We stress that in contradiction with what happens in space dimensions $n \geq
3$, there is no strict bound on the number of bound states with the same
structure as the semi-classical estimate for large coupling constant and give,
in two dimensions, examples of weak potentials with one or infinitely many
bound states. We derive bounds for one and two dimensions which have the
"right" coupling constant behaviour for large coupling.Comment: Talk given by A. Martin at Les Houches, October 2001, to appear in
  "Few-Body Problems

Khuri, N. N.

Martin, A.

Wu, T. T.

English

arXiv

We stress that in contradiction with what happens in space dimensions $n \geq 3$, there is no strict bound on the number of bound states with the same structure as the semi-classical estimate for large coupling constant and give, in two dimensions, examples of weak potentials with one or infinitely many bound states. We derive bounds for one and two dimensions which have the "right" coupling constant behaviour for large coupling.We stress that in contradiction with what happens in space dimensions $n \geq 3$, there is no strict bound on the number of bound states with the same structure as the semi-classical estimate for large coupling constant and give, in two dimensions, examples of weak potentials with one or infinitely many bound states. We derive bounds for one and two dimensions which have the right coupling constant behaviour for large coupling

Khuri, N.N.

Martin, Andre

Wu, Tai Tsun

CERN Document Server

CERN-TH/2001-330BOUND STATES IN n DIMENSIONS (Especially n = 1 and n = 2)N.N. KHURIDepartment of Physics, The Rockefeller UniversityNew York, NY 10021, U.S.A.Andre´ MARTINTheoretical Physics Division, CERNCH - 1211 Geneva 23, SwitzerlandandLaboratoire de Physique Theorique ENSLAPPF - 74941 Annecy-le-Vieux, FranceTai Tsun WUGordon McKay Laboratory, Harvard UniversityCambridge, MA 02138-2901, U.S.A.andTheoretical Physics Division, CERNCH - 1211 Geneva 23, SwitzerlandTalk given by A. Martin at the Workshop“Critical Stability of Few-Body Quantum Systems”, Les Houches, October 2001to appear in “Few-Body Problems”.ABSTRACTWe stress that in contradiction with what happens in space dimensions n  3, there isno strict bound on the number of bound states with the same structure as the semi-classicalestimate for large coupling constant and give, in two dimensions, examples of weak potentialswith one or innitely many bound states. We derive bounds for one and two dimensions whichhave the \right" coupling constant behaviour for large coupling.CERN-TH/2001-330November 20011 IntroductionFirst, I would like to apologize for the fact that I am speaking on something which, in tworespects, is outside the subject of this workshop:(i) I speak of two-body bound states while we speak here of few-bodies which means at leastthree;(ii) the most important results I present are in one- and two-space dimensions.My excuse for (i) is that control of two-body bound states is useful for more body boundstates. For instance the proof that the system proton-electron-muon is unstable uses two-bodyresults. For (ii) it is well known that two-space dimensions are very often realized in condensedmatter physics.2 Large coupling constant behaviourIf we take a potential gV , the number of negative energy bound states in n dimensions is givenby the semi-classical expression for large gNtotal ’ cn gn/2∫dnxjV −(x)jn/2 , (1)where V − designates the negative part of the potential, and cn the semi-classical constant isgiven bycn =2−npi−n/2Γ(1 + n2)(2)This holds for any n, provided V is suciently smooth and suciently rapidly decreasingat innity. For n = 1, it was established by Chadan [1] and for general n by Martin [2] andsome years later by Tamura [3].3 Strict bounds for n  3, absence of bounds for n = 1, 2For n  3, it was established by Lieb [4], Cwikel [5] and Rozenblum [6], and later by others [7]that besides the asymptotic estimate (1) there is a strict bound of the same formNtotal < bn gn/2∫dnxjV −(x)jn/2 (3)where bn is strictly larger than cn, even for very large n, as shown by Glaser, Grosse and Martin[8].On the contrary, in one and two dimensions, the situation is drastically dierent.1For n = 1 and n = 2 an arbitrarily weak attractive potential has at least one bound state.For n = 1 this is easy to demonstrate using a gaussian wave function. For n = 2 it is also truebut more delicate [9]. The most elegant proof is given by Yang and De Llano [10] using a trialfunction exp−(r + r0)α, α very close to zero.It is in fact sucient, both for n = 1 and n = 2 to have an arbitrarily weak, globallyattractive potential, i.e., such that ∫V (x)dnx < 0 (4)to have a bound state.One can go further than that, i.e. construct examples in which an arbitrarily weak potentialhas innitely many bound states. One such example, inspired by Jean-Marc Richard, is, in twodimensionsV = −1∑n=1gnδ(jx− xnj − 1) (5)gn decreasing, g1 arbitrarily small, xn adequately chosen. One can manage to have (5) satisfying∫jV jd2x <1 , (6)and even∫jV j(ln(2 + jxj))1−d2x <1 , (7) > 0 arbitrarily small.One can construct other examples where the delta function is replaced by a nite circular ditch.However, this or these bound states which occur for arbitrarily weak potentials are incrediblyweakly bound. We have been able to show that the ground state energy,  = −κ2, satises, fora potential gVκ < exp−2pi − g∫ln+(1x)VR d2x− 12g∫ jV −jd2xg∫ jV −jd2x (8)where ln+(t) = ln(t) for t > 1, =0 for t < 1, and VR is the circular decreasing rearrangementof jV −j.If some of you do not know what is a circular decreasing rearrangement, let me say thatseeing Mont Blanc it is easy to understand. You take the \Carte Vallot". The Mont Blanchas level lines. You replace these level lines by circles centered at the origin enclosing the samearea as the original level lines. In this way you manufacture the rearranged Mont Blanc. Youmay have to group together disconnected level lines if necessary.In a recent preprint, Nieto [11] has calculated that in units where h = c = 2m = 1, asquare well of radius 1, with a strength 0.1 has a bound state with an energy of 10−18 which,incidentally, is a particular case of (8) except for the non-dominant terms.24 Bounds on the number of bound states in one dimen-sionHere and in the next sections we use extensively the fact that the number of negative energybound states is equal to the number of nodes of the zero energy wave function. In one dimensionlet x1x2 . . . xK . . . xK+1 . . . xN be these nodes and assume that xK < 0 < xK+1. Since theSchro¨dinger equation−d2ψdx2+ V (x)ψ = 0looks like a radial reduced three-dimensional equation with angular momentum zero, we canuse known bounds and apply them to the intervals −1 ! xK , xK+1 ! 1. So the Bargmanbound [12]N` <12`+ 1∫ 10r V −(r)dr(q) (9)which, for ` = 0 reduces toN0 <∫ 10r V −(r)dr ,gives usN −K − 1 < ∫1xK (x− xK)V −(x)dx<∫10 xV−(x)dxandK <∫ 0−1jxjV −(x)dxso thatN(m = 0) < 1 +∫ +1−1jxjV −(x)dx (10)Similarly, the bound obtained by one of us (A.M.) in three dimensions for ` = 0 [13]:N(3dim, ` = 0) <[∫ 10r2jV −jdr∫ 10jV −jdr]1/4(11)leads toN(m = 0) < 1 +p2[∫ +1−1xjV −jdx∫ +1−1jV −jdx]1/4(12)This bound has the advantage that if we replace V by gV it is linear in g, like the semi-classicalestimate for large g.5 Bounds in two-space dimensionsWe start with the radial case for which the reduced Schro¨dinger equation at zero energy is[− d2dr2+m2 − 1/4r2+ V (r)]u = 0 (13)3It is in fact more convenient in the m = 0 case to work directly with the non-reduced form:−d2ψdr2− 1rdψdr+ V (r)ψ = 0 (14)Details of our aproach will be given in a future publication. Let me just say that we integratethe Schro¨dinger from one node to the next and nd a lower bound for the quantities∫ xK+1xKln(xxK)V −(x)x dxand ∫ xK+1xKln(xK+1x)V −(x)x dx .From this we getN(m = 0) < 1 +∫ 10r∣∣∣∣ln(Rr)∣∣∣∣ jV −(r)j dr (15)where R is arbitrary. If we callI(R) =∫ 10rj ln(Rr)j jV −(r)jdr (16)we can minimize with respect to R, and get Imin = I(R0) such that∫ R00rjV −(r)jdr =∫ 1R0rjV −(r)jdr (17)It is interesting to compare Imin with the bound on N(m = 0) obtained years ago by Newton[14] and Seto^ [15].N(m = 0) <12∫r dr r0 dr0 jV −(r)j(V −(r0)j j ln(rr′)j∫r dr jV −(r)j (18)It is easy to see, from the mean value theorem that the right-hand side of (18) is larger than1/2Imin. It is less trivial to prove that it is less or equal to Imin. So we get12Imin < N(Newton − Seto^)  Imin (19)which means that for m = 0 our result is not quite as good, but simpler. However, both boundsare optimal in the sense that one can approach arbitrarily close to saturation.For m 6= 0, the easiest thing to do is notice that the equation (13) looks like a radial reducedSchro¨dinger equation in three dimensions with an angular momentum ` = m− 1/2. Thus, form 6= 0 we can use the Bargmann bound, which is valid not only for integer ` but any real` > −1/2. However, if we want to calculate the TOTAL number of bound states, we nd,summing up the contributions from the various values of m that if we take a potential gV wend a bound which behaves like g ln g for large g, i.e., increases qualitatively faster than the4semi-classical bound. To avoid this we use a trick which was invented many years ago by Glaser,Grosse and one of us (A.M.) [8]. We transform Eq. (13) by making the change of variablesz = ln r, v = r2V , (20)into (− d2dz2+ v(z))φ(z) = −(m2 − 14)φ(z) (21)this is a one-dimensional Schro¨dinger equation whose eigenvalues areei = −(m2i −14)(22)Each mi (seen from the point of view of Eq. (13)), is on a \Regge trajectory", which meansthat if we disregard the m = 0 bound states we have bound states at1, 2, . . . , [mi]− 1, [mi]where [mi] is the integer part of mi.Each bound state has a multiplicity two corresponding to jmj. Hence the total number ofm 6= 0 bound states is less than2∑mi < 2 2p3∑ jeij1/2 , (23)since ei/m2i  3/4. Finding a bound on the right-hand side is a standard moment problem inone dimension rst considered by Lieb and Thirring [16]. Weidl [17] has been able to prove theinequality ∑ jeij1/2 < C∫jv(z)jdz ,and Hundertmark, Lieb and Thomas [18] have found the best possible constant in this inequality∑ jeij1/2 < 12∫ +1−1jv(z)j dz (24)Hence using (20), (23) and for instance our bound (15) for m = 0 we get for the totalnumber of bound states in n potential gVNTOT < 1 + g∫r drj ln(Rr)j jV −rj+ g2p3∫r drjV −(r)j (25)i.e., a bound linear in g.5We can get now a bound on the total number of bound states for a potential withoutsymmetry: choose an origin and deneB(r) = −infθV (r, θ)B(r) = 0, ifinfθV (r, θ) > 0 (26)(with obvious notations in V ).From the monotonicity of the eigenvalues with respect to the potential we getNTOT < 1 + g∫r drj ln(Rr)jB(r)+ g2p3∫r dr B(r) (27)It is clear that this bound will be good if V has only one singular point which can be chosenas origin. Otherwise it may be very bad or even meaningless. Our conjecture is that there mustbe a bound likeNTOT < 1 + C1∫d2x VR(x)(ln jx0xj)++ C2∫d2xjV −(x)j(ln j xx0j)++ C3∫d2xjV −(x)jwhere VR is the rearrangement of V−.6 AcknowledgementsWe are grateful to K. Chadan, J.-M. Richard and W. Thirring for crucial informations.References[1] K. Chadan, Nuovo Cimento 58A (1968) 191.[2] A. Martin, Helv.Phys.Acta 45 (1972) 140[3] H. Tamura, Proc. Japan Acad. 50 (1974) 19.[4] E. Lieb, Bull.Am.Math.Soc. 82 (1978) 751, and Proc. AMS, Symp. Pure Math. 36 (1980)241.6[5] M. Cwikel, Trans. AMS 224 (1977) 93.[6] G.V. Rozenbljum, Dokl. AN SSSR 202 NS (1972) 1012;Izv. VUZO Mathematika N1 (1978) 75.[7] P. Li and S. T. Yau, Comm. Math. Phys 88 (19983) 309;Ph. Blanchard, J. Stubbe and J. Rezende, Lett. Math. Phys. 14 (1987) 215;J.G. Conlon, Rocky Mountain J. Math. 15 (1985) 117.[8] H. Grosse, V. Glaser and A. Martin, Comm.Math.Phys. 59 (1978) 197.[9] B. Simon, Ann.Phys. 97 (1976) 279.[10] K. Yang and M. de Llano, Am. J. Phys 57 (1989) 85.[11] M. Nieto, hep-th/0106233, LA-UR-01-3190 Preprint, Theoretical Division, Los AlamosNational Laboratory.[12] V. Bargmann, Proc. Nat. Acad. Sci. USA 38 (1952) 961.[13] A. Martin, Comm.Math.Phys. 55 (1977) 293.[14] R.G. Newton, J. Math. Phys. 3 (1962) 867.[15] N. Seto^, Publ. RIMS, Kyoto Universisty (1979) 429.[16] E.H. Lieb and W. Thirring, in \Studies in Mathematical Physics", essays in honour of V.Bargmann, E. Lieb, B. Simon, A Wightman eds., Princeton University Press, Princeton(1978), p. 269.[17] T. Weidel, Comm.Math.Phys. 178 (1996) 135.[18] D. Hundertmark, E.H. Lieb and L.E. Thomas, Adv. Theor. Math. Phys. 2 (19998) 719.7

Bound States in n Dimensions (Especially n = 1 and n = 2)

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