Upper and lower limits on the number of bound states in nonrelativistic quantum mechanics

(S-wave bound states)

Low level introduction:

Roughly speaking, a bound state exists between two bodies if the distance between them stays finite for all time. Think for example to the system Sun-Earth well described by classical mechanics (no need to think about general relativity at this level). If this two bodies stay "close" to each other this is because the Sun act on the Earth through gravitational force. In many case, the force, which is a vector, derives from a potential which is a scalar function. In the case of the gravitational force, this potential depends only on the distance between the two bodies. So the potential will be an ordinary function that will depend on the coordinate only as a function of

The (effective) potential that act on the Earth has the following shape (suppose the Sun fixed during the motion, and think that only the Earth move around it, because normally the motion is around what is called the center of mass of the system, but since the sun is much heavier than the Earth, the center of mass coincide almost with the position of the Sun):

Given the initial conditions, the energy will be fixed. This is the horizontal curve draw inside the potential curve. In classical mechanics, the body can only move where its energy is above the potential (simply because E - Veff is proportional to the square of the impulsion of the body, and then this quantity has to be positive). So the motion is restricted between r- and r+. The generic shape for the trajectory of the Earth is an ellipse. r- and r+ are the two radii of the ellipse. If the energy is above zero, then the shape is a branch of hyperbole. Indeed (see on the graph), the body comes from r = infinity, is attracted toward the Sun until r = r- and then move away to infinity, this is a scattering state is contrast to a bound state, like most comets. There is two special curves. One is when the energy coincides with the minimum of Veff. In that case (see on the graph), r- = r+, and the curve is a circle. The other one is when the energy is just equal to zero, then we have a parabola. In this two last cases, the initial conditions have to be chosen with an infinite precision; this is more a mathematical limit than a physical situation.

The important message of this introduction is that there is no restriction on the energy (the horizontal line)  in classical mechanics for bound states (except that min Veff < E < 0). That is to say that the mean value of the distance between the two bodies can be whatever you want; it depends only on the initial conditions. In quantum mechanics, the energy is discretized and only special values can exist. That is to say that, in a semi-classical language, only special values for the mean distance between the two bodies can exist. Moreover, if the potential decrease at infinity more rapidly than 1/r^2, then there exist only a finite number of these special values for the energy.

The goal is then to try to find upper and lower limits on this finite number of possible energies just by knowing the potential, that is to say without solving the problem, without solving the Schrödinger equation.

Conditions on the potential to have a finite number of bound states:

We will only consider space with 3 spatial dimensions and potentials that depend only on the distance r between the two particles (central potentials). The condition for the existence of a finite number of bound states is given by these two relations:

where epsilon is an arbitrary positive number. In particular, we see that a potential 1/r has an infinite number of bound states. This potential is just the electric potential between a proton and an electron like in the hydrogen atom; there are then an infinite number of bound states in this case, this is well known.

We will always use units such as

[It is always possible to have such units. The simplest example is when you set c = 1, where c is the velocity of light. This simply means that you measure all the velocity in comparison with the velocity of light. So v = 0.5 simply means that your velocity is half the velocity of light].

Historical survey:

There exists a large amount of results concerning upper and lower limits on the number of bound states in a central potential; we will only present the most important ones. We will also restrict to the case of a vanishing angular momentum (the centrifugal potential equals zero). So in the following, N, means the number of S-waves bound states.

The first upper limit was obtained by BARGMANN in 1952 [1]. In 1961, SCHWINGER [2] reobtained this upper limit but using a more general method. This upper limit is nowaday called BARGMANN-SCHWINGER upper limit (BS in the graphs below) and it reads:

where the minus sign on the potential means that we consider only the negative par of the potentiel by setting its positive part equal to zero.

In 1965, CALOGERO [3] and COHN [4] have obtained for a monotonous potential (and thus completely negative) the following upper limit, so-called CALOGERO-COHN upper limit (CC in the graphs below),

This formula is clearly better then the previous one, at least when there are enough bound states, since this last formula exhibit the correct dependence on the strength, g, of the potential: it behaves like the square root of g

Indeed, it has been shown by CHADAN in 1968 [5], that if the potential is written as

Then behaves as follow as g goes to infinity (this is an exact asymptotic behavior)

We see that the CALOGERO-COHN upper limit is particularly interesting since it exhibits the correct integral operator. There is only the factor 2.

[I have then decided to study the proof of Calogero to see if some improvement were possible. The 27 mars 2002, I have sent en e-mail to Calogero to work on this problem together; I had arguments to convince him that indeed it was probably possible to remove this factor 2 (with eventually some price to pay). This is not a trivial thing since his upper bound were sharp (best possible in the sense of Bargmann), this means that there exist a potential (a square well in this case) that turns the inequality into equality. So it was not obvious a priori that this upper bound could be improved. Indeed, I first wrote to André Martin (see below) to work on this problem and he used this argument to say that it was impossible to improve the CALOGERO-COHN upper bound and declined my invitation to work on this problem. I will not enter into details but his argument is not correct.]

In 1965, Calogero obtained a lower limit on the number of S-wave bound states, called CALOGERO lower limit (C in the graphs below) [3]. This limit is applicable to all potential having a finite number of bound states but we give here only the version that applies to monotonous potentials

where rho is the solution of the equation

It is clear that rho does not depend on the coupling constant, g, on the potential; consequently the lower limit has the correct law behavior for g.

There exists a weaker version of this lower limit applicable to monotonous which are regular at the origin (C0 in the graph below)

In 1977, Martin obtained an upper limit on the number of S-wave bound states, called MARTIN upper limit (M in the graphs below) [6]. This upper limit is applicable to all potential such as its negative part is integrable from 0 to infinity.

this upper limit has the correct law behavior for g. This conclude our historical survey

New Results for monotonous central potential

The results I have obtained in collaboration with Calogero were derive in spring 2002, the paper was written in August and finally published in 2003 [7]. This paper treat only monotonous central potential, generalisation to larger classes of potentials can be found in [8,9]. We start with an upper limit

where the two quantity p and q are define with these two relations

and

These two relations are unambiguous definition of p and q provide the potential possesses at least one zero energy bound states since a necessary condition for the existence of bound states is (this is the CALOGERO upper limit with N = 1)

So we have an upper bound with the correct asymptotic behavior. The price to pay is the logarithmic term which contains these two radii p and q (remark that when the potential is regular at the origin V(p) can be replaced by V(0), the upper limit is then less stringent but simpler). In addition, the method used to obtain this upper limit provides also lower limits with the correct asymptotic behavior. For a regular potential it reads

For a singular potential at the origin it reads

TESTS

Now we will test these new upper and lower limits and compare them to bounds obtained previously and to exact results. The first potential we consider is a simple exponential potential (the number of bound states is known exactly in terms of the number of zeros of a Bessel function in some interval).

The potential is written like this (R in coupling constant) because it can then be shown easily that the number of bound states depends only on g and not on R. The results are summarized in the following graph

Clearly the new upper and lower limits are very good and it seems difficult to be able to improve this result. Indeed, the bounds have the correct asymptotic behavior and it appears that this regime is reached very fast. Only when there is 1 or 2 bound states in the potential, some other bounds can be a bit better. As soon as there is 3 or 4 or more bounds states, no other bounds can give better results than those obtained with the new ones.

The next test is done with a Pöschl-Teller potential (for which the spectrum, and thus the number of bound states, is known exactly).

The results are

Same conclusions: the new bounds are very good. The last test is done with the Hülthen potential, a singular potential (thus the bounds C0 and M cannot be used). For this potential, the spectrum is also exactly known and thus also the number of bound states as a function of g. The potential reads

The results are

Bibliography

[1] V. BARGMANN, "On the number of bound states in a central field of force", Proc. Nat. Acad. Sci. U.S.A. 38, 961-966 (1952).

[2] J. SCHWINGER, "On the bound states for a given potential", Proc. Nat. Acad. Sci. U.S.A. 47, 122-129 (1961).

[3] F. CALOGERO, "Upper and lower limits for the number of bound states in a given central potential", Commun. Math. Phys. 1, 80-88 (1965).

[4] J. H. E. COHN, "On the number of negative eigenvalues of a singular boundary value problem", J. London Math. Soc. 40, 523-525 (1965).

[5]  K. CHADAN, "The asymptotic behavior of the number of bound states of a given potential in the limit of large coupling", Nuovo Cimento A58, 191-204 (1968).

[6] A. MARTIN, "An inequality on S-wave bound states with correct coupling constant dependence", Commun. Math. Phys. 55, 293-298 (1977).

[7] GENORB and F. CALOGERO, "Upper and lower limits for the number of S-wave bound states in an attractive potential", J. Math. Phys. 44, 1554-1575 (2003).

[8] GENORB and F.  CALOGERO, "Upper and lower limit on the number of bound states in a central potential", J. Phys. A 36, 12021-12063 (2003).

[9] GENORB and F. CALOGERO, "Lower limit in semiclassical form for the number of bound states in a central potential", Phys. Lett. A 321, 225-230 (2004).