Near-threshold quantization and level densities for potential wells with weak inverse-square tails

Near-threshold quantization and level densities for potential wells with weak inverse-square tails

Michael J. Moritz, Christopher Eltschka, and Harald Friedrich Physik-Department, Technische Universita¨t Mu¨nchen, 85747 Garching, Germany

共Received 13 March 2001; published 6 July 2001兲

For potential tails consisting of an inverse-square term and an additional attractive 1/r^m term, V(r)

⬃关ប²/(2M)兴关(␥/r²)⫺(␤^m⫺2/r^m)兴, we derive the near-threshold quantization rule n⫽n(E) which is related to the level density via ␳⫽dn/dE. For a weak inverse-square term, ⫺¹4⬍␥⬍³4 共and m⬎2), the leading contributions to n(E) are n

E→0

⫽A⫺B(⫺E)^冑␥⫹1/4, so ␳ has a singular contribution proportional to (⫺E)^{冑␥⫹1/4⫺1} near threshold. The constant B in the near-threshold quantization rule also determines the strength of the leading contribution to the transmission probability through the potential tail at small positive energies. For␥⫽0 we recover results derived previously for potential tails falling off faster than 1/r². The weak inverse-square tails bridge the gap between the more strongly repulsive tails, ␥⭓3/4, where n(E)

E→0

⫽A⫹O(E) and ␳ remains finite at threshold, and the strongly attractive tails, ␥⬍⫺1/4, where n^E⫽^→0Bln(⫺E/A), which corresponds to an infinite dipole series of bound states and connects to the behavior n

E→0

⫽A⫹B E^{(1/2)⫺(1/m)}, describing infinite Rydberg-like series in potentials with longer-ranged attractive tails falling off as 1/r^m, 0⬍m⬍2. For␥⫽⫺1/4共and m⬎2) we obtain n(E)

E→0

⫽A⫹C/ln(⫺E/B), which remains finite at threshold.

DOI: 10.1103/PhysRevA.64.022101 PACS number共s兲: 03.65.Ge, 03.65.Sq, 03.65.Xp I. INTRODUCTION

Recent intense activity related to cold atoms and their interactions has led to increased interest in threshold proper- ties of atomic and molecular potentials. One property of in- terest is the level density ␳(E), which is closely related to the quantization rule,

n⫽n共E兲, ␳共E兲⫽dn

dE. 共1兲

The quantization rule in Eq. 共1兲 implies that the potential supports a bound state at each energy for which the function n(E) is an integer. The level density ␳(E) is defined such that␳(E)dE is the共expected兲number of energy levels in the energy interval (E,E⫹dE).

The behavior of n(E) near threshold, E→0, depends sen- sitively on the behavior of the potential V(r) at large dis- tances, r→⬁. Potentials with an attractive tail falling off as

V共r兲 ⬃

r→⬁

⫺ ប² 2M

␤^m⫺2

r^m 共2兲

support an infinite number of bound states if the共not neces- sarily integer兲power m lies in the range 0⬍m⬍2. Semiclas- sical approximations become increasingly accurate for E

→0 in this case关1兴, so the near-threshold quantization rule and level density can accurately be derived via straightfor- ward WKB methods. This leads to a simple generalization of the well-known formula applicable for Coulombic potentials (m⫽1), namely,

n⫽A⫹F共m兲

␲ ^共␬␤兲^1⫺2/m with F共m兲⫽

冑

␲ 2m

⌫

冉

^{m1 ⫺12}

冊

⌫

冉

^m1⫹1

冊

^,

共3兲 i.e., n⫽A⫹B(⫺E)^{(1/2)⫺(1/m)}. The associated level density is characterized by a singularity proportional to (⫺E)^{⫺(1/2)⫺(1/m)} near threshold. Note that the coefficient of the leading energy-dependent term in Eq. 共3兲is completely determined by the power m and the strength parameter␤ ^of the leading 1/r^mterm in the potential tail. The constant A on the right-hand side of Eq. 共3兲 depends on the potential at shorter distances, where it may deviate from the asymptotic 1/r^m form; for Coulombic potentials it is usually called the

‘‘quantum defect’’关1兴.

For potentials falling off faster than 1/r², there is at most a finite number of bound states and conventional WKB quan- tization fails near threshold, but for a sufficiently deep well there may be a region of moderate r values in the well where WKB wave functions are very accurate solutions of the Schro¨dinger equation for near-threshold energies. There has recently been considerable progress in the understanding of near-threshold properties in such cases关2–9兴.

For a potential tail falling off asymptotically as 1/r²,

V共r兲 ⬃

r→⬁ ប² 2M

␥

r², 共4兲

the near-threshold properties depend sensitively on the strength parameter␥. An attractive tail of sufficient strength,

␥⬍⫺1/4, supports an infinite ‘‘dipole series’’ of bound states, the energies of which behave as

E_n ⫽

n→⬁

⫺A exp

冉

^⫺

^冑

^{␥2⫹␲n1/4}

冊

^共5兲

towards threshold. Such attractive 1/r² potential tails can oc- cur through a dipole-monopole interaction and its coupling to the orbital angular momentum, as, e.g., in the interaction of an electron and an excited hydrogen atom 关10–12兴. The ratio E_n/E_n⫹1 of successive energy eigenvalues in such a dipole series depends only on the strength parameter␥ ^{of the} 1/r² term and is simply exp(2␲^/

冑

␥⫹1/4), but the absolute values of the energies as determined by the constant A in Eq.

共5兲depend on the nature of the potential at short distances, where it necessarily deviates from the 1/r²behavior. Explicit expressions for A have recently been derived for the case that the potential tail consists of a sufficiently strong attractive 1/r²term together with a shorter-ranged contribution propor- tional to 1/r^m关13兴.

In Ref.关13兴we discussed the near-threshold quantization rule for strongly attractive inverse square tails, but we did not do so for potential wells with weakly attractive or repul- sive inverse-square tails, which support at most a finite num- ber of bound states. It is the aim of this paper to close this gap and so to arrive at a comprehensive overview over the nature of near-threshold quantization rules and level densi- ties for any strength of the 1/r² term in the potential tail.

In Sec. II we study weak共attractive or repulsive兲1/r²tails for which the strength parameter␥ defined according to Eq.

共4兲lies between⫺1/4 and⫹3/4. In Sec. III we focus on the limiting case␥⫽⫺1/4 at which the inverse-square potential ceases to support an infinite dipole series of bound states.

This case is of very general importance since it corresponds to the s-wave centrifugal potential in the radial Schro¨dinger equation for a two-dimensional system. A summary also mentioning the connection to more strongly repulsive or at- tractive tails is given in Sec. IV.

II. WEAK INVERSE-SQUARE TAILS We consider potential tails of the following form:

V共r兲⫽ ប²

2M

冉

^{r␥2⫺␤mrm⫺2}

冊

^{, ␤⬎0, ⫺14⬍␥⬍34. 共6兲}

The term proportional to 1/r^mis attractive and, in this section and the next, the 共not necessarily integer兲 power m is as- sumed to be greater than 2. This guarantees that the WKB approximation becomes increasingly accurate towards small r values, so in a potential well obtained by supplementing Eq. 共6兲with an appropriate short-ranged repulsive contribu- tion there can be a range of moderate r values in the well where the WKB approximation is accurate for near-threshold energies; beyond this WKB region the potential is assumed to be given by Eq. 共6兲. Such a potential well can support a large but at most finite number of bound states.

At energy E⫽⫺ប²␬^2/(2M) the Schro¨dinger equation beyond the WKB region reads

冉

^{drd22⫺r␥2⫹␤rmm⫺2⫺␬2}

冊

^{␺共r兲⫽0. 共7兲}

Asymptotically, r→⬁, this reduces to

冉

^{drd22⫺r␥2⫺␬2}

冊

^{R⫽0, 共8兲}

which posesses analytical solutions depending only on ␬^r.

The solution which decays as required for a bound state is

R共␬^r兲⫽ie^i␮␲/2

冑

␬^rH_␮⁽¹⁾共i␬^r兲 ⬃^␬

r→⬁

冑

␲^2exp共⫺␬^r兲, 共9兲 where H_␮⁽¹⁾is the Hankel function as defined in关14兴, and its order␮ is related to␥ ^by

␮⫽

冑

␥⫹1/4. 共10兲 The range of strength parameters␥ defining the ‘‘weak’’共at- tractive or repulsive兲 inverse-square term in Eq. 共6兲 corre- sponds to the following range of values of␮^:

0⬍␮⬍1. 共11兲

For small arguments ␬^{r Eq.} 共9兲becomes

R共␬^r兲^␬⫽

r→0

冑

²

sin共␲␮兲

冋

^{共␬⌫共r/21兲⫺(1/2)␮⫺␮兲 ⫺共␬⌫共r/21兲⫹(1/2)␮⫹␮兲}

册

⫻关1⫹O„共␬^r兲²…兴. 共12兲 At intermediate values of r the potential is deep and we neglect the energy term in the Schro¨dinger equation共7兲,

冉

^{drd22⫺r␥2⫹␤rmm⫺2}

冊

^{M共r兲⫽0. 共13兲}

Two linearly independent solutions of Eq. 共13兲are M_⫾共r兲⫽

冑

^rJ_⫾␯共z兲, z⫽ 2

m⫺2

冉

^␤r

冊

^{(m⫺2)/2, 共14兲}

where the abbreviation␯ ^{stands for}

␯⫽ 2␮ m⫺2⫽ 2

m⫺2

冑

^␥⫹14, 共15兲 and J_⫾␯ are the ordinary Bessel functions of order ␯ ^and

⫺␯, respectively关14兴. The most general solution of Eq.共13兲 is a linear superposition,

M共r兲⫽A_⫹M_⫹共r兲⫹A_⫺M_⫺共r兲. 共16兲 For large r corresponding to small z, the wave functions共14兲 behave as

M_⫾共r兲 ⫽

r→⬁ ␤^{⫾␮r(1/2)⫿␮}

⌫共1⫾␯兲共m⫺2兲^⫾␯

冋

^1⫹O

冠 冉

^␤r

冊

^m⫺2

冡 册

^{. 共17兲}

The leading r dependence of M_⫹and M_⫺corresponds to the r dependence of the leading near-threshold terms of Eq.

共12兲, and, for a given 共small兲 value of␬ we can determine the ratio of the coefficients A_⫹and A_⫺in Eq.共16兲by requir- ing the ratio of the coefficients of r^(1/2)⫺␮and r^(1/2)⫹␮in Eq.

共16兲to be the same as in Eq.共12兲. This yields A_⫺

A_⫹⫽⫺共␬␤^/2兲^2␮ 共m⫺2兲^2␯

⌫共1⫺␯兲⌫共1⫺␮兲

⌫共1⫹␯兲⌫共1⫹␮兲

⫽⫺ ␲²共␬␤^/2兲^2␮

sin共␲␮兲sin共␲␯兲共m⫺2兲^2␯␮␯关⌫共␮兲⌫共␯兲兴². 共18兲 For certain values of m in the range 2⬍m⭐2⫹2␮^{, the pa-} rameter ␯⫽2␮^/(m⫺2) can be an integer, so the gamma function⌫(1⫺␯^{) in Eq.}共18兲becomes singular, as expressed by the factor sin(␲␯) in the denominator in the second line.

The reason for this singularity is the linear dependence of the two Bessel functions in Eq. 共14兲when ␯ is exactly an inte- ger; one of the solutions M_⫹or M_⫺should be replaced, e.g., by

冑

^rY_␯(z) in this case. In the derivation below, however, the results obtained for the near-threshold quantization rule and the level density for noninteger␯ are free of singularities and can be continued through possible integer values, see, e.g., Eq.共33兲.

The wave function 共9兲 with the near-threshold behavior 共12兲 is an accurate solution of the Schro¨dinger equation共7兲 as long as the shorter-ranged term ␤^m⫺2/rm in the potential is negiligible compared to the longer-ranged term␥^/r2,

rⰇ ␤

兩␥兩^1/(m⫺2). 共19兲

The wave function 共16兲with the asymptotic behavior given by Eq.共17兲is an accurate solution of the Schro¨dinger equa- tion共7兲as long as the energy term␬²is negiligible compared to the longer-ranged term in the potential,

rⰆ

冑

兩␥兩

␬ ^{. 共20兲}

We can match the wave functions共16兲and共9兲, respectively 共17兲and共12兲if there is a region of r values, where the two conditions 共19兲and共20兲are fulfilled simultaneously. Such a region exists when

␬␤Ⰶ兩␥兩^m/(2m⫺4), 共21兲 i.e., as long as ␥⫽0, the matching procedure described above is well justified in the limit␬→0.

Towards smaller values of r, the argument z of the Bessel functions in Eq. 共14兲 becomes large, and their asymptotic expansion yields

M_⫾共r兲 ⫽

z→⬁

冑

␲^2rzcos

冉

^{z⫿␯2␲⫺␲4}

冊

^{, 共22兲}

so the superposition 共16兲becomes

M_␬共r兲 ⫽

z→⬁

冑

_␲^2rz

冋

^A⫹cos

冉

^{z⫺␯2␲⫺␲4}

冊

⫹A_⫺cos

冉

^{z⫹␯2␲⫺␲4}

冊册

⬀

冑

^rzcos

冉

^{z⫺␲4⫹␦}

冊

^{, 共23兲}

where␦ is a phase angle given by

tan␦⫽⫺1⫺A_⫺/A_⫹

1⫹A_⫺/A_⫹tan

冉

^␲␯2

冊

^{. 共24兲}

We have included a subscript label ␬ on the wave function 共23兲 in order to emphasize the dependence on␬, which en- ters via the ␬-dependent ratio 共18兲obtained by matching to the asymptotic solution R(␬^r).

The r dependence of both amplitude and phase of the wave function 共23兲is that of the WKB wave function,

␺WKB共r兲⬀ 1

冑

^p共r兲cos

冉

^1ប

^冕

^rrinp共r

^⬘

^兲dr

^⬘

^⫺␾2in

冊

^{, 共25兲}

at threshold, E⫽0, when the potential near r is given by the shorter-ranged term in Eq. 共6兲 alone. In Eq. 共25兲, p(r)

⫽

冑

²M„E⫺V(r)…is the local classical momentum and␾in

is the reflection phase at the inner classical turning point 关13,15兴, which is assumed to be a smooth function of energy, i.e., of ␬², near threshold. At E⫽0 we have p(r)⫽p₀(r)

⫽

冑

⫺2MV(r), and p₀(r)⫽ប␤^{(m⫺2)/2/rm/2}when the poten- tial is dominated by the term proportional to 1/r^m. Hence

1 ប

冕

^rin

r

p₀共r

⬘

兲dr

⬘

⫽const⫺z, 共26兲

with z as defined in Eq.共14兲.

If the WKB region overlaps with a range of r values where the potential is dominated by the 1/r^m term, then the quantization condition can be formulated by matching the wave functions 共25兲 and 共23兲 in this range of overlap. We expect the WKB wave function here to depend smoothly 共analytically兲on the energy E, so, to order less than ␬^{2, we} can assume E⫽0 in Eq. 共25兲. Equating the cosines in Eqs.

共25兲and共23兲leads to the quantization condition 1

ប

冕

^rin r

p₀共r

⬘

兲dr

⬘

⫺␾in

2 ⫽n␲⫺z⫹␲

4 ⫺␦⫹O共E兲. 共27兲 The subscript zero on the local momentum p indicates that the action integral on the left-hand side is to be taken at threshold, E⫽0. In the range of overlap, where the WKB approximation is accurate and the potential is dominated by the 1/r^m term, the expression

I₀⫽

def1 ប

冕

r_in

r

p₀共r

⬘

兲dr

⬘

⫹z⫺␾in

2 ⫺␲

4 共28兲

is independent of r. Using Eq.共24兲 we can thus rewrite the quantization condition共27兲as

n␲⫽I₀⫹␦⫹O共E兲

⫽I₀⫺arctan

冋

^tan

冉

^␲␯2

冊

^{11⫺⫹AA⫺}⫺^/A/A^⫹_⫹

册

^{⫹O共E兲. 共29兲}

With the help of the identity,

arctan

冋

^tan

冉

^␲␯2

冊

^{11⫺⫹AA⫺}⫺^/A/A^⫹_⫹

册

⫽␲␯

2 ⫺arctan

冋

^{1⫹共A共⫺A}⫺^/A/A^⫹兲_⫹^sin兲cos^共␲␯共␲␯^兲 兲

册

^{, 共30兲}

this is seen to correspond, to leading orders in␬^{, to}

n⫽I₀

␲^⫺

␯

2⫺ ␲共␬␤^/2兲^2␮

sin共␲␮兲共m⫺2兲^2␯␮␯关⌫共␮兲⌫共␯兲兴²

⫹O„共␬␤兲^4␮…⫹O共␬²兲, 共31兲 where we have expressed the 共small兲 quantity A_⫺/A_⫹ in terms of␬ according to Eq.共18兲.

The quantization condition共31兲has the form

n ⫽

E→0

A⫺B共⫺E兲^␮, 共32兲 and the correction to the right-hand side is of order E if 2␮⬎1 and of order (⫺E)^2␮ if 2␮⬍1. The constant A

⫽I₀/␲⫺␯/2 represents the threshold value of the quantum number that determines the total number of bound states sup- ported by the potential. The constant B is a ‘‘tail parameter’’

that depends only on the tail of the potential beyond the WKB region. For the potential tail 共6兲we have

B⫽ ␲„M␤^2/共2ប²兲…^␮

sin共␲␮兲共m⫺2兲^2␯␮␯关⌫共␮兲⌫共␯兲兴², 共33兲 as derived above. The explicit value of B depends on the further term that complements the ␥^/r2 contribution in the potential tail and guarantees that there is a region of r values in the well where WKB wave functions become accurate solutions of the Schro¨dinger equation at near-threshold ener- gies. The power ␮⫽

冑

␥⫹1/4 in Eq. 共32兲 depends only on the strength ␥ ^{of the 1/r2} term, and not on the properties of the shorter-ranged contribution to the potential tail.

The near-threshold level density follows from Eq.共32兲,

␳共E兲⫽dn dE ⬃

E→0

B␮兩E兩^␮⫺1. 共34兲

Since 0⬍␮⬍1, the level density becomes infinite near threshold, even though the total number of bound states re- mains finite.

In Ref. 关13兴we calculated the leading near-threshold be- havior of the transmission probability P_Tthrough a centrifu- gal barrier consisting of two terms as in Eq.共6兲. For positive energies E⫽ប²k²/(2M), the leading energy dependence of the transmission probability is,

P_T ⫽

E→0 4␲²共k␤^/2兲^2␮

共m⫺2兲²␮␯关⌫共␮兲⌫共␯兲兴². 共35兲 As pointed out in 关13兴, this result holds both for repulsive inverse-square terms (␥⬎0) and for weakly attractive terms,

⫺1/4⬍␥⬍0. The leading energy dependence in Eq. 共35兲is P_T⬀k^2␮⬀E^␮. For three-dimensional systems, the centrifugal potential for the angular momentum quantum number l cor- responds to ␮⫽l⫹1/2, so the proportionality to E^␮ simply expresses Wigner’s threshold law. 共Note that Wigner’s threshold law also works for weakly attractive inverse-square potentials formally corresponding to⫺¹2⬍l⬍0.兲For poten- tial wells with weak inverse-square tails, the leading energy dependence of the near-threshold quantization rule 共31兲 is proportionality to ␬^2␮⬀(⫺E)^␮, and the coefficient of this term is closely related to the corresponding coefficient in Eq.

共35兲. Comparing Eqs. 共31兲 and 共35兲 shows that the coeffi- cient of k^2␮in the expression共35兲for the transmission prob- ability through the potential tail is just 4␲^sin(␲␮) times the coefficient of ␬^2␮ in the near-threshold quantization rule 共31兲. This factor depends only on the strength␥ ^{of the 1/r2} term and not on the properties of the shorter-ranged contri- bution to the potential tail.

For a potential well with a tail vanishing faster than 1/r² asymptotically, it has been shown previously 关4 –9兴 that the near-threshold quantization rule has the form

n ⫽

E→0

A⫺B

冑

⫺E⫹O共E兲, 共36兲 where A and B are constants, and that the leading behavior of the transmission probability through the potential tail is 关7,16兴

P_T⫽4␲^B

冑

^E, 共37兲 where the constant B is the same in both Eqs.共36兲and共37兲 and depends only on the potential tail beyond the WKB re- gion. These results correspond exactly to those derived above, Eqs. 共31兲–共35兲, for the special case of vanishing strength of the inverse-square term,␥⫽0, ␮⫽1/2. The ex- pression 共33兲 is equivalent to the formula 共2兲 in Ref. 关8兴 corresponding to Eq. 共9兲 in Ref. 关4兴, and Eq. 共35兲 above corresponds to Eqs. 共35兲 and 共36兲 in Ref. 关16兴 when ␮

⫽1/2.共Generalizations to potential tails with significant de- viations from the homogeneous form proportional to 1/r^m beyond the WKB region are discussed in Ref.关7兴.兲

The derivations in this section have been limited to strength parameters in the range ⫺¹4⬍␥⬍³4 corresponding to ‘‘weak’’ inverse-square tails. The upper end of this inter-

val,␥⫽3/4, corresponds to␮⫽1, so the leading terms pro- portional to (⫺E)^␮ in Eqs. 共31兲 and 共32兲 are of the same order, viz., O(E), as terms neglected by replacing the action integral in Eq. 共27兲 by its value at threshold. For repulsive inverse-square potentials with ␥⫽3/4 or larger, we can ex- pect the near-threshold quantization rule to be of the form

n ⫽

E→0

const⫹O共E兲, 共38兲 so the resulting level density ␳⫽dn/dE remains finite at threshold.

The lower end of the range of strength parameters dis- cussed in this section is ␥⫽⫺1/4 corresponding to ␮⫽0.

This special case is the subject of the next section.

III. THE SPECIAL CASE␥ÄÀ1Õ4

The near-threshold quantization rule共32兲and the formula 共34兲for the near-threshold level density become meaningless for ␥⫽⫺1/4 corresponding to␮⫽0. An inverse-square po- tential behaving asymptotically as⫺¹4ប²/(2Mr²) represents the limit for which the potential will no longer support an infinite dipole series of bound states. This case is of central importance for two-dimensional problems, where the cen- trifugal potential in the radial Schro¨dinger equation is (l²⫺1/4)ប²/(2Mr²) (l⫽0,1,2, . . . ), rather than l(l

⫹1)ប²/(2Mr²) applicable in the three-dimensional case.

For s waves (l⫽0) in two dimensions, the centrifugal poten- tial is attractive, and the strength parameter according to the definition 共4兲is precisely␥⫽⫺1/4.

As in Sec. II, we assume that there is a region of moderate r values where the WKB approximation is accurate at near- threshold energies, and that the potential is described by the following two terms beyond this WKB region:

V共r兲⫽⫺ ប²

2M

冉

^{1/4r2 ⫹␤rmm⫺2}

冊

^{, m⬎2. 共39兲}

At energy E⫽⫺ប²␬^2/(2M) the Schro¨dinger equation beyond the WKB region is given by Eq. 共7兲with␥⫽⫺1/4.

Asymptotically, r→⬁, this reduces to Eq. 共8兲, and for ␥

⫽⫺1/4 the solution which decays as required for a bound state is

R共␬^r兲⫽i

冑

␬^rH0

(1)共i␬^r兲 ⬃^␬

r→⬁

冑

␲^{2 exp共⫺}␬^r兲, 共40兲 where H₀⁽¹⁾ is the Hankel function of order zero. For small arguments we now have

R共␬^r兲 ⬃^␬

r→0

⫺ 2

␲

冋

^ln

冉

^␬2r

冊

^⫹␥E

册 ^冑

^{␬r关1⫹O„共␬r兲2…兴,}

共41兲 where␥E is Euler’s constant关14兴.

At intermediate values of r the potential is deep and we neglect the energy term in the Schro¨dinger equation共7兲and consider Eq.共13兲. For␥⫽⫺1/4, this equation is solved关14兴

by functions of the form

冑

^rC0(z), whereC0 is共any兲 Bessel function of order zero, and the argument z is as already de- fined in Eq.共14兲. The most general solution M (r) of Eq.共13兲 共with␥⫽⫺1/4) can be expressed as a linear superposition of two linearly independent solutions generated, e.g., by the Bessel functions J₀(z) and Y₀(z),

M共r兲⫽

冑

^r关A_JJ₀共z兲⫹A_YY₀共z兲兴. 共42兲 For large values of r corresponding to small values of z this solution behaves as

M共r兲 ⬃

r→⬁

冑

^r

再

^AJ⫹2A␲Y

冋

^ln

冉

^2z

冊

^⫹␥E

册冎

^{, 共43兲}

which contains the same leading r dependence 关⬀

冑

^r(const

⫺ln r)兴as the near-threshold form共41兲of the solution of Eq.

共8兲. Matching the two leading terms of M (r) and R(␬^r) leads to the following result for the ratio of the coefficients in the expression 共42兲for M (r):

A_J A_Y⫽⫺1

␲

冋

^m␥E⫹2ln

冉

^{共␬␤m/2⫺兲(m2⫺2)/2}

冊册

^{. 共44兲}

As in Sec. II, this matching is justified if there is a region of r values where both Schro¨dinger equations 共8兲 and 共13兲 are simultaneously accurate approximations of the full Schro¨dinger equation共7兲, leading to the condition共21兲 共with

␥⫽⫺1/4), which is increasingly well fulfilled towards threshold.

Towards smaller values of r 共larger z), the asymptotic expansion of the Bessel functions in Eq. 共42兲yields

M_␬共r兲 ⬀

z→⬁

冑

^rz

冋

^AAJYcos

冉

^z⫺␲4

冊

^⫹sin

冉

^z⫺␲4

冊册

⬀

冑

^rzcos

冉

^{z⫺␲4⫹␦}

冊

^{, 共45兲}

where the phase angle ␦ is now given by

cot␦⫽1

␲

冋

^{m␥E⫹2 ln}

冉

^{共␬␤m/2⫺兲(m2⫺2)/2}

冊册

^{. 共46兲}

In the region of overlap, where the WKB approximation is accurate and the potential is dominated by the 1/r^m term, matching the right-hand side of Eq. 共45兲to the WKB wave function共25兲now yields

n␲⫽I₀⫹␦⫽I₀⫹arccot

再

^␲1

冋

^{m␥E⫹2 ln}

冉

^{共␬␤m/2⫺兲(m2⫺2)/2}

冊册冎

^,

共47兲 where I₀ is again given by Eq.共28兲and does not depend on the value of r, as long as r is chosen in the region where the WKB approximation is accurate and the potential is domi- nated by the 1/r^mterm. Near threshold,␬→0, the argument of the arcus-cotangent on the right-hand side of Eq. 共47兲

becomes a large共negative兲number, so the leading contribu- tion to the near-threshold quantization rule is

n^␬→⫽

0I₀

␲^⫹

1

m␥E⫹2ln

冉

^{共␬␤m/2⫺兲(m2⫺2)/2}

冊

⫽I₀

␲^⫹

1

a₀⫹共m⫺2兲ln共␬␤兲, 共48兲

with a₀⫽m␥E⫺2 ln(m⫺2)⫺(m⫺2)ln 2. In terms of energy,

n ⫽

E→0I₀

␲^⫹

1 a₀⫹m⫺2

2 ln共⫺E/B兲

⫽I₀

␲^⫹

2/共m⫺2兲

ln共⫺E/B兲⫹O

冉

^{关ln共⫺1E/B兲兴2}

冊

^{, 共49兲}

with B⫽ប²/(2M␤²). The leading contribution to the near- threshold level density is

␳共E兲⫽dn dE ⫽

E→0 m⫺2 2共⫺E兲

冉

^{a0⫹m⫺2 2}1^{ln共⫺E/B兲}

冊

²

⫽ 2/共m⫺2兲

共⫺E兲关ln共⫺E/B兲兴²⫹O

冉

^{共⫺E兲关ln共⫺1 E/B兲兴3}

冊

^.

共50兲 IV. SUMMARY AND DISCUSSION

We have studied potential wells with tails behaving as

V共r兲⫽ ប²

2M

冉

^{r␥2⫺␤rmm⫺2}

冊

^{. 共51兲}

For m⬎2 the near-threshold properties depend crucially on the strength parameter␥ of the inverse-square term. For a weak inverse-square term with ⫺¹4⬍␥⬍³4, the near- threshold quantization rule has the general form,

n ⫽

E→0

A⫺B共⫺E兲^冑␥⫹1/4⫹O共E兲, 共52兲 where A and B are constants. The related level density ␳

⫽dn/dE is characterized by a singularity proportional to (⫺E)^{冑␥⫹1/4⫺1} near threshold; it becomes infinite, even TABLE I. Summary of near-threshold quantization rules for potential wells with tails given by Eq.共51兲.

For m⬎2,␥⬎⫺1/4 the table also contains the leading near-threshold behavior of the transmission probabil- ity PTthrough the tail. The symbols A and B refer to constants appropriate to the respective cases. Whereas A generally depends on properties of the whole potential well, the parameter B depends only on the properties of the tail共51兲. The last column lists equations containing explicit expressions for the constants A and/or B in the respective cases.

0⬍m⬍2, any␥

n^E→0⫽A⫹B共⫺E兲(1/2)⫺(1/m) Eq.共3兲

m⬎2,␥⬍⫺¹4 n⫽

E→01

2␲

冑

␥⫹1/4 ln共⫺E/A兲 Eq.共5兲; Eqs.共30兲and共38兲in关13兴

m⬎2,␥⫽⫺¹4

n⫽

E→0

A⫹2/关共m⫺2兲ln共⫺E/B兲兴 Eqs.共48兲and共49兲

m⬎2,⫺¹4⬍␥⬍³4

n⫽

E→0

A⫺B共⫺E兲^冑␥⫹1/4 Eqs.共31兲and共33兲 P_T^E→0⫽ 4␲B sin共␲

冑

␥⫹1/4兲E^冑␥⫹1/4 Eqs.共35兲and共33兲; Eq.共57兲in关13兴

m⬎2,␥⭓³4 n^E→0⫽A⫹O共E兲 P_T⬀

E→0

E^冑␥⫹1/4 Eq.共35兲; Eq.共57兲in关13兴

though the number of bound states in the well is finite. For positive energies just above threshold, the probability for transmission through the potential tail is, to leading order, 4␲^Bsin(␲

冑

␥⫹1/4)E^冑␥⫹1/4. The same constant B appears in the quantization rule and the transmission probability, and it depends only on the tail of the potential beyond the region where the WKB approximation is accurate. Results derived previously for potential tails falling off faster than 1/r² fit into this scheme if we put␥⫽0, corresponding to vanishing strength of the inverse-square term.

For repulsive inverse-square terms beyond the range re- ferred to above, ␥⭓3/4, the leading energy dependence of n(E) near threshold is of order O(E), and the level density remains finite. This means that for a potential well with a repulsive inverse-square tail at least as strong as the p-wave

centrifugal term in two-dimensional systems, there is no sin- gular contribution to the level density near threshold. The transmission probability through the barrier is still given by Eq. 共35兲and obeys Wigner’s threshold law.

For attractive inverse-square tails with␥⫽⫺1/4, the lead- ing energy dependence of n(E) near threshold is propor- tional to 1/ln(⫺E). For more strongly attractive inverse- square tails it is proportional to ln(⫺E), and the potential well supports an infinite dipole series of bound states, as described by Eq. 共5兲. This behavior connects to the case of longer-ranged potential tails which fall off slower than 1/r², see Eq. 共3兲.

These results are summarized in Table I. It covers all po- tential tails of the form 共51兲with arbitrary values of ␥^{, m}

⬎0, and␤⬎0.

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