõ❡s ❞♦ t✐♣♦ ❦✐♥❦✳
