INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 37 (2004) 11243–11257
PII: S0305-4470(04)82171-1
Critical strength of attractive central potentials Fabian Brau1 and Monique Lassaut2 1 Groupe de Physique Nucl´ eaire Th´eorique, Universit´e de Mons-Hainaut, Acad´emie Universitaire Wallonie-Bruxelles, B-7000 Mons, Belgique 2 Groupe de Physique Th´ eorique, Institut de Physique Nucl´eaire, F-91406 Orsay CEDEX, France
E-mail:
[email protected] and
[email protected]
Received 16 June 2004, in final form 24 September 2004 Published 3 November 2004 Online at stacks.iop.org/JPhysA/37/11243 doi:10.1088/0305-4470/37/46/010
Abstract We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yield several sequences of upper and lower limits on the critical value, gc() , of the coupling constant (strength), g, of the potential, V (r) = −gv(r), for which a first -wave bound state appears, which converges to the exact critical value. PACS numbers: 03.65.−w, 03.65.Ge, 02.30.Rz
1. Introduction Since the pioneering works of Jost and Pais in 1951 [1] and Bargmann in 1952 [2], the determination of upper and lower limits on the number of bound states of a given potential, having spherical symmetry V (r), in the framework of non-relativistic quantum mechanics is still of interest. A fairly large number of results of this kind can be found in the literature for the Schr¨odinger equation (see, for example, [3–18]) and for results applicable to one and two dimension spaces (see, for example, [19–23]). An important theorem for classifying these results was found by Chadan [8] and gives the asymptotic behaviour of the number of -wave bound states as the strength, g, of the central potential V (r) = gv(r) goes to infinity: g 1/2 ∞ dr v − (r)1/2 as g → ∞, (1) N ≈ π 0 where the symbol ≈ means asymptotic equality, V − (r) = gv − (r) and v − (r) = max(0, −v(r)) (see also [24] for a generalization of relation (1)). This result implies that any upper and lower limit on N which could yield cogent results should behave asymptotically as g 1/2 . More importantly, relation (1) gives the functional of the potential, that is to say, the coefficient in front of g 1/2 that appears in the asymptotic behaviour. Upper and lower limits on the number 0305-4470/04/4611243+15$30.00 © 2004 IOP Publishing Ltd Printed in the UK
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of -wave bound states featuring the correct g 1/2 dependency was first obtained in [6]. Upper and lower limits on N featuring the correct asymptotic behaviour (1) was first derived in [17, 18]. In practice, the asymptotic regime is reached very quickly when the strength of the potential is large enough to bind two or three bound states. The situation is completely different when one considers the transition between zero and one bound state. In contrast to one and two dimension cases where any attractive potential, satisfying adequate integrability conditions, has at least one bound state, in the three-dimensional case, the potential acquires a bound state only if it is attractive (negative) enough. Thus, for central potential, for example, there exists a ‘critical’ value, gc() , of the coupling constant (strength), g, of the potential, V (r) = gv(r), for which a first -wave bound state appears. The determination of this critical value requires to solve the zero energy Schr¨odinger equation [3, 25, 26]. To circumvent the exact calculation of the Jost function at zero energy, upper and lower bounds are very helpful. From now on, use will be made of the standard quantum-mechanical units h ¯ = 2µ = 1 where µ is the reduced mass of the particles. In 1976, Glaser et al obtained a strong necessary condition for the existence of a -wave bound state in an arbitrary central potential in three dimensions [9]: ∞ dr 2 − (p − 1)p−1 (2p) [r V (r)]p 1. (2) (∀p 1) 2p−1 p 2 (2 + 1) p (p) 0 r This relation yields a lower limit on the critical value gc() , by making a minimization over p 1, which was shown to be very accurate (see, for example, [9, 16, 28]). Other necessary conditions for the existence of bound states can be found in the literature (see, for example, [7, 11, 28] and for reviews see [10, 17, 18]), but in general, the relation (2) yields the strongest restriction on gc() (in some cases, the relations obtained in [28] can, however, be better). Sufficient conditions for the existence of bound states, yielding upper limits on the critical value of the strength of the potential, are scarcer. Two sufficient conditions for the existence of at least one bound state with angular momentum have been found by Calogero in 1965 [5, 6] ∞ R dr r|V (r)|(r/R)2+1 + dr r|V (r)|(r/R)−(2+1) > 2 + 1, (3) (∀R > 0) 0
R
and
(∀R > 0)
∞
R
dr|V (r)|[(r/R)2 + (r/R)−2 R 2 |V (r)|]−1 > 1.
(4)
0
These two conditions apply provided the potential is nowhere positive, V (r) = −|V (r)|. The most stringent conditions are obtained by minimizing the left-hand sides of (3) and (4) over all positive values of R. Some other sufficient conditions for the existence of bound states can be found in the literature (see, for example, [16, 28] and for reviews see [10, 17, 18]). A sufficient condition which does not require the spherical symmetry for the potential V was proposed in 1980 by Chadan [29] (see also [30]). When the potential is central and purely attractive, the inequality Tr K (2) Tr K (1) ,
(5)
K (1) (r, r ) = inf(r, r )+1 sup(r, r )− V (r ) ∞ K (n) (r, r ) = ds K (1) (r, s)K (n−1) (s, r ),
(6)
where
0
Critical strength of attractive central potentials
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implies the existence of at least a bound state for the potential V (r). In the case where V (r) has some changes of sign, condition (5), is replaced by Tr K (4) Tr K (2) ,
(7)
which implies that one of the potentials ±V (r) has at least one bound state. Recently, an upper limit on the critical strength has been found, originating from a variational technique [31]: −1 ∞ ∞ y gc() λ dxF (2p − 1; x) dy F (p; y)y −λ dz F (p; z)zλ , (8) 0
0
0
with F (q; x) = x v(x) , v(x) 0, λ = + 1/2 and q > 0 which was found to be very accurate. Clearly more accurate upper limits could be obtained but depending strongly on the choice of the trial wavefunction. This paper follows a different scheme. To circumvent the difficulty to guess a trial (strongly potential dependent) wavefunction for variational methods we propose upper and lower limits originating from iterative procedures, all designed to converge towards the exact result. The methods proposed by Chadan enter this category of iterative convergent procedures. Also, Lassaut and Lombard [16] worked in this sense some years ago, but the procedure was ∞ constrained by the condition gc() 0 dr r|V (r)| < 2(2 + 1), which is not verified when the convexity of the potential is too high or when the angular momentum increases. We thus propose in this paper sequences of upper and lower limits of the critical value, gc() , which converge towards the critical coupling constant without any restriction on the possible values of gc() . The advantage of our procedure, based upon the Riesz theorem [32, 33] in what concerns the upper limits, is that there is no need to start from a conveniently chosen wavefunction. Indeed, the improvements are simple for these sequences: one just needs to calculate the next order. Note that the basic idea of sequences of lower limits for gc() have already been explored in [28]. The paper is organized as follows. In section 2 we derive the upper limits on the critical value gc() . In section 3 the algorithm for generating both lower and upper limits is discussed. In section 4 our proposal of upper and lower bounds are tested against the exact values for common potentials. Our conclusions are presented in section 5. q
(q+1)/2
2. Upper limits on the critical strength From now on, we assume that V (r) is locally integrable and such that ∞ dr r|V (r)| < ∞,
(9)
0
remembering that we consider purely attractive potentials namely satisfying V (r) 0. Following Birman and Schwinger [3, 25, 26] the critical values of the strength of the potential correspond to the occurrence of an eigenstate with a vanishing energy. In this paper we consider the zero energy Schr¨odinger equation that we write into the form of an integral equation incorporating the boundary conditions ∞ dr g (r, r )V (r )u (r ), (10) u (r) = − 0
where g (r, r ) is the Green function of the kinetic energy operator and is explicitly given by g (r, r ) =
1 r +1 r − , 2 + 1 < >
(11)
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where r< = inf(r, r ) and r> = sup(r, r ). An important technical difficulty appears if the potential possesses some changes of sign (see relation (12) below). This is overcome for the derivation of necessary conditions, or of upper bounds on the number of bound states, by replacing the potential by its negative part V (r) → V − (r) = max(0, −V (r)). Indeed, the potential V − (r) is more attractive than V (r) and thus a necessary condition for the existence of bound states in V − (r) is certainly a valid necessary condition for V (r). This procedure can no longer be used to obtain sufficient conditions. For this reason we consider potentials that are nowhere positive, V (r) = −gv(r), with v(r) 0. To obtain a symmetrical kernel we now introduce the function ψ (r) as follows ψ (r) = v(r)u (r). (12) Equation (10) becomes
∞
ψ (r) = g
dr K (r, r )ψ (r ),
(13)
0
where the symmetric kernel K (r, r ) is given by K (r, r ) = v(r)g (r, r ) v(r ).
(14)
Relation (13) is thus an eigenvalue problem with a symmetric kernel and, for each value of , the smallest characteristic numbers are just the critical values gc() . The higher characteristic numbers correspond to the critical values of the strength for which a second, a third, . . . , -wave bound state appears. The kernel (14) acting on the Hilbert space L2 (R) is a Hilbert–Schmidt kernel [34] for the class of potentials defined by (9), i.e. satisfies the inequality ∞ ∞ dx dy K (x, y)K (x, y) < ∞. (15) 0
0
Consequently the eigenvalue problem (13) always possesses at least one characteristic number [35] (in general, this problem has an infinity of characteristic numbers). We propose now to solve the eigenvalue problem (13) using iterative methods. 2.1. Iterative power method Let us write, for simplicity, relation (13) under the form ψ = g K ψ ,
(16)
where K denotes the symmetric linear operator, operating on the Hilbert space L (R), which is in this paper the integral operator generated by the so-called Birman–Schwinger [3, 25] kernel K , equation (14). Since the kernel K (r, r ) is Hilbert–Schmidt, K is a compact operator [27]. As K is symmetric the Riesz theorem applies [32, 33]. For each value of the angular momentum , the set of eigenvalues 1/gp , 1 p, (which in the present case are all positive) can be ordered according to a sequence tending to zero, 1/g1 1/g2 · · · 1/gp · · · 0. There exists an orthonormal basis in L2 (R), labelled by ϕp (r), p 1, each ϕp (r) being associated with 1/gp , and for each function φ (r) ∈ L2 (R) 2
∞ ∞ ∞ 1 K φ = K φ |ϕp ϕp = φ |K ϕp ϕp = φ |ϕp ϕp , (17) g p=1 p=1 p=1 p ∞ where the symbol f |g denotes the scalar product 0 dr f (r)g(r). For the sake of simplicity we have dropped the indices () which should appear on gp and on ϕp (r).
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The positivity of the eigenvalues originates [33] from the fact that K is positive i.e. φ |K φ 0. (18) (∀φ ∈ L2 (R)) In the present case, where the potential has the spherical symmetry, there is no degeneracy and we have strict inequalities for the eigenvalues 1 1 1 > > ··· > > · · · > 0. (19) g1 g2 gp This is due to the fact that the eigenstates are solutions of a linear second order equation with constraints at the origin and infinity. Now, we introduce the iterated kernel K(n) (s, t) of K (s, t) ∞ (n) K (s, t) = duK (s, u)K(n−1) (u, t) n 2, (20) 0
with K(1) (s, t) = K (s, t).
(21) (n) K φ ,
We can then compute the scalar product between φ and ∞ (n) (n) 1 φ |ϕp 2 . φ K φ = K φ φ = n g p=1 p
and find (22)
Therefore, we obtain the following convexity-type relation:
∞ (n+1) (n−1) (n) 2 1 1 1 2 φ K φ φ K φ − φ K φ = − φ |ϕp 2 φ |ϕq 2 , n−1 g (g g ) g p q p q p 0 2 βn+2 βn βn+1 .
(A.9)
Then, for every λ > 0, the series G(g) =
∞
βp g p ×
p=0
∞
λq g q =
q=0
∞
γn g n
(A.10)
n=0
convergent for g < R˜ = inf(R, 1/λ) is such that γ0 = 1 and has the property (∀n 0) (∀n 0)
γn > 0 2 γn+2 γn γn+1 .
(A.11)
The proof of the lemma originates from the definition of γn γn =
n
βp λn−p
(A.12)
p=0
which asserts that γ0 = 1 and γn > 0 since λ and βn are positive (see (A.9)). The second property in (A.11) is satisfied if and only if the following relation is satisfied for n 0: 2 n+2 n+1 n βp λn+2−p βq λn−q − βp λn+1−p 0. (A.13) p=0
q=0
p=0
Inequality (A.13) is equivalent to the requirement that (βn+2 − βn+1 λ)
n
2 βp λn−p − βn+1 0,
(A.14)
p=0
which is manifestly satisfied when (∀n, 1 p n)
βn+2 βp − βn+1 βp+1 0.
(A.15)
Relation (A.9) and the positivity of βn imply that this latter inequality, which originates from the iteration of (A.9) when n is lowered up to p, is verified. Now we use the fact that, for any g < g1 = gc(0) the Jost function can be written as [27] ∞
g 1− (A.16) f0 (g, 0) = gn n=1
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where the gn still denote the characteristic numbers of the eigenvalue problem considered. −p converges for every integer Product (A.16) exists for g < g1 when each series ∞ n=0 gn p 1, which is true when the (positive) trace of the iterated kernel K0(n) (r, r ) is finite. Equation (A.16) implies that for any g < g1 we have ∞ N ∞
∞
g p g p 1 = lim SN SN = . (A.17) = N→∞ f0 (g, 0) n=1 p=0 gn gn n=1 p=0 Since all the quantities in (A.17) are positive we can write, for g < g1 SN =
∞
sp(N) g p
(A.18)
p=0
where s0(N) ≡ 1. Now assuming that for some N, the following property holds:
(N) 2 (N) (N) sp − sp+1 0 (∀p 0) sp+2
(A.19)
according to the lemma, the property still holds for N + 1 as well. Note that the radius of p convergence R˜ of the lemma is always minorated by g1 > 0. For N = 1, sp(1) is simply 1 g1 p k−p and property (A.19) is valid. For N = 2, sp(2) = k=0 g1−k g2 and it can be verified that property (A.19) is again valid. Therefore, for every N 2 property (A.19) is also valid and in particular for N going to infinity. The comparison between (A.2) and (A.17) shows that ψn (0) = sn(∞) . From relation (A.19) we obtain (∀n 0)
(0) (0) ψn+1 ψn+2 , ψn+1 (0) ψn (0)
(A.20)
which, with definition (46) of αn , proves that the sequence αn is decreasing. Since αn is decreasing and positive it converges towards some α 0. On the other hand we know that the radius of convergence of the series 1/f (g, 0) is g1 . Using the d’Alembert rule (0)/ψn (0) = α for the series of positive numbers (ψn (0))n0 we have 1/g1 = limn→∞ ψn+1 (0) so that αn converges towards 1/g1 = 1/gc . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Jost R and Pais A 1951 Phys. Rev. 82 840 Bargmann V 1952 Proc. Natl Acad. Sci. USA 38 961 Schwinger J 1961 Proc. Natl Acad. Sci. USA 47 122 Calogero F 1965 Commun. Math. Phys. 1 80 Calogero F 1965 J. Math. Phys. 6 161 Calogero F 1965 J. Math. Phys. 6 1105 Calogero F 1965 Nuovo Cimento 36 199 Chadan K 1968 Nuovo Cimento A 58 191 Glaser V, Grosse H, Martin A and Thirring W 1976 Studies in Mathematical Physics—Essays in Honor of Valentine Bargmann (Princeton, NJ: Princeton University Press) p 169 Simon B 1976 Studies in Mathematical Physics—Essays in Honor of Valentine Bargmann (Princeton, NJ: Princeton University Press) p 305 Martin A 1977 Commun. Math. Phys. 55 293 Lieb E H 1980 Proc. Am. Math. Soc. 36 241 Chadan K, Martin A and Stubbe J 1995 J. Math. Phys. 36 1616 Chadan K, Kobayashi R, Martin A and Stubbe J 1996 J. Math. Phys. 37 1106 Blanchard Ph and Stubbe J 1996 Rev. Math. Phys. 8 503 Lassaut M and Lombard R J 1997 J. Phys. A: Math. Gen. 30 2467 Brau F and Calogero F 2003 J. Math. Phys. 44 1554 Brau F and Calogero F 2003 J. Phys. A: Math. Gen. 36 12021
Critical strength of attractive central potentials [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
[38] [39]
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Newton R G 1962 J. Math. Phys. 3 867 Glaser V, Grosse H and Martin A 1978 Commun. Math. Phys. 59 197 Newton R G 1983 J. Operat. Theor. 10 119 Aktosun T, Klaus M and van der Mee C 1998 J. Math. Phys. 39 4249 Chadan K, Khuri N N, Martin A and Wu T T 2003 J. Math. Phys. 44 406 Martin A 1972 Helv. Phys. Acta 45 140 Birman S 1961 Math. Sb. 16 63 Simon B 1971 Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (Princeton, NJ: Princeton University Press) Newton R G 1982 Scattering Theory of Waves and Particles 2nd edn (Berlin: Springer) Brau F 2003 J. Phys. A: Math. Gen. 36 9907 Chadan K 1980 Views on Contemporary Physics (Paris: Editions du CNRS) p 29 Chadan K and Grosse H 1983 J. Phys. A: Math. Gen. 16 955 Brau F 2004 J. Phys. A: Math. Gen. 37 6687 Schwartz L 1970 Topologie G´en´erale et Analyse Fonctionelle (Paris: Hermann) Riesz F and Nagy B 1953 Functional Analysis (New York: Frederick Ungar) Boll´e D, Chadan K and Karner G 1986 J. Phys. A: Math. Gen. 19 2337 Tricomi F G 1965 Integral Equations (New York: Interscience) pp 102–6 Mikhlin S G and Smolitskiy K L 1967 Approximate Methods for Solutions of Differential and Integral Equations (New York: Elsevier) p 276 Kolom´y J 1960 Comment. Math. Univ. Carolina 1 18 ˇ Pˇest. Mat. 86 148 Kolom´y J 1961 Cas. ˇ Pˇest. Mat. 86 308 Kolom´y J 1961 Cas. Riesz F 1910 Math. Annalen 69 449 Riesz F 1910 Math. Annalen 33 232 Tricomi F G 1965 Integral Equations (New York: Interscience) p 118
