312Scattering theory8.5TheLevinsontheorem313whereHthotandardnotationforHanetfunctionsotfrkindThematching condition at ro forthe logarithmic derivativeof theand order X.Thus, the tighit-hand aide of the matching conditiou (8.5.12)fisnction is,ofcourse,readsasM191s(Ma1981y(20)1aWra+1[A(r)(r.0)C(8.5.17)2roH(icro)Far1AsE-0-,theright-hand sideof (8.517)reducesto(Ma RnySome limiting casesof Eq.(8.5.6)are easily dealtwith,Forexampls free particle,which corresponds to avanishing value ofμ,one findat(A+)告=PA(8.5,18)being the Bessel function of first kind and order )whereasittends.to.KasEoo.Moreover,the solution (8.5.138)wnatr.-VekrJa(ar),(8.5)satisfies.the conditionwhenE>0,andCketacelt18w(iro)1[g(0.0)oe(n0)(8.5.19)JA(iATO)2roSERH(Mar)(j)4for which the right-hand side tends toMamwithxam+In the interval [re（+）=成E-0-(8.5.20)ofEtwooscllatingsolutions.ofEq-(8.5.6)exiet,sothatthegensolution readsMa9）（）19850Thus, no bound state exists when μ = 0.IfAx(oa)decreasesacromethevaluepx asμincreases,anoverlspmkr[a(kr)ce6a(k,u)-Na(kr)sinba(k,p)],(83VE,A(r,P) =1betweentwo ranges of variation of thelogarithmicderivative on the twosides of ro oocurs.Bearing in mind the SturmLiouville theorem expressedwhere &a(k,p) is the phase shift and N is theNeumann function ofnby (8.5.10) and (8.5.11),such an overlap means that thematching condi1.Thematching conditiou (8.5.12)leadsto a veryuseful formulafocttion (8.5.12) can only be sstisfied by one particular value of the energy,phase shift, upon using the result (8.5.14),Le.9and hence a scattering state is turned into a bound state.In general, as(kro) [Aa(E,n) -(-双)(I1μ increases fromOtol,each.timeA,decreasee acrosspa,a scatteringtanda(k,u)m(8.5.15)astate isturned into a bound statefor the above reasons.In contrast,eachMA(O[A(E,)--)time Ax(O,p)increnses across pa,abound stateisturned intoa scatter-ing state.The number of bound states is then equal to the number ofEquation (8.5.15)providesthekeytool for proving theLevinson thgtimee.thatAx(0)decreasesncrouspaasyrangesfromtolminustherem,jointly witha careful analysis ofmatching conditions.Here wesbammber of times thnt A(o) incresses scros the.valne Pa.Thenexttaskrequire that the phase shift is determined with respect to the phase slinis now to provethat this difference equalsox(o),the phuse shift at zeroa(k,0)forafree particle,whre,bydefinition,onechooses&a(k)=momentum,divided bys.With such a convention, thephase shift is determined completely ssFor this purpose we have to eyalitate tana(k.p)whenk increasesfrom 0to1.becauseIr E≤0, the only square-integrable solution of Eq.(8.5.6) is(2)(Malars)8a(0 n)-m ba(ku).VE.AA+ORFKH(iKr),(8.5.10)11
11 (20)
Scatteringtheory314By virtue of theexact result (8.5.15)and of the limitingbehaviourofBesselfunctions at small argument, onefinds,to lowest order in kro,[A(0,以)-(+(kro)24(Ma 198s)(15b)tanox(c.)-22A(>)A(O.+(X-量）告A(0.m)+(X-(ero)2A(8.5.21)XAP(A)A(0)-e)where we have neglected,for simplicity,higher-order terms inkro in thedenominator.Notethettan 5y(k,p)tends to0 ask0 (since入≥0),and hencea(o,p) is always equal to an integer muiltipleof .This means that thephase shift changes discontinuously.Besides, the exact formula (8.5.15)(Ma 1983)（24)式前面can beused to provethat thephase shift increases monotonically as thelogarithmicderivativeA(E,)decreases.Thus,&(O,)jumpsbyifforksuficientlysmall,tanSa(k)changessignasA(E)dcreases.summary,whenμrangesfrom0to1,wheneverAa(0,μ)decreasesfromnear and larger than the value px to smaller than pa,thedenominstarin (8.5.21)changes signfrompositivetonegative,jeadingtoa jumpof6(0μ)equsl.to.w.Incontrast,wheneverA,(O.μ)incressesacrosspx,(Ma 1985)(26)6a(0.j) jumpsby-.Bearing in mindwhatwesaid afterEq.(8.5.20),we conclude that &x(O) divided by is indeed equal to the number ny afbound states:2615k(0) =nam,(Ma1)(8.522)which ls.theform ofthe Levinson theorem for.central potentials in R3obeying (8.5.2)and (8.5.3) (Ma1985).(Ma 1985).8.e Scatteringfrom singular potentialsThe consideration of singular potential scattering equations was moti-vatod, in the19Ds,by theneed to obtain new ideas and techniques thstcould beextendedtoquantumfieldtheory(Bastai etal.1963,KhuriandPals 1964,DeAlfaro and Regge1955,Calogero1967,Frank et al.1971,Graeber andDarr1977,Enss 1979,Dolinszky1980),especially forthecaee of fieldtheorieswhere conventional perturbative methods fail to provideconsistentpicture.Ourrender,however,isnotassumedtoknogquantum feld theory,and he/she will not need itto understand what wearegoingtosay.The remarkable feature of singular potentials (which can be used toiefine them)is that they lead to differential equations with non-Fuchsiansingularities (c.appendix5.B)bothasr→0andasroo,Weare12
12 (15b) (26) (24)式前面
Jost函数方法证明Levinson定理讨论有球对称势的薛定方程R"(k,r) + [k2 - ^U(r) - l(l + 1)r-2 R(k,r) =0k2 = 2ME/h2,U(r) = 2MV(r)/h2drr|U(r)l < 0,0U(r）在原点比r-2更少奇异在无穷远比r收敛更快13
13 Jost 函数方法证明Levinson定理 U(r)在原点比 更少奇异 在无穷远比 收敛更快 讨论有球对称势的薛定谔方程 −2 r −2 r
Jost函数方法证明Levinson定理讨论有球对称势的薛定方程R"(k,r) + [k2 - ^U(r) - l(l + 1)r-2 R(k,r) =0k2 = 2ME/h2,U(r) = 2MV(r)/h2αdrr2|U(r)/ < drr|U(r)l < 0,100U(r）在原点比r-2更少奇异在无穷远比r-3收敛更快14
14 Jost 函数方法证明Levinson定理 U(r)在原点比 更少奇异 在无穷远比 收敛更快 讨论有球对称势的薛定谔方程 −2 r