n 次中心对称量子图的可约性.pdf

Advances in Applied Mathematics A?,2024,13(4),1862-1874Published Online April 2024 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2024.134175n g%f?5 ppp ZZZ?n?U9vF2024c3?28FF2024c4?23FuF2024c4?30F?+?L?n g%fm?)9 n g%f?f1?)J?#?g?f?K=z?K?f?e:c+Lm?)ff1?A?Reducibility of n Subcentrally SymmetricQuantum GraphsKai Zhang,Jia ZhaoSchool of Science,Hebei University of Technology,TianjinReceived:Mar.28th,2024;accepted:Apr.23rd,2024;published:Apr.30th,2024AbstractIn this paper,according to the irreducible representation of the group,the decompo-sitions of the space of square integrable functions on n subcenter-symmetric quantum:p,Z.n g%f?5J.A?,2024,13(4):1862-1874.DOI:10.12677/aam.2024.134175pZgraphs and the quotient graph of n subcenter-symmetric quantum graphs are given.It provides a new idea for the decomposition of the long determinant of quantumgraphs,and transforms the spectrum problem of the original quantum graph into thespectrum problem of the quotient graph,which lays a foundation for the research ofisospectral quantum graphs.KeywordsGroup Representation,Function Space Decomposition,Differential Operator,Quantum Graph,Secular Determinant,EigenvalueCopyright c?2024 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.3?f?|?N,3n+?f.f3!nz?+k2A,1936c,Pauling 1 gfkf?gdf.Cc5$LkN?(?A5?5,Xp?5,3Ek?2,Ac,d$Lf,f5-.XE?u,NnX?.IA?X,df?.y,f+?9:.um?)k(,XCarlson?8.EvL:v?8,:v?dv v?8?8,=dv=|Ev|.|E|L8 E?,e?:8 V 8 E?k,K k.e z ej Lej,=Lej(0,+,K .?Lej=+,?.?Laplace f/Xe:H:f 7 d2fdx2,f3ejPf|ej,f|ejw30,Lej?.f3 v?:vAvf|e1(v).f|e|Ev|(v)+Bvf0|e1(v).f0|e|Ev|(v)=0,Av Bv m dv?,m?.X3 v?Kirchhoff:f|e(v)=f|e0(v),e,e0 Ev,XeEvf0|e(v)=0,=Av=11.11000,Bv=000.000111.2.1.k?m L2()d3z?f|e|,kfk2L2()=XeEkf|ek2L2(e),=L2()m L2(e)?.?e5+9+L?.2.2.?G+,V 6=0 F?5m,GL(V)V k_5C|?+,G?GL(V)?+?G 3 F?L.2.3.e+G?z?G,?a?(?$P)a?(?$P),KG+,aG?)?,n?+PGn=ai|i=1,2,n,an?u?e Gn=ia|i=1,2,n,na?u?e.DOI:10.12677/aam.2024.1341751864A?pZ2.4.Cn(s)Lds=(s1,s2,sk)?kn:?,u s?skv 1 sk n/2.e i j sk(mod n),K:vi,vj?.3.n g%m?)n?+k n gEL,P ng,?LL 1:Table 1.Irreducible representations of cyclic groups of order nL 1.n?+?Leaa2a3an11111112123n1312462(n1).n1n12(n1)3(n1)n gEL,?!?n g%m?)9Laplace f?),dm?)?n g%?.3.1.k+Gn,eN?(ai,x)Gn 7 aix ve5:(1)+:?ai Gn,N?x 7 aix?g?V?,?x ,?ekex=x,?ai,al Gn,x ,(aial)x=ai(alx);(2)Y5:?ai Gn,?g?N?x 7 aixY?;(3)5:ex ,aix=x,Kai=e;(4)l5:?x ,3x?U,kai6=e,aix/U;(5)?(?:aiu aiv?=?u v.(6)?5:3f?e,+?,=Laie=Le.ek+Gn?3 w v:(1)Gnk+?3w?8CX?,=aiGnaiw=,(2)w?B?,=ai6=al Gn?aiwalwkk?:?:,o w Gnu?.5,?.3v Kirchhoff?:v?Y,dXJkv Kirchhoff?:?du3T vk:kY?,L5,3J:k Kirchhoff?:,?:J:,J:UC?f?.1?k+G6?C6(1,2)?,?E?,3C6(1,2)z?:?%?L?J:,dC?5?,JLDOI:10.12677/aam.2024.1341751865A?pZ,C6(1,2)?:vj?OP ej,1,ej,2,ej,dvj,:v1?J:9v1?C6(1,2)?.Figure 1.(a)The circulant graph,C6(1,2),with dummy vertices(b)The fundamental domain of C6(1,2)1.(a)J:?C6(1,2)(b)C6(1,2)?n3.2ng%nk+Gn,Gn?L?m L2(n)?)L2(n)=MnXt=1Ft,nm Ft?Ftd3?w?Ft|wL:Ft|aiw=t(ani)Ft|w.y.?F L2(n),”?wkm,zz,o3?F|wdmk?5L,km?L,=F|w=Fe1,Fe2,Fem.mL2(n)?mnt=1Ft?N?P,Gn?1 t?LkF?Ft Ft?N?Pt,t=1,2,n,PtF|e1=Ft|e1=t(e)F|e1+t(a)F|e1+m+t(an1)f1+m(n1)n,PtF|e2=Ft|e2=t(e)F|e2+t(a)F|e2+m+t(an1)f2+m(n1)n,.PtF|emn=Ft|emn=t(e)F|emn+t(a)F|em+t(an1)fm(n1)n,(3.1)DOI:10.12677/aam.2024.1341751866A?pZKN?P?,nXt=1t(e)=n,nXt=1t(ai)=0(i=1,2,n 1),F=F1+F2+F3+Fn1+Fn,kFk2L2(n)=nXt=1kFtk2Ft,dN?P?N?.d(3.1)Ft|aiw=t(ani)Ft|w.?3.3k+G4?4 g%4m?),?w4k8,2.Figure 2.(a)4with partial edges marked(b)The fundamental domain w4 2.(a)IP?4(b)?w4G4?LL 2:Table 2.Irreducible representations of cyclic groups of order 4L 2.4?+?Leaa2a31111121i-1i31-11-141i-1i?F L2(4),dG4?o?L?Ft Ft,t=1,2,3,4,DOI:10.12677/aam.2024.1341751867A?pZF1=F1|e1,F1|e2,F1|e24=F|e1+F|e7+F|e13+F|e194,F|e2+F|e8+F|e14+F|e204,F|e24+F|e6+F|e12+F|e184,F2=F2|e1,F2|e2,F2|e24=F|e1+iF|e7 F|e13 iF|e194,F|e2+iF|e8 F|e14 iF|e204,F|e24+iF|e6 F|e12 iF|e184,F3=F2|e1,F2|e2,F2|e24=F|e1 F|e7+F|e13 F|e194,F|e2 F|e8+F|e14 F|e204,F|e24 F|e6+F|e12 F|e184,F4=F2|e1,F2|e2,F2|e24=F|e1 iF|e7 F|e13+iF|e194,F|e2 iF|e8 F|e14+iF|e204,F|e24 iF|e6 fe12+iF|e184,Ft|aiw4=t(a4i)Ft|w4.n3.43ng%n?Laplace fHD(H)=F,F0,F00L2(n),F 3 n?:?v Kirchhoff:,Hj?du3m Ft?Laplace fHt?,=H=nXt=1Ht,fHt?D(Ht)=Ft,F0t,F00t Ft,Ft3 n?:?v Kirchhoff:Ft3?ej el=aiej?J:v?vFt|ej(v)=t(ai)Ft|el(v),F0t|ej(v)+t(ai)F0t|el(v)=0.(3.2)df?)?XeX:(H)=nt=1(Ht).y.D(H)L2(n),odn3.2?f?),=DOI:10.12677/aam.2024.1341751868A?pZD(H)=nXt=1D(Ht).?Ft|ei Ft,HtFt|ei=F00t|ei Ft,Ht(Ft)Ft.n?H=nXt=1Ht.y.?e k+G0,1,2,sG0?L,Laplace f?0(H)=st=10(Ht),s G0?D?L,o?G0/sd D?L.:?1?,L/?#?:6u s,G0/s?0(Hs)?.e?Ek+G4?4k+G12?3?4?,+?LL,d4 3?4?d?,3.Figure 3.(a)The fundamental domain w4of 4(solid line)(b)The quotientgraph of 4 3.(a)4?w4()(b)4?3.5z:v Kirchhoff?4(2),?w43 G4?e(=)?,3=L,:v1 v6?1/#?:v1(6),:C,dd?4?,d3.34ko.1?L?:,+?3?u?L,aF|e1(v1):=iF|e1(v1)=F|e7(v6),aF0|e1(v1):=iF0|e1(v1)=F0|e7(v6),(3.3)DOI:10.12677/aam.2024.1341751869A?pZF 3 v6?v Kirchhoff?F|e6(v6)=F|e7(v6),F0|e6(v6)+F0|e7(v6)=0,(3.4)(3.3)(3.4)F|e1(v1)+iF|e6(v6)=0,F0|e1(v1)iF0|e7(v6)=0,Av1(6)=1i00!,Bv1(6)=001i!,do?L?3 v1(6)?o:?,:Kirchhoff,d1?Av1(6)=1100!,Bv1(6)=0011!,d2?Av1(6)=1i00!,Bv1(6)=001i!,d3?Av1(6)=1100!,Bv1(6)=0011!,d4?Av1(6)=1i00!,Bv1(6)=001i!.?e5y Ft3 4?G4/tEvA?:.BO,v1:?4zwm 0,1,F k3z:v Kirchhoff,F|e1(0)=F|e24(1),F|0e1(0)+(F|0e24(1)=0,F|e7(0)=F|e6(1),F|0e7(0)+(F|0e6(1)=0,F|e13(0)=F|e12(1),F|0e13(0)+(F|0e12(1)=0,F|e19(0)=F|e18(1),F|0e19(0)+(F|0e18(1)=0,(3.5)3 F2?c6P mi(i=1,2,6),m1(0)=F|e1(0)+iF|e7(0)F|e13(0)iF|e19(0)4,DOI:10.12677/aam.2024.1341751870A?pZm6(1)=F|e6(1)+iF|e12(1)F|e18(1)iF|e24(1)4,d(3.5)?m1(0)=im6(1),m01(0)+i(m06(1)=0,=F23 v1(6)?ve:Av1(6)=1i00!,Bv1(6)=001i!.ey F23 v2 v3?Ov Kirchhoff,m1(1)=m2(0)=m3(0),(m01(1)+m02(0)+m03(0)=0,m2(1)=m4(0),(m02(1)+m04(0)=0,n v4 v5v Kirchhoff.n,F23vd 2?:,nF1,F3,F4n.4.n g%f1?51?f?A?,?Lz 10?O?n g%fn?1?,dm?)n?1?),n1?n?1?,l?f?K=z?K.?n?1?,?1,?Vg.k?.3k,z?j j?“,d Lj=Lj.j z,-j?:o(j)?I xj=0 j?:t(j)?I xj=Lj.?,u j?=j,o(j)?I xj=0,t(j)?I xj=Lj.X 4dxjxj?,xj=Lj xj.Figure 4.The pair of bonds j and j associated with the interval 0,Lj 4.m 0,Lj?j jzejz 0,Lj,?k 6=0,Laplace fA?f00=k2f 3z?)?fj(x)=ajeikx+ajeikLjx,z 10=k26=0 Laplace f?A?,?=?k DOI:10.12677/aam.2024.1341751871A?pZe?,(k):=det(I SD(k)=0,(4.1)(k)f?1?,SD(k)2|E|?,D(k)?,D(k)j,j=eikLj,S?,val=Sl,j aj,lj?1,2,|E|,1,2|E|.3?4?9:,?e5 a,b,c?4 g%4(5)?1?9 41?).Figure 5.The rotation-invariantgraph 4of side lengths a,b,c 5.a,b,c?4 g%4k:Kirchhoff?4?1?4(k)=16561(9v4av4bv4c v4av4b 8v4av2bv2c v4av4c+v4a 8v2av4bv2c 8v2av2bv4c+8v2av2b+8v2av2c v4bv4c+v4b+8v2bv2c+v4c 9)(9v4av4bv4c v4av4b 8v4av2bv2c v4av4c+v4a+8v2av4bv2c+8v2av2bv4c 8v2av2b 8v2av2c v4bv4c+v4b+8v2bv2c+v4c 9)(81v8av8bv8c 18v8av8bv4c+v8av8b 144v8av6bv6c+16v8av6bv2c 18v8av4bv8c+84v8av4bv4c 2v8av4b+16v8av2bv6c 16v8av2bv2c+v8av8c 2v8av4c+v8a 18v4av8bv8c+20v4av8bv4c 2v4av8b+160v4av6bv6c 32v4av6bv2c+20v4av4bv8c 296v4av4bv4c+20v4av4b 32v4av2bv6c+160v4av2bv2c 2v4av8c+20v4av4c 18v4a+v8bv8c 2v8bv4c+v8b 16v6bv6c+16v6bv2c 2v4bv8c+84v4bv4c 18v4b+16v2b 144v2bv2c+v8c 18v4c+81),(4.2)va=eikLa,vb=eikLb,vc=eikLc.DOI:10.12677/aam.2024.1341751872A?pZ4U+)?o:?1?O14(k)=19(9v4av4bv4c v4av4b 8v4av2bv2c v4av4c+v4a 8v2av4bv2c 8v2av2bv4c+8v2av2b+8v2av2c v4bv4c+v4b+8v2bv2c+v4c 9),24(k)=19(9v4av4bv4c v4av4b 8v4av2bv2c v4av4c+v4a v4bv4c+v4b+8v2bv2c+v4c 9),34(k)=19(9v4av4bv4c v4av4b 8v4av2bv2c v4av4c+v4a+8v2av4bv2c+8v2av2bv4c 8v2av2b 8v2av2c v4bv4c+v4b+8v2bv2c+v4c 9),44(k)=19(9v4av4bv4c v4av4b 8v4av2bv2c v4av4c+v4a v4bv4c+v4b+8v2bv2c+v4c 9).w4(k)?O14(k),24(k),34(k),44(k),1?4m)A,?n?1?)n(k)=nYt=1nn(k).?a?“ZP“,P“?E?4f,P“gG%?,4?|?.?,%a-,6?P“Uz%,?.78Ig,7(No.12001153)z1 Brooks,H.(1940)Diamagnetic Anisotropy and Electronic Structure of AROMATIC Molecules.TheJournal of Chemical Physics,8,939-949.https:/doi.org/10.1063/1.17506082 Carlson,R.(2000)Nonclassical Sturm-Liouville Problems and Schr odinger Operators on Radial Trees.Electronic Journal of Differential Equations,71,1-24.3 Solomyak,M.(2003)On the Spectrum of 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Graphs.Doctoral Thesis,TexasAM University,College Station,TX.10 Berkolaiko,G.(2017)An Elementary Introduction to Quantum Graphs.Geometric and Computa-tional Spectral Theory,700,41-72.https:/doi.org/10.1090/conm/700/14182DOI:10.12677/aam.2024.1341751874A?