In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by an energy-dependent phase shift. The $1∕x2$ potential on $0 confounds all of our intuitions and expectations. Resolving its paradoxes requires sophisticated theoretical machinery: regularization, renormalization, anomalous symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a level accessible to advanced undergraduates.

1.
Of course, $V(x)$ is unbounded as $x→0$, but this by itself is not problematic—the same is true of the Coulomb potential and the delta function well. Our study of the $1∕x2$ potential was inspired by (and in the early parts modeled on)
S. A.
Coon
and
B. R.
Holstein
, “
Anomalies in quantum mechanics: The $1∕r2$ potential
Am. J. Phys.
70
,
513
519
(
2002
).
2.
Is there any other potential with this defect? In other words, is there another function of position that contains no dimensional constants and has the same units as $1∕x2$? There is—the two-dimensional delta function, $δ2(r)=δ(x)δ(y)$. Our story can be told using the two-dimensional delta function, but because the extra dimension makes the details a little more cumbersome and because the $1∕x2$ potential is arguably more realistic, we prefer to consider it as the model. In fact, for states with circular symmetry the two-dimensional delta function is mathematically equivalent to the $1∕x2$ potential with $α=1∕4$ [Eq. (3)]—see Exercise 1 in Ref. 30. For discussion of the two-dimensional delta function see
L. R.
, and
J.
Godines
, “
An analytical example of renormalization in two-dimensional quantum mechanics
,”
Am. J. Phys.
59
,
935
937
(
1991
);
R.
Jackiw
, “
Delta-function potentials in two- and three-dimensional quantum mechanics
,” in
M. A. B. Bég Memorial Volume
, edited by
A.
Ali
and
P.
Hoodbhoy
(
World Scientific
,
Singapore
,
1991
), pp.
25
42
.
3.
Notice that if $ψ(x)$ is normalizable, so too is $ψβ(x)$.
4.
K. S.
Gupta
and
S. G.
Rajeev
, “
Renormalization in quantum mechanics
,”
Phys. Rev. D
48
,
5940
5945
(
1993
).
5.
G.
Arfken
and
H.-J.
Weber
,
Mathematical Methods for Physicists
(
,
Orlando
(
2000
), 5th ed., Chap. 11. The other solution, $xIig(κx)$, diverges for large $x$.
6.
I. S.
and
I. M.
Ryzhik
,
Tables of Integrals, Series, and Products
(
,
San Diego
, (
1980
), Eqs. (6.576.4) and (8.332.3). Incidentally, states with different $κ$ are not orthogonal.
7.
One might ask what the usual approximation schemes have to say about this potential. Not much. For the general power law $V(x)=axν$ on $0, the WKB approximation yields $En≈a[(n−1∕4)ℏ2π∕maΓ(1∕ν+3∕2)∕Γ(1∕ν+1)]2ν∕(ν+2)$ [see
D. J.
Griffiths
,
Introduction to Quantum Mechanics
(
Prentice Hall
,
Englewood Cliffs, NJ
,
2004
), 2nd ed., Problem 8.11], and the exponent is infinite when $ν=−2$. The variational principle only confirms what we already knew—that the ground state is lower than every negative energy.
8.
Reference 6, p.
962
, Eqs. (3) and (4).
9.
Some authors use a plus sign in Eq. (24), which adds $π∕2$ to the phase shift. We prefer the minus sign, because it reduces to $δ=0$ when the potential is zero.
10.
For a list of accessible references see
C. V.
Siclen
, “
The one-dimensional hydrogen atom
,”
Am. J. Phys.
56
,
9
10
(
1988
).
11.
Indeed, because allowed energies must exceed $Vmin$, $E1>−a∕ϵ2$.
12.
For these parameters $κ1ϵ=1.024645$, $κ2ϵ=0.350972$, $κ3ϵ=0.122830$, and $κ4ϵ=0.043089$.
13.
For the ground state this inequality would appear to require $g⪡3$ (see Fig. 5), but in practice the approximation is good up to $g=3$. For the excited states $κ$ is smaller, and the approximation is valid for even higher $g$.
14.
Reference 5, Eqs. (11.112) and (11.118), and Ref. 6, Eqs. (8.331) and (8.332).
15.
This holds for $g<3$, as we can easily confirm by comparing the graph of Eq. (30) with Fig. 5. For larger values of $g$ the approximation itself is invalid for the ground state. Incidentally, Eq. (29) has solutions for negative $n$, but these are spurious, because they violate the assumption $κϵ⪡1$.
16.
The limiting case $g=0$ is obviously problematic—indeed, $K0(z)$ has no zeros for positive $z$. For $∣z∣⪡1$, $Γ(1+z)≈1−Cz$, where $C=0.577215$ is Euler’s constant, so for small $gargΓ(1+ig)≈−Cg$, and Eq. (30) is replaced by $κn=(2∕γϵ)exp(−nπ∕g)$, where $γ≡exp(C)=1.781072$. See Ref. 6, Eq. (8.321.1), and p.
xxviii
.
17.
We used the identity $Hig(2)(x)=[H−ig(1)(x)]*=e−πg[Hig(1)(x)]*$ (valid for real $x$ and real $g$). See Ref. 6, p.
969
.
18.
We used Ref. 6, Eq. (8.405.1), to express $Hν(1)$ in terms of $Jν$ and $Nν$, Eq. (8.440) to approximate $Jν$, Eq. (8.443) to approximate $Nν$, and Eq. (8.332.3) to eliminate $Γ(1−ig)$.
19.
Reference 6, Eqs. (8.331) and (8.332.1).
20.
Reference 6, Eqs. (8.441.1) and (8.444.1), and p.
xxviii
.
21.
Note that this is a correlated limit in which $ϵ$ and $g$ both go to zero in such a way as to hold $κ1$ in Eq. (42) fixed. This is not the same as going straight to $g=0$ and then letting $ϵ→0$ [Eq. (41)], which only reproduces the limiting value $π∕4$.
22.
This is not the only way to tame the $1∕x2$ potential. Other regularizations have been proposed. See, for example,
C.
Schwartz
, “
Almost singular potentials
,”
J. Math. Phys.
17
,
863
867
(
1976
);
H. E.
Camblong
,
L. N.
Epele
,
H.
Fanchiotti
, and
C. A.
Garca Canal
, “
Dimensional transmutation and dimensional regularization in quantum mechanics, I. General theory,” and “II. Rotational invariance
,”
Ann. Phys.
287
,
14
100
(
2001
). Our approach follows Ref. 4. It is important in principle to demonstrate that all regularizations lead to the same physical predictions. If they do not, the theory is non-renormalizable and there is very little that can be done with it.
23.
Of course, if we could detect several bound states, or measure the phase shifts at sufficiently high energy, then we could map out any departures from the $1∕x2$ potential. The question is whether we can make any sense out of the pure $1∕x2$ potential; renormalization offers a means for doing so. By the way, something very similar happens in quantum electrodynamics, where the theory, naively construed, yields an infinite mass for the electron. The introduction of a cutoff renders the mass finite but indeterminate. We take the observed mass of the electron as input and use it to eliminate any explicit reference to the cutoff. The resulting renormalized theory has been spectacularly successful, yielding by far the most precise (and precisely confirmed) predictions in all of physics.
24.
The classic example of an anomaly is the decay of the neutral pion, $π0→γ+γ$, which could not occur without the breaking of chiral symmetry.
25.
See, for instance,
E.
Zeidler
,
Applied Functional Analysis: Applications to Mathematical Physics
(
Springer
,
New York
,
1997
), pp.
116
117
.
26.
Note the logical structure here: We choose$DA$, but $DA†$ is then determined—it is the space of functions $ϕ$ (in $L2$) such that if $ψ$ is in $DA$ then Eq. (52) holds.
27.
H.
Weyl
, “
Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen
,”
Math. Ann.
68
,
220
269
(
1910
);
J.
von Neumann
, “
Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren
,”
Math. Ann.
102
,
49
131
(
1929
);
M. H.
Stone
, “
On one-parameter unitary groups in Hilbert space
,”
Ann. Math.
33
,
643
648
(
1932
).
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N. I.
Akhiezer
and
I. M.
Glazman
,
Theory of Linear Operators in Hilbert Space
(
Dover
,
New York
,
1993
);
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Reed
and
B.
Simon
,
Methods of Modern Mathematical Physics
(
,
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,
1975
);
G.
Hellwig
,
Differential Operators of Mathematical Physics
(
,
,
1964
).
29.
G.
Bonneau
,
J.
Faraut
, and
G.
Valent
, “
Self-adjoint extensions of operators and the teaching of quantum mechanics
,”
Am. J. Phys.
69
,
322
331
(
2001
).
30.
V. S.
Araujo
,
F. A. B.
Coutinho
, and
J. F.
Perez
, “
,”
Am. J. Phys.
72
,
203
213
(
2004
).
31.
There exist pathological functions that are square-integrable and yet do not go to zero at infinity, but there is no penalty for excluding them here. See, for example,
D. V.
Widder
,
(
Dover
,
New York
,
1998
), 2nd ed., p.
325
.
32.
We assume here that $α<1∕4$; for $α>1∕4$ we could run the same argument using $ψ=ϕ=u+$ to obtain the same result.
33.
More explicitly, functions in $DH$ vanish if $0⩽x⩽ϵ$ or $x⩾τ$ for arbitrarily small $ϵ$ and arbitrarily large $τ$.
34.
We follow the treatment in Ref. 30, where the special case $α=1∕4$ is posed as an exercise.
35.
Of course, the eigenvalues of a Hermitian operator are real, so $ϕ+$ and $ϕ−$ cannot be in $DH$; rather, the eigenfunctions we seek lie in $DH†$.
36.
Mathematicians usually take $η=1$, but this choice offends the physicist’s concern for dimensional consistency. In any case, it combines with other arbitrary constants at the end.
37.
This approximation assumes $Re(ig)>−1∕2$, which is fine as long as $α>0$ (the potential is attractive). It is of some interest to explore self-adjoint extensions of the repulsive$1∕x2$ potential, but we shall not do so here.
38.
Here $x0$ is simply a convenient packaging of the arbitrary constants $m$, $A±$, $λ$, and $η$. It is clear from Eq. (73) [if not from Eq. (74)] that $x0$ carries the dimensions of length, and hence the choice of a particular self-adjoint extension entails breaking the scale invariance that led to all the difficulties in Sec. II.
39.
This case violates our assumption in Ref. 37, so it should be taken with a grain of salt. See Ref. 30, Example 1, for a more rigorous analysis.
40.
This result agrees with Eq. (80) of Ref. 30, with $r→x$ and $φ→ψ∕x$.
41.
The term “self-adjoint extension” is potentially misleading, because at first sight it appears to involve a contraction, not an expansion, of the domain. The point is that you must start out with a Hermitian operator, and $H$ is not Hermitian with respect to the set of functions that satisfy the boundary condition $ψ(0)=0$. That is why we first had to restrict the domain (see Ref. 33), and the “extension” is with respect to that much more limited domain.
42.
By changing the boundary conditions it could be argued that we are radically altering the physical system, albeit at a single point. The process is analogous (in some cases identical) to adding a delta function to the potential, and it is hardly surprising that this changes the spectrum of allowed states. But the question was whether there is anything we could do to salvage the $1∕x2$ potential, and if the remedy is necessarily radical, so be it.
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and
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, “
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