Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order
The paper presents the derivation of the asymptotic behavior of -zeros of the modified Bessel function of imaginary order Kᵢᵥ(). This derivation is based on the quasiclassical treatment of the exponential potential on the positive half-axis. The asymptotic expression for the -zeros (zeros with respe...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2021
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| Цитувати: | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order. Yuri Krynytskyi and Andrij Rovenchak. SIGMA 17 (2021), 057, 7 pages |
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| author | Krynytskyi, Yuri Rovenchak, Andrij |
| author_facet | Krynytskyi, Yuri Rovenchak, Andrij |
| citation_txt | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order. Yuri Krynytskyi and Andrij Rovenchak. SIGMA 17 (2021), 057, 7 pages |
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| description | The paper presents the derivation of the asymptotic behavior of -zeros of the modified Bessel function of imaginary order Kᵢᵥ(). This derivation is based on the quasiclassical treatment of the exponential potential on the positive half-axis. The asymptotic expression for the -zeros (zeros with respect to order) contains the Lambert function, which is readily available in most computer algebra systems and numerical software packages. The use of this function provides much higher accuracy of the estimation compared to known relations containing the logarithm, which is just the leading term of () at large . Our result ensures accuracy sufficient for practical applications.
|
| first_indexed | 2026-03-19T06:50:25Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 057, 7 pages
Asymptotic Estimation for Eigenvalues
in the Exponential Potential and for Zeros of Kiν(z)
with Respect to Order
Yuri KRYNYTSKYI and Andrij ROVENCHAK
Department for Theoretical Physics, Ivan Franko National University of Lviv, Ukraine
E-mail: yurikryn@gmail.com, andrij.rovenchak@gmail.com
URL: https://physics.lnu.edu.ua/en/employee/krynytskyi-yu,
https://physics.lnu.edu.ua/en/employee/rovenchak-a
Received May 15, 2021, in final form June 01, 2021; Published online June 10, 2021
https://doi.org/10.3842/SIGMA.2021.057
Abstract. The paper presents the derivation of the asymptotic behavior of ν-zeros of the
modified Bessel function of imaginary order Kiν(z). This derivation is based on the quasi-
classical treatment of the exponential potential on the positive half axis. The asymptotic
expression for the ν-zeros (zeros with respect to order) contains the Lambert W function,
which is readily available in most computer algebra systems and numerical software pack-
ages. The use of this function provides much higher accuracy of the estimation comparing to
known relations containing the logarithm, which is just the leading term of W (x) at large x.
Our result ensures accuracies sufficient for practical applications.
Key words: quasiclassical approximation; exponential potential; ν-zeros; modified Bessel
functions of the second kind; imaginary order; Lambert W function
2020 Mathematics Subject Classification: 33C10; 81Q05; 81Q20
1 Introduction
The present work originates from an attempt to analyze the accuracy of numerical computa-
tions of the energy eigenvalues in steep potentials, one of which is the exponential potential.
Additionally to purely mathematical interest linking this problem to finding the modified Bessel
function zeros, such potentials appear in several physical problems, including quantum wells
in semiconductors [15, 24, 26] and various cosmological models [16, 18].
Even though the mathematical formulation of the problem is quite straightforward and can
be relatively simply reduced to well-known modified Bessel functions, advancing to practical
applications appears unexpectedly problematic. To be specific, the eigenvalues in the exponential
potential are expressed via zeros of the modified Bessel function of the second kind (known also
as the Macdonald function) Kiν(z) of imaginary order. The computation of the Kiν(z) zeros with
respect to order, known as ν-zeros, is not readily implemented in modern software even though
algorithms for this were proposed decades ago [8, 10]. Moreover, available asymptotic expansions
for large zeros reported in the literature [4, 9, 20] provide rather inaccurate estimations not
applicable for direct calculations. Our aim is to fill in this gap.
The paper is organized as follows. In Section 2, the quasiclassical approximation to the
quantization in the exponential potential is considered. The equivalence of this problem with
the problem of finding zeros of the Bessel Kiν(z) function is used in Section 3 to obtain the
asymptotic estimation for these zeros. Numerical comparison of the obtained asymptotics and
previously suggested expression for zeros is made in Section 4. Brief discussion in Section 5
concludes the paper.
mailto:yurikryn@gmail.com
mailto:andrij.rovenchak@gmail.com
https://physics.lnu.edu.ua/en/employee/krynytskyi-yu
https://physics.lnu.edu.ua/en/employee/rovenchak-a
https://doi.org/10.3842/SIGMA.2021.057
2 Yu. Krynytskyi and A. Rovenchak
2 Quasiclassical approximation
Consider a potential given by
U(x) =
{
U0e
2x/a for x > 0,
+∞ for x ≤ 0.
The Hamiltonian for x > 0 thus reads:
H =
p2
2m
+ U0e
2x/a.
The Bohr–Sommerfeld quantization condition is given by
x2∫
x1
√
2m[E − U(x)] dx = π~
(
n+
3
4
)
, (1)
where x1 and x2 are the classical turning points and the 3/4 correction originates from 1/2 as
a contribution due to the hard wall at x1 = 0 and another 1/4 contribution from the turning
point x2 [7, 12, 21, 25]. The quantization condition thus reads
√
2m
a
2
ln E
U0∫
0
√
E − U0e2x/a dx =
√
2ma
(√
E artanh
√
1− U0
E
−
√
E − U0
)
= π~
(
n+
3
4
)
, (2)
where artanh z denotes the principal value of the inverse hyperbolic tangent. To approach an
analytical solution of this equation we will expand the functions of E into series taking into
account that values of E are large. Simple transformations yield√
E
U0
ln
2
e
+
√
E
U0
ln
√
E
U0
+O
(√
U0
E
)
=
π~
(
n+ 3
4
)
√
2mU0a2
. (3)
Solutions for the transcendental equation
bX +X lnX = P
are given by
X =
P
W (ebP )
,
where W (z) is the Lambert function being the solution to
W eW = z.
Finally, for the spectrum we obtain
En =
~2
2m
[
π
a
(
n+
3
4
)]2 1
W 2
(
2
e
π~
(
n+ 3
4
)
√
2mU0a2
) . (4)
Note that here, as in the quantization condition (1), n enumerates the wavefunction nodes, i.e.,
starts from n = 0 at the ground state.
Asymptotic Estimation for Eigenvalues in the Exponential Potential 3
0
1
2
3
4
5
6
50 100 150 200 250 300 350 400 450 500
f(x)
W(x )
ln x
2 terms
3 terms
x
Figure 1. Comparison of the Lambert W function and its approximation by one, two, and three terms
in the large x series (5).
For large arguments, the Lambert W function can be expanded as a series in the following
form [11]
W (z) = ln z − ln ln z +
ln ln z
ln z
+ · · · . (5)
The leading order of the spectrum for large n is thus
En ∝
n2
ln2 n
. (6)
Series (5), however, converges very slowly and even the first two terms do not provide acceptable
approximation to be used in practical calculations. For illustration, see Figure 1. While the
approximation with three terms seems satisfactory from the figure, this is an artifact of the
vertical scale being too coarse. The accuracy of such an approximation is still far from good, as
we will see in Section 4.
Presently, the Lambert W is well implemented in computer algebra systems, such as Maple,
Mathematica, Maxima, etc., and numerical software packages, including Perl, Python, R,
GNUPlot, etc. It can be calculated using a fast converging iterative procedure with Newton’s
method [17].
3 Zeros of Kiν(z)
Expressions for eigenvalues of the Schrödinger equation with exponential potentials of the e−x
type have been long known to involve Bessel function zeros [3, 6, 19, 23]. A generalization
of the respective approaches for the positive sign in the exponential is straightforward and can
be considered a textbook problem, cf. [2, 22]. Below we will briefly recall the derivation chain
for consistency.
The stationary Schrödinger equation for our problem reads(
− ~2
2m
d2
dx2
+ U0e
2x/a
)
ψn(x) = Enψn(x)
and the boundary conditions are
ψn(0) = ψn(∞) = 0.
4 Yu. Krynytskyi and A. Rovenchak
Introducing the dimensionless variable ξ = 2x/a and using further ~2/
(
2ma2
)
as the unit of
energy we obtain the following equation(
−4
d2
dξ2
+ ueξ
)
ψn(ξ) = εnψn(ξ),
where
u = U0
2ma2
~2
, εn = En
2ma2
~2
.
Changing the variable y =
√
u eξ/2 we ultimately arrive at
y2
d2ψn(y)
dy2
+ y
dψn(y)
dy
− (y2 − εn)ψn(y) = 0.
The solutions of the above equation are given by the modified Bessel functions with the imaginary
index iν = i
√
εn, namely
ψn(y) = AnIiν(y) +BnKiν(y).
To satisfy the boundary condition ψn(x = ∞) = ψn(y = ∞) = 0 we have to get rid of the
function Iiν(y) that tends to infinity as y →∞. The wave functions are thus
ψn(y) = BnKiν(y).
Another boundary condition, ψn(x = 0) = ψn(y =
√
u) = 0 yields
Kiν(
√
u) = 0.
The eigenvalues εn are thus defined by the imaginary zeros iνn of Kiν(
√
u):
Ki
√
εn(
√
u) = 0.
Simple expressions for νn are not known. Implicit expressions for them involve, in partic-
ular, Airy function zeros [5, 13, 14] while the asymptotic behavior of νn as n → ∞ is known
only in rather rough estimations [4, 9, 20] being weakly convergent due to the lnn and ln lnn
dependences. From our asymptotic estimation (4) for eigenvalues, shifting n → n − 1 to start
counting zeros from unity, we have the following asymptotic behavior of the zeros of Kiν(z):
νasymp
n =
π
(
n− 1
4
)
W
(
2π
(
n− 1
4
)
e z
)[1 +O
(
lnn
n2
)]
, n = 1, 2, 3, . . . . (7)
As we will see in the next section, this expression gives values quite close to the real ones even
for small n. For instance, already the first zero of Kiν(1) is approximated with the relative error
less than 1%. For Kiν(2), the first zero has the relative error of 3% dropping below 1% already
at the third zero. The accuracy of the approach is demonstrated in Figure 2. The remainder
term can be estimated from (3).
Asymptotic Estimation for Eigenvalues in the Exponential Potential 5
10
−10
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
1 10 100 1000
z = 0.5
z = 1
z = 2
z = 5
z = 10
n
Figure 2. Relative error |νasymp
n /νn − 1| of the asymptotic expression (7) for zeros νn of Kiν(z).
4 Comparison of asymptotic expressions for Kiν(z) zeros
Zeros of Kν coincide with those of the Hankel function H
(1)
ν in view of the relation [1, 14]
Kν(z) =
iπ
2
eiπν/2H(1)
ν
(
zeiπ/2
)
.
So, one can use equally results for Kν and H
(1)
ν available in the literature. Some closed-form
asymptotic results can be obtained from both classical [9, 20] and new [4] works. They are as
follows
paper [20]: νMK
n =
π
(
n+ 1
4)
ln
(
π
(
n+ 1
4
)
e z
)[1 +O
(
ln lnn
lnn
)]
; (8a)
paper [9]: νCn =
πn
ln
(
3πn
e z
)[1 +O
(
ln lnn
lnn
)]
; (8b)
paper [4]: νBK
n ∼ πn
lnn
. (8c)
In Figure 3 we compare the above asymptotics with our result expressed via the Lambert W
function (7) and its expansion (5) with one, two, and three terms.
As we can see, keeping the logarithmic term only cannot provide sufficient accuracy even
for as large ν-zero orders as n ' 103–104. The situation slightly improves with three terms
in expansion (5). The asymptotic containing exact W (x) values yields the best result.
Curiously, the function V (x) solving(
artanh
√
1− 1
V
−
√
1− 1
V
)
V = x, (9)
that is, the exact quantization condition (2), gives for the ν-zeros of Kiν(z)
νn = zV
(
π
(
n− 1
4
)
z
)
(10)
that provides only a slightly better extimation comparing to equation (7), as can be seen from
Figure 3. The use of a similar function was suggested in [2].
6 Yu. Krynytskyi and A. Rovenchak
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10 100 1000 10000
ν
n
via W
log term
2 terms
3 terms
ν
n
via V
ν
MK
n
ν
C
n
ν
BK
n
n
Figure 3. Relative errors |νasymp
n /νn−1| of several asymptotic expressions (7) (including expansion (5)),
(8), and (9)–(10) for zeros νn of Kiν(1).
5 Discussion
One can show that for the xk potential on the positive half-axis the leading order of the eigenval-
ues is En ∝ n2k/(2+k) [25]. Obviously, in the limit of k → ∞ the energy levels are proportional
to n2 corresponding to a particle in a box (the infinite square well). Considering the first
correction to this dependence and the asymptotic relation (6) we have
En ∝
n2 n−4/k for U(x) ∝ xk,
n2
ln2 n
for U(x) ∝ e2x/a.
This agrees with the fact that the exponential potential at large x is steeper than any power-law
dependence xk hence the deviation from the square box n2 for the exponential potential should
be weaker that any power nα→−0 implying a logarithmic law.
The obtained asymptotic expressions for energy levels and directly related ν-zeros of the
modified Bessel functions contain the Lambert W function. Nowadays, the calculation of this
function is quite an easy task for most software so the appearance of W (x) should not repel one
from using these asymptotics. As we have shown, the substitution of the Lambert function with
its leading logarithmic term cannot provide satisfactory numerical accuracy even for rather high
energy levels En or zeros νn for n ∼ 103–104.
We should remark that preserving the 3/4 correction in the quantization condition (yielding
the −1/4 shift in the expression for ν-zeros) is essential for the accuracy of our approach not
only for low values of n but in the whole domain n > 0. This correction allows keeping the
relative error of the asymptotics as low as O
(
lnn
n2
)
.
To summarize, we propose an asymptotic estimation for the eigenvalues in the exponen-
tial potential and for the ν-zeros of the modified Bessel function of the second kind with and
imaginary order, which are of sufficient accuracy to be used in practical calculations. This
estimations are expressed in particular via the Lambert W function being presently well imple-
mented in modern software. Further attempts to improve the approach based on the numerical
solution of the equation for the quasiclassical quantization condition only lead to an insignificant
increase in the accuracy. Our asymptotic is also useful as an initial guess for the numerical com-
putation of ν-zeros of a given order as well as and estimation for distances between consecutive
zeros.
Asymptotic Estimation for Eigenvalues in the Exponential Potential 7
For convenience, we uploaded to the online repository at https://doi.org/10.5281/
zenodo.4573305 the data for first 10 000 zeros for several values of z together with their asymp-
totic estimations according to our formula.
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1 Introduction
2 Quasiclassical approximation
3 Zeros of K i nu(z)
4 Comparison of asymptotic expressions for K i nu(z) zeros
5 Discussion
References
|
| id | nasplib_isofts_kiev_ua-123456789-211366 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T06:50:25Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Krynytskyi, Yuri Rovenchak, Andrij 2025-12-30T15:57:13Z 2021 Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order. Yuri Krynytskyi and Andrij Rovenchak. SIGMA 17 (2021), 057, 7 pages 1815-0659 2020 Mathematics Subject Classification: 33C10; 81Q05; 81Q20 arXiv:2103.01732 https://nasplib.isofts.kiev.ua/handle/123456789/211366 https://doi.org/10.3842/SIGMA.2021.057 The paper presents the derivation of the asymptotic behavior of -zeros of the modified Bessel function of imaginary order Kᵢᵥ(). This derivation is based on the quasiclassical treatment of the exponential potential on the positive half-axis. The asymptotic expression for the -zeros (zeros with respect to order) contains the Lambert function, which is readily available in most computer algebra systems and numerical software packages. The use of this function provides much higher accuracy of the estimation compared to known relations containing the logarithm, which is just the leading term of () at large . Our result ensures accuracy sufficient for practical applications. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order Article published earlier |
| spellingShingle | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order Krynytskyi, Yuri Rovenchak, Andrij |
| title | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order |
| title_full | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order |
| title_fullStr | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order |
| title_full_unstemmed | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order |
| title_short | Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kᵢᵥ() with Respect to Order |
| title_sort | asymptotic estimation for eigenvalues in the exponential potential and for zeros of kᵢᵥ() with respect to order |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211366 |
| work_keys_str_mv | AT krynytskyiyuri asymptoticestimationforeigenvaluesintheexponentialpotentialandforzerosofkivwithrespecttoorder AT rovenchakandrij asymptoticestimationforeigenvaluesintheexponentialpotentialandforzerosofkivwithrespecttoorder |