Fock型空间上的Schödinger测不准关系.pdf

2023,43A(5):13211332http:/Fock.m?Sch odinger OXoc?_(U9“?U9 300387):T?Fock.m?f?Sch odinger OX,?wL,?2?z 4?Fock m Heisenberg.OX?z 16?(J.T?(J2?f/,?f?g/?O?.c:Fock m;O?n;?f;Segal-Bargmann C;?f.MR(2010)Ka:47B38;32A37a:O177.1zI:A?:1003-3998(2023)05-1321-121O?nf?n,o/,*f?U?(/.O?n?n?,3N!&?n9?+?.O?n?:9;(J Folland?n5.Fock ma-?Xm,f!&?n!fn?kXX.;?Fock md Bargmann2,?25,5f?79,13,15,17.Cc5,o?f?,?1.?z 10 Ko?Fock mEf?5?1?.2015 c,Y4 Fock mO?n(,35I:6uL?;Fock m?(?n?25.+L Segal-Bargmann C,N?Sch odinger m(m)?O?n3 Fock m73aq(J2,?l)m?kg?dfn?.2018 c,z 11 aq?2 Fock m?(?n.2020 c,?L?ffn)A?f,?Fock.m?O?n,?3(?,=?E,?S“L,z 16,n 2.2.d?,)c?12?Fock m Heisenberg(X?r/?2.?Lf?,X/?f9f?5|?Sch odinger.O?n,(?wL,l?2?z 4?(J,?z 16?(J.?(Jwm5?KC?O?n3 Fock.m?A.?k5,Au?XvF:2022-09-20;?F:2023-04-10E-mail:;78:Ig,7(12101451)Supported by the NSFC(12101451)1322n?Vol.43 Am,X Hardy m,Dirichlet m?.d?,?Fock.m*m?OX.fm?OXCc5f&E+?9K,?3)m?L.,?,?f?f?#?O?.5?Fock m?fg?,d?(4.1)z 4,16?(J.?(J2?f.?(?SXe:1 2!0?Fock.m,?f9?,?zk?(J;31 3!,Fock.m?Sch odinger O?n,(?;31 4!,Fock.m?#?(g/)O?.2?k?,C E8,H(C)L C?X|?8,R 8.u?0,C?pdd(z)=e|z|2dA(z),dA(z)=dxdy C?Lebesgue.|pd,Fock.m?.u?0,3 C u d?N|mL2(C,d)=?f:ZC|f|2d 0,Fock.m F2,-Fock m,L2(C,d)?4fm,LF2:=(f H(C):kfk2,:=?ZC|f|2d?1/2?f.aq/,u?ES?wnn=1,W k?S?wnn=1?m f,?W vWen=wn+1en+1,n=0,1,2,.5 2.2Fock.m F2?|?5?Xeen(z)=rnn!zn,n=0,1,2,.?f=Pn=0 xnen F2,?f D 3 f?Df(z)=ddz Xn=0 xnrnn!zn!=Xn=0p(n+1)xn+1en(z).No.5oc?:Fock.m?Sch odinger OX1323d,?fS Fock.m F2?S?nn=1?f.z 16?f T?f TvTen=un+1en+1,n=0,1,2,.(2.2)n 2.1?f=Pn=0 xnen F2,K?f T 9?f Tuf?LOTf=Xn=1xnunen1,Tf=Xn=0 xnun+1en+1.2.3?A H?5f,u F2?1?f,P A?hAi,hAi=hAf,fi.(2.3)2.4u5f A B,f fA,B=AB BA,A,B+=AB+BA.(2.4)f?LAB=12A,B+i2(A,B/i),(2.5)5f A B?CovAB=12hA,B+i hAihBi=12hA hAi,B hBi+i.(2.6)2.5u?,0,u5f A B?COVAB=12hA ,B +i.(2.7)e0?Fock mO?n?(J.Fock muO?n?1N(J3z 4.n A-f F2,Ku?a,b R kkf0(z)+zf(z)af(z)k kf0(z)zf(z)ibf(z)k kfk2,?=?3?c E C?f(z)=C exp?c 12(c+1)z2+a ibcc+1z?.2018c,z 11 3 Fock.m F2?aq(J:?0,k?1f0(z)+zf(z)af(z)?2,?1f0(z)zf(z)+ibf(z)?2,1kfk22,?=?3 c R+C0 C,?f(z)=C0exp?(c 1)2(c+1)z2+(a ibc)c+1z?.1324n?Vol.43 Au?f T 9f T,z 16/T T5f?E?gfA=T+T,B=i(T T),?Fock.m?XeOX.n Bb?T?S?unn=1?f,?S?v|un|2=n,n=1,2,?.K?0,a,b R 9 f F2,kk(A a)fk2,k(B b)fk2,kfk22,?=?3?c E C?f=CXn=0 xnen,X xn(n 1)v?z 16,n 2.2.?,=?E,?S“X.,z 14 fnE|?mf T 9f W?5|?(?n.n C-UM=aT+bW,a1,a2,b1,b2?,u?1?f,k2UM12UM214(a1b2 a2b1)2+Cov2UM1UM2,2UM=h(UM)2i hUMi2,CovUM1UM2 UM1 UM2?.?=?f(t)=?i(a1 a2)(b1 b2)?14ei?a2a1b1b2?(thT)22+i(Wit+i,?f T?E?gf A=T+T B=i(T T),?A,B m?OX.p,?f A B?5|?(?n.kXe?Cn,Cn+1=1+b(n+1)/2cXp=1M(n,p)Xq=1pm=1g(m,n,p,q)Q Hg(m,n,p,q)g(m 1,n,p,q)2.(3.1)H(x)m?,H(x)=1,x 0,0,x?f,?S?v|un|2=n,n=1,2,?,Kgf A=T+T,B=i(T T)?fvA,B=2iI,I F2?f.e?n.n 3.1?T?S?unn=1?f,?S?v|un|2=n,n=1,2,p?.gf A=T+T,B=i(T T)?5|,=UM=aA+bB.Ku?,0,M1:=(a1,b1),M2:=(a2,b2)9 F2?1?f,kXe Sch odinger OXk(UM1)fk22,k(UM2)fk22,2(a1b2 a2b1)2+COV2UM1UM2,(3.6)?=?f=C Xn=0 xnen,(3.7)X xnvx0=a1 ca2+(ib1 ib2c)c2(c)u1u2(c)2 h(a1 ca2)2(ib1 ib2c)2ix1=c2u2(c)2(c)2 h(a1 ca2)2(ib1 ib2c)2ixn+1=Cn+1nYj=1Aj,(3.8)p c?ua1+ib1a2+ib2?E,Aj=ca1 ca2+(ib1 ib2c)uj+1,Cn+1,Q,M(n,p),g(m,n,p,q)H(x)(3.1)(3.5),C v|C|2=?Pn=0|xn|2?1.1326n?Vol.43 Ay?z?,PO1=UM1,O2=UM2,(3.9)d)?(2.3),kk(UM1)fk22,k(UM2)fk22,=hO1f,O1fi hO2f,O2fi=?O21f,f?O22f,f?|hO2f,O1fi|2=|hO1O2f,fi|2=|hO1O2i|2.(3.10),d(2.5)hO1O2i=12hO1,O2+i+i2hO1,O2/ii,O1,O2+=UM1,UM2+2UM1 2UM2+2,d(2.7),f?hO1,O2+i=2COVUM1UM2.?dn 3.1,kO1,O2=UM1,UM2=(a1b2 a2b1)A,B=(a1b2 a2b1)(2iI).d,hO1O2i=COVUM1UM2+i2hUM11,UM2/ii,(3.11)(3.10)(3.11),=?Fock.m?Sch odinger.OXk(UM1)fk22,k(UM2)fk22,?COVUM1UM2+i2?UM1,UM2?/i?2=COV2UM1UM2+2(a1b2 a2b1)2.?=?3E c?(UM1)f=c(UM2)f,(3.12)kfk=1.k)(3.12),=(a1A+b1B )Xn=0 xnen=c(a2A+b2B )Xn=0 xnen,(3.13)“f A,B?Lza1(T+T)+ib1(T T)Xn=0 xnen=ca2(T+T)+ib2(T T)Xn=0 xnen.No.5oc?:Fock.m?Sch odinger OX1327dn 2.1?f?NL,za1 ca2+(ib1 ib2c)Xn=0 xn+1un+1en+a1 ca2(ib1 ib2c)Xn=1xn1unen=(c)Xn=0 xnendu enn=0 F2?|?5?,)?e|a1 ca2+(ib1 ib2c)x1u1=(c)x0;a1 ca2+(ib1 ib2c)xn+1un+1+a1 ca2(ib1 ib2c)xn1un=(c)xn,n 1.(3.14)e?1a?:(1)e3 c?a1 ca2+(ib1 ib2c)=0,c=0,dk xn=0,n 0,v f?1?,T;(2)e3 c=?a1 ca2+(ib1 ib2c)=0,?c 6=0,d|(3.14)C?x0=0,a1 ca2(ib1 ib2c)xn1un=(c)xn,n 1,d)?xn=0,n 0,f?1 g,d;(3)?c?a1 ca2+(ib1 ib2c)6=0,|(3.14)S?xn?nS“X,LC/?xn+1=ca1 ca2+(ib1 ib2c)un+1 xna1 ca2(ib1 ib2c)una1 ca2+(ib1 ib2c)un+1 xn1,B,Pxn+1=Anxn+Bnxn1,(3.15)An=ca1 ca2+(ib1 ib2c)un+1,Bn=(ib1 ib2c)(a1 ca2)una1 ca2+(ib1 ib2c)un+1,nS“X(3.15)cX8z?n,Pxn+1=Cn+1nYj=1Aj,(3.16)d Cnv#?S“XCn+1=Cn+BnAnAn1 Cn1,C0=1,C1=1.BnAn An1=(ib1 ib2c)(a1 ca2)un a1 ca2+(ib1 ib2c)2un+1una1 ca2+(ib1 ib2c)un+1(c)21328n?Vol.43 A=|un|2h(ib1 ib2c)2(a1 ca2)2i(c)2=n Q.dz 6?48,Cn?wLd(3.1).?,Cn?L“(3.16).q?1?,f?8z?n,X xn kfk,d5(3.13)vkK,=?(3.7).y.uz 16?n 2.2,aq/,?wL(S“L).3.1u?gf A=T+T,B=i(T T),u?0,a,b R F2?1?f,kk(A a)fk k(B b)fk ,?=?3 c R,C C,?f=CXn=0 xnen,c 6=1 C v f?1,X xnvx0=(1+c)c2u1(a ibc)(a ibc)2(1 c2),x1=c2(a ibc)2(a ibc)2(1 c2),xn+1=Cn+1nYi=1Ai.Q=(1c2)(aibc)2,Cn+1,M(n,p),g(m,n,p,q),H(x)d(3.1)(3.5),C?v|C|2=?Pn=0|xn|2?1?E.?f?A?f D,=-A=D+D,B=i(D D),kXeO?,S5?KC3mO?3 Fock.m?A(.3.2-UM=aA+bB,Ku?,0 a,b,a1,a2,b1,b2 R 9 F2?1?f,kk(UM1)fk22,k(UM2)fk22,(a1b2 a2b1)2+COV2UM1UM2,(3.17)?=?3 c,C C?f(z)=C exp?(a1 ca2 ib1+icb2)2(a1 ca2+ib1 icb2)z2+ca1 ca2+ib1 icb2z?,(3.18)a2bb1 a1bb2,c 6=a1+ib1a2+ib2 C?f?1?E.yu?1?f F2vn?a,b,a1,a2,b1,b29,0,dn 3.1 (3.17).?=?3E c?(UM1)f=c(UM2)f,No.5oc?:Fock.m?Sch odinger OX1329=(a1A+b1B )f=c(a2A+b2B )f,=(a1 ca2)(f0(z)+zf(z)+(b1 cb2)(i(f0(z)zf(z)=(c)f,?n?(a1 ca2+ib1 icb2)f0(z)+z(a1 ca2 ib1+icb2)(c)f=0,e c=a1+ib1a2+ib2,Kk)f=0,d f?1 g,.e c 6=a1+ib1a2+ib2,Kd)?)f(z)=C exp?(a1 ca2 ib1+icb2)2(a1 ca2+ib1 icb2)z2+ca1 ca2+ib1 icb2z?,C?E.d f F2,klimzf(z)exp?12|z|2?=0,3m F2?7 C=0|a1 ca2 ib1+icb2|a1 ca2+ib1 icb2|,-c=a+bi,Kk|a1(a+bi)a2 ib1+i(a+bi)b2|a1(a+bi)a2+ib1 i(a+bi)b2|,=(a1 aa2)bb22+a2b+(b1 ab2)2(a1 aa2)+bb22+a2b (b1 ab2)2,ka2bb1 a1bb2.y.A/,=m F2?ff,=Af(z)=f0(z)+zf(z),Bf(z)=i(f0(z)zf(z),(3.19)3.2?n A?e.3.3-f F2?1?,K?0,a,b R kkf0+zf afk2kf0 zf ibfk2 COV2AB+1,(3.20)?=?3E c C?f(z)=C exp?(1+ic)2(1 ic)z2+a+bc1 icz?,b 0,c 6=i C v f?1.1330n?Vol.43 An 3.1,?F2mgfUM1=a1A+b1B,UM2=a2A+b2Bm?OX.?!?,u Dodonov?,Fock.m?N A B 5|?OX.n 3.2(Dodonov?3)?Tj(1 j m)E Hilbert m?m gf,KkXe?OXmXk=12Tj 2mXjk(ajbk akbj)2.5 3.1T(JSm?5|?OX3 Fockm?,Ly/,z 11.Dodonov?,p2y.4g/?OXL?Egf5OX,?!3 Fock m?f?OX.n 4.1?T?S?unn=1?f,?S?v|un|2=n,n=1,2,p?,Tf.u F2?1?f=Pn=0 xnen,kkTfk+kTfk.(4.1)y dn?|a|+|b|a b|,kkTfk2+kTfk2|hTf,Tfi hTf,Tfi|=|hTTf,fi hTTf,fi|=|h(TTf TTf),fi|.dn 2.1?kTfk2+kTfk2?*TXn=1xnunen1 TXn=0 xnun+1en+1,f+?=?*Xn=0 xn+1un+1un+1en+1Xn=0 xnun+1un+1en,f+?No.5oc?:Fock.m?Sch odinger OX1331d|un|2=n?=?*Xn=0 xn+1(n+1)en+1Xn=0 xn(n+1)en,f+?=?*Xn=0 xnen,Xn=0 xnen+?=kfk2=,qdu kTfk 0,kTfk 0,(kTfk+kTfk)2 kTfk2+kTfk2,?kkTfk+kTfk.y.?f,k 4.1?T?S?unn=1?f,u F2?1?f=Pn=0 xnen,kkTfk2+kTfk2Xn=1x2n?|un+1|2|un|2?.AO/,?f?f,k 4.2-f F2,uA?f D,f,?ff z.kkDfk2+kzfk2 kfk2.z1 Alpay D,Colombo F,Sabadini I,Salomon G.The Fock space in the slice hyperholomorphic set-ting/Bernstein S,Kahler U,Sabadini I,Sommen F.Hypercomplex Analysis:New Perspectives and Ap-plications Trends in Mathematics.Cham:Birkh auser,2014:43592 Bargmann V.On a Hilbert space of analytic functions and an associated integral transform.Comm PureAppl Math,1961,14:1872143 Chen B,Lian P.Uncertainty relations for multiple operators without covariances.J Physics A:Math Theory,2022,55(9):0953034 Y,.Fock m?O?n,I:,2015,45(11):18471854Chen Y,Zhu K H.Uncertainty principles for the Fock space.Sci Sin Math,2015,45(11):184718545 Folland G B,Sitaram A.The uncertainty principle:A mathematical survey.J Fourier Anal Appl,1997,3:2072386 Gonoskov I.Closed-form solution of a general three-term recurrence relation.Adv Difference Equ,2014,2014(1):1127 Hai P V,Khoi L H.Complex symmetry of weighted composition operators on the Fock space.J Math AnalAppl,2016,433:175717718?,?.Fock m9f.I:,2015,45(11):17591778Hu Z J,Lv X F.Fock spaces and relative operators.Sci Sin Math,2015,45(11):175917789 Le T.Normal and isometric weighted composition operators on the Fock space.Bull Lond Math Soc,2014,46(4):8478561332n?Vol.43 A10 Lian P,Liang Y.Weighted composition operator on quaternionic Fock space.Banach J Math Anal,2021,15(1):12011 Kechrimparis S,Weigert S.Geometry of uncertainty relations for linear combinations of position andmomentum.J Physics A:Math Theory,2017,51(2):02530312,)c,“.Fock m5f?(?n.?“?,2017,53(2):133136Qu F F,Deng G T,Wen Z H.Uncertainty principles for the linear operators on Fock space.Journal ofBeijing Normal University,2017,53(2):13313613 Seip K,YoussfiE H.Hankel operators on Fock spaces and related Bergman kernel estimates.J Geom Anal,2013,23:17020114 Shi J,Liu X,Zhang N.On uncertainty principles for linear canonical transform of complex signals viaoperator methods.Signal Image Video Process,2014,8(1):859315 Wang X,Cao G,Zhu K H.BMO and Hankel operators on Fock-type spaces.J Geom Anal,2015,25(3):1650166516?,_.Fock.m?2(?n.n?,2020,40A(6):14091419Wu H G,Liang Y X.The generalized uncertainty principle on Fock-type space.Acta Math Sci,2020,40A(6):1409141917 Zhu K H.Analysis on Fock Spaces.New York:Springer-Verlag,2012The Sch odinger Uncertainty Relation in the Fock-Type SpacesLi WenxinLian PanLiang Yuxia(School of Mathematical Sciences,Tianjin Normal University,Tianjin 300387)Abstract:In this paper,the Sch odinger uncertainty relation for the unilateral weighted shift operatorson Fock space is established,and the explicit expression when the equality attained is given,whichfurther extends the Heisenberg uncertainty relation on Fock space established in 4 and overcomes thedifficulty in 16.In addition,we generalize the uncertainty relation to the multiple operators case.Anew uncertainty inequality in the form of non-self adjoint operators is obtained as well.Key words:Fock space;Uncertainty Principle;Unilateral weighted shift operator;Segal-Bargmanntransform;Derivation operator.MR(2010)Subject Classification:47B38;32A37