Exotic atoms in two dimensions^{1}^{1}1PACS: 36.10.Dr,03.65.Ge,03.65.nk
Abstract
We study the behavior of energy levels in two dimensions for exotic atoms, i.e., when a longrange attractive potential is supplemented by a shortrange interaction, and compare the results with these of the one and threedimensional cases. The energy shifts are well reproduced by a scattering length formula , where is the scattering length in the shortrange potential, the square of the wave function at the origin in the external potential, and is related to the derivative with respect to the energy of the solution that is regular at large distances.
1 Introduction
Hadronic atoms give valuable information about strong interactions at low energy. For a review, see, e.g.,[1]. They have also motivated several studies on the behavior of the energy levels in a Schrödinger operator, with a potential , where dominates at large distances, but is superseded by at short distances. The case of exotic atoms corresponds to a world with three dimensions, where (as a negativelycharged hadron orbits near the nucleus and is almost unscreened by the remaining electrons, if any), and describes the shortrange hadronic interaction. But the situation is far more general, and many features do not depend on the Coulomb character of . Nevertheless, we shall use the word “exotic atom” for such a system, “atomic” for the energy domain of the eigenstates of alone, and “nuclear” for any typical energy within alone, for the sake of simplicity.
The spectral problem of exotic atoms [1, 2, 3] differs significantly from the ordinary perturbation theory, for which an expansion of the eigenenergies in powers of is attempted. For exotic atoms, the energies for are often very close to the ones for alone, but perturbation theory usually does not hold. For instance, if is an infinite hard core of small radius, the energies are slightly shifted upwards, but the ordinary perturbative expansion diverges already at the first order. The proper concept here is “radius perturbation theory”, as described by Mandelszweig [4].
In this paper, we discuss how exotic atoms behave in dimensions. It may be noted that the study of exotic atoms in is more straightforward, and already discussed in the literature [3]. The case is more delicate. The leading order term for the energy shift is easily identified, and linked to , where is the scattering length in the shortrange potential. As in the case, the overall coefficient is the square of the wavefunction at the origin in the external potential. However, the scale regularizing this leading term, i.e., the radius leading to is not immediate, but it can be derived from a matching of the solution of which is normalizable to the asymptotic solution emerging from the shortrange term .
The case of dimensions is rather special in spectral problems, as it corresponds to the largest value of for which an attractive potential, however weak, always holds at least one bound state, see, e.g., [5, 6]. ^{6}^{6}6More precisely, what is sufficient is that the integral of the potential over the whole space is positive. Hence, for , if is attractive, immediately develops its own bound state, which becomes the ground state of the Hamiltonian. However, this process is less effective for than for , and the spectrum, as a function of evolves more slowly.
This paper is organized as follows. In Sec. 2 and Sec. 3, we give a brief reminder about the cases of and space dimensions respectively, with particular emphasis on the phenomenon of level rearrangement and on the scattering length (hereafter referred to as SL) formula for the energy shifts. In Sec. 4, we present the results for the case of dimensions. The theoretical framework is presented in Sec. 5, before the final discussion in Sec. 6.
2 Exotic atoms in three dimensions
There is an abundant literature about exotic atoms in three dimensions, motivated by experiments with pionic, kaonic and antiprotonic atoms [1, 2]. The simplest model consists of a twocomponent potential
(1) 
where is a longrange interaction with one or several bound states. Genuine exotic atoms correspond to . The second term, with an explicit strength introduced for the ease of the discussion, accounts for the shortrange interaction. The main results are:

the shift is usually rather small, although can be very large at short distances,

the shift is usually well described by the approximate formula
(2) where is the normalized wave function for , and the scattering length in alone. In case is Coulombic, one recovers the wellknown SL formula by Deser, Golberger, Bauman and Thirring, and Trueman [7, 8]
(3) where is the Bohr radius and the principal number for the energy in alone or in the total potential. Many improvements and further corrections to this formula have been discussed in the literature [9, 10].

When is varied, the shift usually varies very slowly, except near the specific values , , …, where the energy levels change very rapidly, and a level rearrangement occurs: near , the energy drops toward very large (negative) values in the nuclear domain, and is replaced in the upper part of the spectrum by the next level, which in turn is replaced by the next one, etc. An example is given is Fig. 1. Further examples are provided, e.g., in [3]. The critical values correspond to the coupling thresholds for which the shortrange interaction starts supporting a first or an additional bound state.
3 Results in one dimension
An example of spectrum of exotic atom in is shown in Fig. 2. It consists again of a superposition of two square wells, the strength of the shortrange one being varied. The main differences, as compared to the more familiar case are:

As soon as slightly departs from zero, the atomic ground state energy immediately drops towards the range of the nuclear energies.

As a coupling threshold in is reached and further increases, a plateau is observed; the corresponding energy drops, and, by rearrangement, a upper level makes another plateau near the same value. This plateau in the sector of the even parity states, corresponds to an unperturbed energy level in the odd sector of . Indeed, the orthogonality with the ground state forces a zero in the wave function near , and mimics an odd state.

The Deser–Trueman formula, if translated for , reads
(4) The presence of the scattering length in the denominator can be understood by dimensional analysis. Also, weaker the shortrange interaction , more flat the zeroenergy wave function, and thus larger the scattering length , defined (as for ) as the abscissa where the asymptotic zeroenergy wave function vanishes.
4 Results for two dimensions
The calculation can be repeated for the isotropic (i.e., azimuthal quantum number ) states with . If the atomic spectrum is examined for increasing values of the strength of the shortrange interaction, a pattern of level rearrangement is clearly identified, see Fig. 3.
The behavior of the ground state is displayed again in Fig. 4, where it is compared to the and cases. The trend is clearly intermediate between the plateau of and the immediate falloff of .
For small values of , we can easily identify the following behavior for the energy shift
(5) 
If one plots, as in the example shown in Fig. 5, as a function of , one hardly distinguishes the exact values from the results of a linear fit.
5 Derivation of the energy shift
5.1 General formula
There are many approaches to the SL formula for , and various corrections and generalizations, see, e.g., [7, 8, 4, 9, 10, 3] and references there. For the case, the following simple minded derivation is just based on the matching condition between solutions of the Schrödinger equation that are regular at short and at large distances.
For the sake of clarity, one can identify several approximations that are made when solving the boundstate problem in the potential :

dominates at large distances

the energy in alone is a smooth function of the boundary condition enforced at ,

and the energy term can be neglected at very small distances, where dominates.
Let us start with Schrödinger equation for the external potential alone, i.e.,
(6) 
where is the reduced radial wave function and we are working in the units with . We denote the solution that is regular at infinity, i.e., at large distance, with and the usual Bessel function. The case of a confining interaction is treated later. At short distance, this solution behaves as
(7)  
for the modified energy and the unperturbed one . The unperturbed energy corresponds to , i.e., a solution that is regular as , and is normalized, leading to a real value at energy , that can be chosen to be positive. For in the neighborhood of , we impose that the solution remains normalized, i.e.,
(8) 
By combining (6) for and , one obtains the exact relation
(9) 
which gives for the energy shift a first relation . It is rather precise. Indeed, if the solution is kept to be normalized as per (8), and if as , the integral of entering (9) is also equal to , up to second order in .
Now, , one can identify the shortrange behavior of with , to obtain
(10) 
which when combined with gives
(11) 
where the denominator can be cast as . This relation gives explicitly the link between the energy and the boundary condition at , expressed by , where is the scattering length in the shortrange interaction alone, in terms of the quantities and linked to the value of the unperturbed solution at the origin.
The effective range correction to the scattering length approximation can be worked out explicitly, but turns out to be very small in most cases. For a positive energy , the solution to the scattering problem in alone is [13, 11, 12]
(12) 
where and are Bessel functions, and
(13) 
involving the scattering length and effective range . This expression is easily translated for negative energies , and, anyhow, the range of which is explored is small as compared to the typical energy scale in the shortrange potential, and thus the solution coming out is safely approximated by its small limit
(14) 
This means one can replace by to probe the contribution of the effective range, which turns out negligible, provided the energy shift remains small as compared to .
5.2 First example: double deltashell
To illustrate (11), we consider as long range interaction an attractive deltashell of strength and radius that can be set to to fix the length scale. The solution can be worked out analytically, in particular is regular at large and at small . The deltashell interaction imposes the continuity of the radial solution near and the proper step in its derivatives, to fix the unperturbed energy . A second deltashell can be implemented at , leading to an explicit transcendental equation for the exact energy , and shift , to be compared to the simple approximate value and the improved given by (11). For , , and , one gets
(15) 
i.e., an almost perfect agreement, when the correction is taken into account.
5.3 Double exponential well
As an example involving smooth potentials, we consider the exponential potential with and take for the longrange interaction, and study the changes due to another exponential interaction, with a much shorter range and a variable strength. The results are displayed in Fig. 6. Again, there is a net gain as compared to the ordinary perturbation theory, and a good agreement with the exact calculation as long as the deviation from the unperturbed energy is not too large.
5.4 Harmonic confinement
The problem is to study how the levels, in particular the ground state, are modified when a harmonic oscillator is supplemented by a shortrange interaction. A recent contribution is by Farrell and van Zyl [14]. They first stressed the property of universality, namely that the energy shift does not depend on the details of the shortrange potential, but instead is governed by the scattering length alone. This is, indeed, a very general property of the exotic atoms, in the general sense define in the introduction [3]. For , the general solution that is regular at large distances can be written as
(16) 
in terms of the confluent hypergeometric function . This expression is simpler, but equivalent to the one given in [14]. From (16), one can calculate explicitly the normalization integral and its derivative. The shortrange behavior of is known and if the ratio of the to coefficients is identified with , one recovers the formula given in [14]. Our prescription (11) corresponds to an approximate, but accurate, solution to the matching equation. For instance, using as an additional potential, with and , one gets, using the same notation as above and for [14],
(17) 
Clearly, the main discrepancy comes from reducing this shortrange interaction to a zerorange ansatz. Once this is accepted, our approximate treatment is nearly exact as compared to the precise matching of to the boundary condition.
6 Summary
In this note, we have studied how the energy levels in a wide potential are modified by a shortrange attraction of increasing strength, focusing on the case of space dimensions, as compared to the and situations.
The energy shifts in a given external potential are well described by the following SL formulas,
(18) 
i.e., a perfect fit is obtained if (and for ) are treated as free parameters. Moreover, can be identified with the first non vanishing coefficient of the shortrange expansion of the radial wave function and is thus proportional to , the wave function at the origin for the state in the external potential alone. The ratio is , the unitsphere area in dimensions.
In the case which is our main concern, a formula has been derived for , namely , where is the coefficient of in the normalized wave function, assumed to be real and positive and to match at energy .
This SL relation becomes more accurate when additional potential becomes more shortranged. In particular, it improves significantly the simple prediction from first order perturbation theory in and .
This study of exotic atoms is intimately linked to the statistical physics of bosons. The common tool is the pseudopotential, which enables one to replace a finite (but short) range interaction by a contact interaction. Deriving the pseudopotential as a function of the scattering length for different values of the space dimension has been extensively discussed. The case of is notoriously delicate, see, e.g., [15, 16, 17, 14] for recent contributions.
Acknowledgements
This work was done with the support of the Indo–French cooperation program CEFIPRA 34044.
References
 [1] A Gal, E Friedman, and C J Batty, On the interplay between Coulomb and nuclear states in exotic atoms, Nuclear Physics A, 606:283–291, 1996.
 [2] A M Badalyan, L P Kok, M I Polikarpov, and Y A Simonov, Resonances in coupled channels in nuclear and particle physics, Phys. Rep. , 82:31–177, 1982.
 [3] M Combescure, A Khare, A K Raina, JM Richard, and C Weydert Level Rearrangement in Exotic Atoms and Quantum Dots, International Journal of Modern Physics B, 21:3765–3781, 2007.
 [4] V B Mandelzweig, Radius perturbation theory and its application to pionic atoms, Nuclear Physics A, 292:333–349, 1977.
 [5] K Yang and M de Llano, Simple variational proof that any twodimensional potential well supports at least one bound state, American Journal of Physics, 57(1):85–86, 1989.
 [6] H Grosse and A Martin, Particle Physics and the Schrödinger Equation; new ed. Cambridge monographs on particle physics, nuclear physics, and cosmology. Cambridge Univ., Cambridge, 2005.
 [7] S Deser, M L Goldberger, K Baumann, and W Thirring, Energy Level Displacements in PiMesonic Atoms, Physical Review, 96:774–776, 1954.
 [8] T L Trueman, Energy level shifts in atomic states of stronglyinteracting particles, Nucl. Phys. A , 26:57–67, 1961.
 [9] J Carbonell, JM Richard, and S Wycech, On the relation between protonium level shifts and nucleonantinucleon scattering amplitudes, Zeitschrift fur Physik A Hadrons and Nuclei, 343:325–329, 1992.
 [10] J Mitroy and I A Ivallov, Quantum defect theory for the study of hadronic atoms, Journal of Physics G Nuclear Physics, 27:1421–1433, 2001.
 [11] F Arnecke, H Friedrich, and P Raab, Nearthreshold scattering, quantum reflection, and quantization in two dimensions, Phys. Rev. A , 78(5):052711, 2008.
 [12] N N Khuri, A Martin, JM Richard, and T T Wu, Lowenergy potential scattering in two and three dimensions, Journal of Mathematical Physics, 50(7):072105, 2009.
 [13] B J Verhaar, J P H W van den Eijnde, M A J Voermans, and M M J Schaffrath, Scattering length and effective range in two dimensions: application to adsorbed hydrogen atoms, Journal of Physics A: Mathematical and General, 17(3):595–598, 1984.
 [14] A Farrell and B P van Zyl, Universality of the energy spectrum for two interacting harmonically trapped ultracold atoms in one and two dimensions, Journal of Physics A Mathematical General, 43(1):015302, 2010.
 [15] C N Yang, Pseudopotential method and dilute hard ”sphere” Bose gas in dimensions 2, 4 and 5, Europhysics Letters, 84:40001, 2008.
 [16] M Li, H Fu, YZ Wang, J Chen, L Chen, and C Chen, Pseudopotential operator for hardsphere interactions in anydimensional space, Phys. Rev. A , 66(1):015601, 2002.
 [17] M Olshanii and L Pricoupenko, Rigorous Approach to the Problem of Ultraviolet Divergencies in Dilute Bose Gases, Physical Review Letters, 88(1):010402, 2002.