Outline1nù©lCþ{o(ÔnÆ2007cSC.S.Wu1nù©lCþ{o(OutlineùÇ:1Sturm-Liouville.§¯KgίKSturm-Liouville.§¯KSturm-Liouville.§¯K¥{¿y2lS-L.§¯Kw©lCþ{©lCþ{°Â©lCþ{uÐC.S.Wu1nù©lCþ{o(OutlineùÇ:1Sturm-Liouville.§¯KgίKSturm-Liouville.§¯KSturm-Liouville.§¯K¥{¿y2lS-L.§¯Kw©lCþ{©lCþ{°Â©lCþ{uÐC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...ReferencesÇÂÁ§5êÆÔn{6§§18.3—18.6ù&§5êÆÔn{6§§9.4nÎ!X1Á§5êÆÔn{6§§10.6C.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnùÇ:1Sturm-Liouville.§¯KgίKSturm-Liouville.§¯KSturm-Liouville.§¯K¥{¿y2lS-L.§¯Kw©lCþ{©lCþ{°Â©lCþ{uÐC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÎDefinitionLÚM½Â3½¼êmS(©)ΧeéuT¼êmS?¿ü¼êuÚv§ðk(v,Lu)=(Mv,u)=Zbav∗Ludx=Zba(Mv)∗udxK¡M´LÎC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.1L=ddxZbav∗dudxdx=v∗uba−Zbadv∗dxudx¤±§uÚvÑ÷v.^y(a)=y(b)§ddxδ−ddxC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.1L=ddxZbav∗dudxdx=v∗uba−Zbadv∗dxudx¤±§uÚvÑ÷v.^y(a)=y(b)§ddxδ−ddxC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.1L=ddxZbav∗dudxdx=v∗uba−Zbadv∗dxudx¤±§uÚvÑ÷v.^y(a)=y(b)§ddxδ−ddxC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqn?ØeM´LΧKéu?¿¼êuÚv§kZbav∗Mudx=Zba(Mu)∗vdx∗=Zbau∗Lvdx∗=Zba(Lv)∗udx¤±L´MÎC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqn?ØeM´LΧKéu?¿¼êuÚv§kZbav∗Mudx=Zba(Mu)∗vdx∗=Zbau∗Lvdx∗=Zba(Lv)∗udx¤±L´MÎC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqn?ØeM´LΧKéu?¿¼êuÚv§kZbav∗Mudx=Zba(Mu)∗vdx∗=Zbau∗Lvdx∗=Zba(Lv)∗udx¤±L´MÎC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.2L=d2dx2©ÜÈ©üg=Zbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx¤±§¼êuÚv÷vhv∗u0−(v∗)0uiba=0§d2dx2ÎÒ´§gC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.2L=d2dx2©ÜÈ©üg=Zbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx¤±§¼êuÚv÷vhv∗u0−(v∗)0uiba=0§d2dx2ÎÒ´§gC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.2L=d2dx2©ÜÈ©üg=Zbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx¤±§¼êuÚv÷vhv∗u0−(v∗)0uiba=0§d2dx2ÎÒ´§gC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~o^ehv∗u0−(v∗)0uiba=0º¼êuÚv÷v!!na.^α1y(a)+β1y0(a)=0α2y(b+β2y0(b)=0(Ù¥|α1|2+|β1|26=0,|α2|2+|β2|26=0)¼êuÚv÷v±Ï^y(a)=y(b)y0(a)=y0(b)C.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~o^ehv∗u0−(v∗)0uiba=0º¼êuÚv÷v!!na.^α1y(a)+β1y0(a)=0α2y(b+β2y0(b)=0(Ù¥|α1|2+|β1|26=0,|α2|2+|β2|26=0)¼êuÚv÷v±Ï^y(a)=y(b)y0(a)=y0(b)C.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~o^ehv∗u0−(v∗)0uiba=0º¼êuÚv÷v!!na.^α1y(a)+β1y0(a)=0α2y(b+β2y0(b)=0(Ù¥|α1|2+|β1|26=0,|α2|2+|β2|26=0)¼êuÚv÷v±Ï^y(a)=y(b)y0(a)=y0(b)C.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqngÎDefinitioneÎLÎÒ´§g§=éuT¼êmS?¿ü¼êuÚv§ðk(v,Lu)=(Lv,u)=Zbav∗Ludx=Zba(Lv)∗udxK¡L´gÎC.S.Wu1nù©lCþ{o(EigenproblemofSturm-LiouvilleEqnMethodofSeparationofVariables...EigenproblemofSelf-AdjiontOperatorsEigenproblemofSturm-LiouvilleEqnDegenerationinEigenproblemofS-LEqnÞ~~13.3L=iddxZbav∗idudxdx=iv∗uba−iZbadv∗dxudxu