Exact solvability of PDM systems with extended Lie symmetries
It is shown that all PDM Schrodinger equations admitting more than five-dimensional Lie symmetry algebras (whose completed list can be found in paper [J.~Math. Phys. 58  (2017), 083508, 16 pp.] are exactly solvable. The corresponding exact solutions are presented. The supersymmetric asp...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2019
|
| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/361 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Download file: |  |
Institution
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552855933550592 |
|---|---|
| author | Nikitin, A. Нікітін, А. |
| author_facet | Nikitin, A. Нікітін, А. |
| author_institution_txt_mv | [
{
"author": "A. Nikitin",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Nikitin, A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | It is shown that all PDM Schrodinger equations admitting more than five-dimensional Lie symmetry algebras (whose completed list can be found in paper [J.~Math. Phys. 58  (2017), 083508, 16 pp.] are exactly solvable. The corresponding exact solutions are presented. The supersymmetric aspects of the exactly solvable systems are discussed. |
| first_indexed | 2026-08-04T01:06:04Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 113�130
ÓÄÊ 517.912:512.816
Exact solvability of PDM systems
with extended Lie symmetries
A.G. Nikitin
Institute of Mathematics of NAS of Ukraine, Kyiv
E-mail: nikitin@imath.kiev.ua
Ïîêàçàíî, ùî óñi ðiâíÿííÿ Øðüîäiíãåðà çi çìiííèì ïàðàìåòðîì ìàñè,
ÿêi äîïóñêàþòü àëãåáðè iíâàðiàíòíîñòi ðîçìiðíîñòi áiëüøå ï'ÿòè (ïîâ-
íèé ñïèñîê òàêèõ ðiâíÿíü íàâåäåíî ó ðîáîòi [J. Math. Phys. 58 (2017),
083508, 16 pp.], ¹ òî÷íî ðîçâ'ÿçíèìè. Çíàéäåíî ó ÿâíîìó âèãëÿäi âiäïî-
âiäíi ðîçâ'ÿçêè òà ïîêàçàíî ¨õ ñóïåðñèìåòðè÷íó ïðèðîäó.
It is shown that all PDM Schr�odinger equations admitting more than �ve-
dimensional Lie symmetry algebras (whose completed list can be found in
paper [J. Math. Phys. 58 (2017), 083508, 16 pp.] are exactly solvable. The
corresponding exact solutions are presented. The supersymmetric aspects
of the exactly solvable systems are discussed.
1. Introduction. Group classification of differential equations con-
sists in the specification of non-equivalent classes of such equations which
possess the same symmetry groups. It is a rather attractive research field
which has both fundamental and application values.
A perfect example of group classification of fundamental equations
of mathematical physics was presented by Boyer [3] who had specified
all inequivalent Schrödinger equations with time independent potentials
admitting symmetries with respect to Lie groups, see also [1, 7, 10],
where particular important symmetries were discussed, and [14], where
the Boyer results were corrected. These old results have a big impact
since include a priori information about all symmetry groups which can
be admitted by the fundamental equation of quantum mechanics. Let
us mention also that the nonlinear Schrödinger equation as well as the
generalized Ginsburg–Landau quasilinear equations have been classified
also [11, 15] as well as symmetries of more general systems of reaction-
diffusion equations [16, 17].
In contrary, the group classification of Schrödinger equations with
position dependent mass (PDM) was waited for a very long time. There
114 A.G. Nikitin
were many papers devoted to PDM Schrödinger equations with particu-
lar symmetries, see, e.g., [5, 8, 20, 21]. But the complete group classifica-
tion of these equations appears only recently in [18] and [13, 19] for the
stationary and time dependent equations correspondingly. A system-
atic search for the higher order symmetries if the PDM systems started
in [12]. So late making of such important job have to cause the blame for
experts in group analysis of differential equations, taking into account
the fundamental role played by such equations in modern theoretical
physics!
Let us remind that the PDM Schrödinger equations are requested for
the description of various condensed-matter systems such as semicon-
ductors, quantum liquids, and metal clusters, quantum wells, wires and
dots, super-lattice band structures, etc.
It happens that the number of PDM systems with different Lie sym-
metries is rather extended. Namely, in [13] seventy classes of such sys-
tems are specified. Twenty of them are defined up to arbitrary parame-
ters, the remaining fifty systems include arbitrary functions.
The knowledge of all Lie groups which can be admitted by the PDM
Schrödinger equations has both fundamental and application values. In
particular, when construct the models with a priory requested symme-
tries we can use the complete lists of inequivalent PDM systems pre-
sented in [19] for d = 2 and [13] for d = 3. Moreover, in many cases
a sufficiently extended symmetry induces integrability or exact solvabil-
ity of the system, and just this aspect will be discussed in the present
paper.
It will be shown that all PDM systems admitting six parametric Lie
groups of symmetries or more extended symmetries are exactly solvable.
Moreover, the complete sets of solutions of the corresponding stationary
PDM Schrödinger equations will be presented explicitly.
There exist a tight connection between the complete solvability and
various types of higher symmetries and supersymmetries. We will see
that extended Lie symmetries also can cause the exact solvability. More-
over, the systems admitting extended Lie symmetries in many cases are
supersymmetric and superintegrable.
2. PDM Schrödinger equations with extended Lie symmet-
ries. In [13] we present the group classification of PDM Schrödinger
equations
Lψ ≡
(
i
∂
∂t
−H
)
ψ = 0, (1)
Exact solvability of PDM systems with extended Lie symmetries 115
where H is the PDM Hamiltonian of the following generic form
H = 1
4
(
mαpam
βpam
γ +mγpam
βpam
α
)
+ V̂ , pa = −i
∂
∂xa
. (2)
Here m = m(x) and V̂ = V̂ (x) are the mass and potential depending
on spatial variables x = (x1, x2, x3), and summation with respect to the
repeating indices a is imposed over the values a = 1, 2, 3. In addition, α,
β and γ are the ambiguity parameters satisfying the condition α+β+γ =
−1.
The choice of values of the ambiguity parameters can be motivated
by physical reasons, see a short discussion of this point in [13].
Hamiltonian (2) can be rewritten in the following more compact form
H = 1
2pafpa + V, (3)
where
V = V̂ + 1
4 (α+ γ)faa + αγ
fafa
2f
(4)
with f = 1
m , fa = ∂f
∂xa
and faa = ∆f = ∂fa
∂xa
.
In the following text representation (4) will be used.
In accordance with [13] there is a big variety of Hamiltonians (4)
generating non-equivalent continuous point symmetries of equation (2).
The corresponding potential and mass terms are defined up to arbitrary
parameters or even up to arbitrary functions.
In the present paper we consider the PDM systems defined up to
arbitrary parameters. Only such systems admit the most extended Lie
symmetries. Using the classification results presented in [13, 18] we
enumerate these systems in the following Table 1, where ϕ = arctan x2
x1
and the other Greek letters denote arbitrary constants parameters, which
are supposed not to be zero simultaneously. Moreover, λ and ω are either
real or imaginary, the remaining parameters are real.
The symmetry operators presented in column 4 of the table are given
by the following formulae
Pi = pi = −i
∂
∂xi
, D = xnpn − 3i
2 ,
Mij = xipj − xjpj , M0i = 1
2
(
Ki + Pi
)
, M4i = 1
2
(
Ki + Pi
)
,
116 A.G. Nikitin
B1
1 = λ sin(λt)M12
(
λ2ϕ+ ν
)
cos(λt), B1
2 =
∂
∂t
B1
1 ,
B2
1 = sin(λt)D − cos(λt)
(
λ ln(r) + ν
λ
)
, B2
2 =
∂
∂t
B2
1 ,
N1
1 = ω cos(ωσt)L3 − sin(ωσt)
(
i∂t − ω2e−σΘ
)
, N1
2 =
∂
∂t
N1
1 ,
N2
1 = ω cos(ωσt)D + sin(ωσt)
(
i∂t − ω2r−σ
)
, N2
2 =
∂
∂t
N1
1 , (5)
where Ki = xnxnpi − 2xiD and indices i, j, k, n take the values 1, 2, 3.
Rather surprisingly, all systems (except ones given in items 4 and 5)
presented in Table 1 are exactly solvable. In the following sections we
present their exact solutions. To obtain these solutions we use some
nice properties of the considered systems like superintegrability and su-
persymmetry with shape invariance. Let us remind that the quantum
mechanical system is called superintegrable if it admits more integrals
of motion than its number of degrees of freedom.
In accordance with Table 1 we can indicate 11 inequivalent PDM
systems which are defined up to arbitrary parameters and admit Lie
symmetry algebras of dimension five or higher. Notice that the systems
fixed in items 4 and 5 admit five dimension symmetry algebras while the
remaining systems admit more extended symmetries.
3. Systems with fixed mass and potentials. Firstly we consider
those systems whose mass and potential terms are fixed, i.e., do not
include arbitrary parameters. These systems are presented in items 1, 2
of Table 1 and others provided the mass does not depends on parameters
and parameters of the potential are trivial.
3.1. System invariant with respect to algebra so(4). Consider
Hamiltonian (3) with functions f and V presented in item 1 of Table 1:
H = 1
2pa
(
1 + r2
)2
pa − 3r2. (6)
The eigenvalue problem for this Hamiltonian can be written in the fol-
lowing form
Hψ = 2Eψ, (7)
where E are yet unknown numbers.
Equation (7) admits six integrals of motion MAB , A,B = 1, 2, 3, 4,
presented in equation (5). Let us write them explicitly
Mab = xapb − xbpa, M4a = 1
2
(
r2 − 1
)
pa − xaxbpb + 3i
2 x
a. (8)
Exact solvability of PDM systems with extended Lie symmetries 117
Table 4. PDM systems with extended Lie symmetries.
no.
inverse
mass f
potential V symmetries
1
(
r2 + 1
)2 −3r2 M41,M42,M43,
M21,M31,M32
2
(
r2 − 1
)2 −3r2 M01,M02,M03,
M21,M31,M32
3 x2
3 ν ln(x3) P1, P2, M12, D + νt
4 r̃3 κx3 + λr̃ P3 + κt, D + it∂t, M12
5 x3
1 λx1 + κx3 P3 + κt, P2, D + it∂t
6 xσ+2
3 κxσ3
P1, P2, M12, D + iσt∂t,
σ 6= 0, 1,−2
7 r̃σ+2eλϕ κr̃σeλϕ
M12 + iλt∂t, P3,
D + iσt∂t, σ 6= 0
8 r̃2 λ2
2 ϕ
2 + µϕ+ ν ln(r̃) B1
1 , B
1
2 , D + νt, P3
9 r̃2eσϕ κeσϕ + ω2
2 e−σϕ N1
1 , N
1
2 , P3, D, K3
10 r2 ν ln(r) + λ2
2 ln(r)2 B2
1 , B
2
2 , L1, L2, L3
11 r2+σ κrσ + ω2
2 r
−σ N2
1 , N
2
2 , L1, L2, L3
Operators (8) form a basis of algebra so(4). Moreover, the first
Casimir operator of this algebra is proportional to Hamiltonian (6) up
to the constant shift
C1 = 1
2MABMAB = 1
2 (H − 9),
while the second Casimir operator C2 = εABCDMABMCD appears to be
zero.
Thus like the Hydrogen atom system (7) admits six integrals of mo-
tion belonging to algebra so(4) and is maximally superintegrable.
Using our knowledge of unitary representations of algebra so(4) is
possible to find eigenvalues E algebraically
E = 4n2 + 5, (9)
where n = 0, 1, 2, . . . are natural numbers.
118 A.G. Nikitin
To find the eigenvectors of Hamiltonian (6) corresponding to eigen-
values (9) we use the rotation invariance of (7) and separate variables.
Introducing spherical variables and expanding solutions via spherical
functions
ψ =
1
r
∑
l,m
φlm(r)Y lm, (10)
we come to the following equations for radial functions(
−
(
r2 + 1
)2( ∂2
∂r2
− l(l + 1)
r2
)
− 4r
(
r2 + 1
) ∂
∂r
− 2r2
)
ϕlm
=
(
4n2 + 1
)
ϕlm,
where l = 0, 1, 2, . . . are parameters numerating eigenvalues of the squa-
red orbital momentum. The square integrable solutions of these equa-
tions are
ϕlm = Cnlm
(
r2 + 1
)−n− 1
2 rl+1F
(
[A,B], [C]− r2
)
, (11)
where
A = −n+ l + 1, B = −n+ 1
2 , C = l + 3
2 .
F(· · · ) is the hypergeometric function and Cnlm are integration constants.
Solutions (11) tend to zero at infinity provided n is a natural number
and l ≤ n− 1.
Thus the system (7) is maximally superintegrable and exactly sol-
vable.
3.2. System invariant with respect to algebra so(1, 3). The
next Hamiltonian we consider corresponds to functions f and V pre-
sented in item 2 of Table 1. The related eigenvalue problem includes the
following equation
Hψ ≡ − 1
2
(
∂a
(
1− r2
)2
∂a + 6r2
)
ψ = Eψ. (12)
Equation (12) admits six integrals of motion Mµν , µ, ν = 0, 1, 2, 3, given
by equation (5), which can be written explicitly in the following form
Mab = xapb − xbpa,
M0a = 1
2
(
r2 + 1
)
pa − xaxbpb + 3i
2 x
a, a, b = 1, 2, 3. (13)
Exact solvability of PDM systems with extended Lie symmetries 119
These operators form a basis of algebra so(1, 3), i.e., the Lie algebra of
Lorentz group.
As in the previous section, the corresponding first Casimir operator
is expressed via the Hamiltonian, namely
C1 = 1
2M
abMab −M0aM0a = 1
2 (H + 9), (14)
while the second one appears to be zero.
Using our knowledge of irreducible unitary representations of Lorentz
group we find eigenvalues of C1 and C2 in the form [2, 9]:
c1 = 1− j2
0 − j2
1 , c2 = 2ij0j1,
where j0 and j1 are quantum numbers labeling irreducible representa-
tions. Since the second Casimir operator C2 is trivial, we have c1 = j0 =
0. So there are two possibilities [9]: either j1 is an arbitrary imaginary
number, and the corresponding representation belongs to the principal
series, or j1 is a real number satisfying |j1| ≤ 1, and we come to the
subsidiary series of IRs. So
j1 = iλ, c1 = 1− j2
1 = λ2 + 1, (15)
where λ is an arbitrary real number, or, alternatively,
0 ≤ j1 ≤ 1, c1 = 1− j2
1 . (16)
In accordance with (14) the related eigenvalues E in (12) are
E = −5− j2
1 . (17)
In view of the rotational invariance of equation (12) it is convenient
to represent solutions in form (10). As a result we obtain the following
radial equations(
−
(
r2 − 1
)2( ∂2
∂r2
− l(l + 1)
r2
)
− 4r
(
r2 − 1
) ∂
∂r
− 2r2
)
ϕlm
= (Ẽ + 4)ϕlm. (18)
The general solution of (18) is
ϕlm =
(
1− r2
)− 1
2−k(Cklmrl+1F
(
[A,B], [C], r2
)
120 A.G. Nikitin
+ C̃klmr
−lF
(
[Ã, B̃], [C̃], r2
))
, (19)
where
A = −k + l + 1, B = −k + 1
2 , C = l + 3
2 ,
à = −k − l, B̃ = −k + 1
2 , C̃ = 1
2 − l, k = 1
2
√
−Ẽ − 5
and is singular at r = 1. However, for C̃klm = 0 and k = j1 the solutions
are normalizable in some specific metric [18].
Thus the system presented in item 7 of Table 1 is exactly solvable too.
The corresponding eigenvalues and eigenvectors are given by equations
(15), (16), (17) and (19), respectively.
3.3. Scale invariant systems. Consider one more PDM system
which is presented in item 3 of the table and includes the following
Hamiltonian: Let us note that the free fall effective potential appears also
one more system specified in Table 1. Thus, considering the inverse mass
and potential specified in item 3 we come to the following Hamiltonian
H = −1
2
(
x3
∂
∂x3
x3
∂
∂x3
+ x3
∂
∂x3
+ x2
3
(
∂2
∂x2
1
+
∂2
∂x2
2
))
+ ν ln(x3). (20)
Equation (12) with Hamiltonian given in (20) can be easily solved by
separation of variables in Cartesian coordinates. Expanding the wave
function ψ via eigenfunctions of integrals of motion P1 and P2:
ψ = exp(−i(k1x1 + k2x2))Φ(k1, k2, x3) (21)
and introducing new variable y = ln(x3) we come to the following equa-
tion for Φ = Φ(k1, k2, x3):
−∂
2Φ
∂y2
+
((
k2
1 + k2
2
)
exp(2y) + 2νy
)
Φ = ẼΦ, (22)
where Ẽ = 2E − 1
4 .
Here we consider the simplest version of equation (22) when parame-
ter ν is trivial
−∂
2Φ
∂y2
+
(
k2
1 + k2
2
)
exp(2y)Φ = ẼΦ. (23)
Exact solvability of PDM systems with extended Lie symmetries 121
This equation is scale invariant and can be easily solved. Its square
integrable solutions are given by Bessel functions
Ψ = CEk1k2
K
i
√
Ẽ
(√
k2
1 + k2
2 ln(x3)
)
,
where CEk1k2
are integration constants and Ẽ are arbitrary real parame-
ters.
It is interesting to note that there are rather non-trivial relations
between the results given in the present and previous sections. Equa-
tion (23) admits six integrals of motion which are nothing but the fol-
lowing operators
P1, P2, K1, K2, M12, D, (24)
which are presented in equations (5).
Like operators (13) integrals of motion (24) form a basis of the Lie
algebra of Lorentz group, and we again can find the eigenvalues of Hamil-
tonian (23) algebraically by direct analogy with the above. We will not
present this routine procedure since there exist strong equivalence rela-
tions between Hamiltonians (23) with zero ν and (6). To find them we
note that basis (24) is equivalent to the following linear combinations of
the basis elements
M01, M02,M04, M41 M42, M12, (25)
whose expressions via operators (24) are given by equation (5). To re-
duce (25) to the set (13) it is sufficient to change subindices 4 to 3, i.e.,
to make the rotation in the plane 43. The infinitesimal operator for such
rotation is given by the following operator
M43 = 1
2 (K3 + P3) = 1
2
(
r2 − 1
)
p3 − x3xbpb + 3i
2 x3,
which belongs to the equivalence group of equations. Solving the cor-
responding Lie equations and choosing the group parameter be equal π2
we easily find the requested equivalence transformations.
One more scale invariant system is presented in item 8 where all pa-
rameters of potential are zero. The relation Hamiltonian looks as follows
H = −r̃ ∂
∂xα
r̃
∂
∂xα
− xα
∂
∂xα
− r̃2 ∂
2
∂x2
3
, α = 1, 2. (26)
122 A.G. Nikitin
Considering the eigenvalue problem for (26) it is convenient to use the
cylindrical variables
r̃ =
√
x2
1 + x2
2, ϕ = arctan
x2
x1
, x3 = z (27)
and expand solutions via eigenfunctions of M12 and P3 = −i ∂∂z :
Ψ = exp[i(κϕ+ ωz)]Φκω(r̃), κ = 0,±1,±2, . . . , −∞ < ω <∞.
As a result we come to the following equations for radial functions Φ =
Φκω(r̃):
−
(
r̃
∂
∂r̃
r̃
∂
∂r̃
+ r̃
∂
∂r̃
+ ω2
)
Φ =
(
Ẽ − κ2
)
Φ.
Square integrable (with the weight r̃) solutions of this equation are
Φκω =
1
r̃
Jα(ωr̃), α = κ2 + 1− Ẽ, (28)
where Jα(ωr̃) is Bessel function of the first kind. Functions (28) are
normalizable and disappear at r̃ = 0 provided α ≤ 0. The rescaled
energies Ẽ continuously take the values κ2 ≤ Ẽ ≤ ∞.
The last scale invariant system which we have to consider is fixed in
item 10 where ν = λ = 0. We will do it later in the end of the following
section.
4. Systems defined up to arbitrary parameters. In previous
section we present exact solutions for systems with fixed potential and
mass terms. In the following we deal with the systems defined up to
arbitrary parameters.
4.1. The system with oscillator effective potential. Let us
consider equation (1) with f and V are functions fixed in item 10 of
Table 1, i.e.,
i
∂ψ
∂t
=
(
−1
2
∂
∂xa
r2 ∂
∂xa
+ ν ln(r) +
λ2
2
ln(r)2
)
ψ.
These equations admit extended Lie symmetries (whose generators
are indicated in the table) being invariant with respect to six-parametri-
cal Lie group. Let us show that they also admit hidden supersymmetries.
Exact solvability of PDM systems with extended Lie symmetries 123
In view of the rotational invariance and symmetry of the considered
equations with respect to shifts of time variable, it is reasonable to search
for their solutions in spherical variables, i.e., in the following form
Ψ = e−iEtRlm(r)Ylm(ϕ, θ), (29)
where ϕ and θ are angular variables and Ylm(ϕ,ϕ) are spherical func-
tions, i.e., eigenvectors of L2 = L2
1 + L2
2 +M2
12 and M12. As a result we
come to the following radial equations(
−r ∂Rlm
∂r
r
∂Rlm
∂r
− r ∂Rlm
∂r
+ l(l + 1) + ν ln(r) +
λ2
2
ln(r)2
)
Rlm = 2ERlm. (30)
Introducing new variable y =
√
2 ln(r) we can rewrite equation (30)
in the following form(
− ∂2
∂y2
+ l(l + 1) + νy +
λ2
2
y2
)
Rlm(y) = ẼRlm(y), (31)
where Ẽ = E − 1
4 .
Let λ 6= 0 then equation (31) is reduced to the 1D harmonic oscillator
up to the additional term l(l+1). The admissible eigenvalues Ẽ are given
by the following formula
Ẽ = n+ l(l + 1),
where n is a natural number. The corresponding eigenfunctions are
well known and we will not presented them here. The same is true for
supersymmetric aspects of the considered system.
If parameter λ is equal to zero then (31) reduces to equation with free
fall potential slightly modified by the term l(l + 1). The corresponding
solutions can be found in textbooks devoted to quantum mechanics. If
both parameters ν and λ are zero, equation (31) is solved by trigono-
metric or hyperbolic functions. The corresponding PDM Schrödinger
equation is scale invariant, i.e., belongs to the class considered in the
previous section.
4.2. The systems with potentials equivalent to 3d oscilla-
tor. Consider now the system represented in item 11 of the table. The
124 A.G. Nikitin
corresponding equation (1) takes the following form
i
∂ψ
∂t
=
(
−1
2
∂ar
σ+2∂a + κr2σ +
ω2
r2σ
)
ψ. (32)
Like in previous section we represent the wave function in the form given
in (29) and came to the following radial equation
−r2σ+2 ∂
2Rlm
∂r2
− (2σ + 4)r2σ+1 ∂Rlm
∂r
+
(
r2σ(l(l + 1) + κ) + ω2r−2σ
)
Rlm = 2ERlm. (33)
Using the Liouville transform
r → z = r−σ, Rlm → R̃lm = z
σ+3
2σ Rlm,
we reduce (33) to the following form
−σ2 ∂
2R̃lm
∂z2
+
(
l(l + 1) + δ
z2
+ ω2z2
)
R̃lm = 2ER̃lm, (34)
where δ = 3
4 (σ + 1)(σ + 3) + 2κ.
Equation (34) describes a deformed 3d harmonic oscillator including
two deformation parameters, namely, σ and κ.
Let
2κ = −σ2 − 3σ − 2,
then equation (34) is reduced to the following form
HlR̃lm ≡
(
−σ2 ∂
2
∂z2
+
(2l + 1)2−σ2
4z2
+ ω2z2
)
R̃lm = 2ER̃lm. (35)
Equation (35) is shape invariant. Hamiltonian Hr can be factorized
Hl = a+
l al − Cl, (36)
where
a = −σ ∂
∂z
+W, a+ = σ
∂
∂z
+W,
W =
2l + 1 + σ
2z
+ ωz, Cl = ω(2l + 2σ + 1).
Exact solvability of PDM systems with extended Lie symmetries 125
The superpartner Ĥl of Hamiltonian (36) has the following property
Ĥl ≡ ala+
l + Cl = Hl+σ + Cl.
Thus our Hamiltonian is shape invariant.
Thus to solve equation (35) we can use the standard tools of SUSY
quantum mechanics and find the admissible eigenvalues in the following
form
En = ω
(
2nσ + l + σ + 1
2
)
= ω
(
2n+ l + 3
2
)
+ δω(2n+ 1), (37)
where δ = σ − 1.
Equation (37) represents the spectrum of 3d isotropic harmonic os-
cillator deformed by the term proportional to δ.
For equation (34) we obtain in the analogous way
En =
ω
2
(
σ(2n+ 1) +
√
(2l + 1)2 + κ̃
)
, (38)
where κ̃ = 8(κ + 1) + σ(σ + 3). The related eigenvectors are expressed
via the confluent hypergeometric functions F :
Rn = e−
ωrσ
2σ rσn−
En
ω F
(
−n, En
σω
− n, ω
σ
r−σ
)
,
where n is integer and En is eigenvalue (38).
4.3. System with angular oscillator potential. The next system
which we consider is specified by the inverse mass and potential presented
in item 8 of the table. The corresponding Hamiltonian is
H = par
2pa +
λ2
2
ϕ2 + σϕ+ ν ln(r̃).
The corresponding eigenvalue equation is separable in cylindrical vari-
ables, thus it is reasonable to represent the wave function as follows
ψ = Ψ(r̃)Φ(ϕ) exp(−ikx3). (39)
As a result we obtain the following equations for radial and angular
variables(
−r̃∂r̃ r̃∂r̃ − r̃∂r̃ + ν ln(r̃) + k2r̃2 − µ
)
Ψ(r̃) = 0 (40)
126 A.G. Nikitin
and (
− ∂2
∂ϕ2
+
λ2
2
ϕ2 + σϕ− µ
)
Φ(ϕ) = 0, (41)
where µ is a separation constant.
For λ nonzero equation (41) is equivalent to the Harmonic oscillator.
The specificity of this system is that, in contrast with (31), it includes
angular variable ϕ whose origin is
0 ≤ ϕ ≤ 2π. (42)
For trivial λ our equation (41) is reduced to equation with free fall
potential, but again for the angular variable satisfying (42).
The radial equation (40) is simple solvable too. In the case k = 0 we
again come to the free fall potential.
4.4. Systems with Morse effective potential. The next system
we consider is specified by the inverse mass and potentials represented
in item 9 of Table 1. The corresponding Hamiltonian is
H = − ∂
∂xa
r̃2eσϕ
∂
∂xa
+ κeσϕ +
ω2
2
e−σϕ.
Introducing again the cylindric variables and representing the wave
function in the form (39) we come to the following equations for the
radial and angular variables(
−
(
∂2
∂y2
+
∂
∂y
)
+ µ+ k2e2y
)
Ψ(r̃) = µΨ(r̃)
and (
−eσϕ
(
∂2
∂ϕ2
+ κ− µ
)
+
ω2
2
e−σϕ
)
Φ(ϕ) = ẼΦ(ϕ). (43)
Dividing all terms in (43) by exp(σϕ) we obtain the following equation(
−
(
∂2
∂ϕ2
+ κ− µ
)
+
ω2
2
e−2σϕ
)
Φ(ϕ) = e−σϕẼΦ(ϕ).
or (
−
(
∂2
∂ϕ2
)
+
ω2
2
e−2σϕ − Ẽe−σϕ
)
Φ(ϕ) = ÊΦ(ϕ), (44)
where we denote Ê = µ− κ.
Exact solvability of PDM systems with extended Lie symmetries 127
Formula (44) represents the Schrödinger equation with Morse poten-
tial. This equation is shape invariant and also can be solved using tools
of SUSY quantum mechanics. We demonstrate this procedure using
another system.
Considering the mass and potential presented in item 6 of Table 1 we
come to the following Hamiltonian
H = 1
2pax
σ+2
3 pa + κxσ3 .
Equation (12) with Hamiltonian (20) can be solved by separation of
variables in Cartesian coordinates. Expanding the wave function ψ via
eigenfunctions of integrals of motion P1 and P2 in the form (21) and in-
troducing new variable y = ln(x3) we reduce the problem to the following
equation for Φ(k1, k2, x3):(
− ∂
∂x3
xσ+2
3
∂
∂x3
+ xσ+2
3 k2 + 2κxσ3
)
Φ = 2EΦ, (45)
where k2 = k2
1 + k2
2.
Dividing all terms in (45) by xσ3 we can rewrite it in the following
form (
− ∂2
∂y2
− (σ + 1)
∂
∂y
− 2E exp(−σy) + k2 exp(2y) + 2κ
)
Φ = 0.
In the particular case σ = 2 we again come to the equation with
Morse effective potential.
One more system which can be related to Morse potential is repre-
sented in item 7 and include the following Hamiltonian
H = 1
2pa exp(λϕ)r̃σ+2pa + ν exp(λϕ)r̃σ.
The corresponding equation (12) is separable in the cylindrical vari-
ables (27) provided σ · λ = 0 and again includes the Morse effective
potential.
Let us return to equation (33) and solve it using approach analogous
to the presented above. In other words, we will change the roles of
eigenvalues and coupling constants.
First we divide all terms in (33) by r2σ and obtain
−r2 ∂
2Rlm
∂r2
− (2σ + 4)r
∂Rlm
∂r
128 A.G. Nikitin
+
(
ω2r−4σ + µr−2σ
)
Rlm = εRlm, (46)
where
ε = −l(l + 1)− 2κ, µ = −2E. (47)
Applying the Liouville transform
r → ρ = ln(r), Rlm → R̃lm = e−
σ+3
2 Rlm
we reduce (46) to a more compact form
HνR̃lm ≡
(
− ∂2
∂ρ2
+ ω2e−2σρ + (2ων + ωσ)e−σρ
)
R̃lm = ε̂R̃lm,(48)
where
ε̂ = ε−
(
σ + 3
2
)2
, ν =
µ
2ω
− σ
2
. (49)
Like (44) equation (48) includes the familiar Morse potential and
so is shape invariant. Indeed, denoting µ = 2ω(ν + σ
2 ) we can factorize
Hamiltonian Hν like it was done in (36) where index l should be changed
to ν and
W = ν − ωe−aρ, Cν = ν2
and the shape invariance is easy recognized.
To find the admissible eigenvalues ε and the corresponding eigen-
vectors we can directly use the results presented in [4], see item 4 of
Table 4.1 there
ε̂ = ε̂n = −(ν − nσ)2,
(
R̃lm
)
n
= y
ν
σ−ne−
y
2L
2( νσ−n)
n (y),
where y = 2ω
σ r
−σ.
Thus we find the admissible values of ε̂n. Using definitions (47)
and (49) we can find the corresponding values of E which are in per-
fect accordance with (38).
Discussion. The results presented above in Section 2 include the
complete list of continuous symmetries which can be admitted by PDM
Schrödinger equations, provided these equations are defined up to arbi-
trary parameters. All such systems appear to be exactly solvable.
Exact solvability of PDM systems with extended Lie symmetries 129
It is important to note that the list of symmetries presented in the
fourth column of the table is valid only for the case of nonzero parameters
defined the potential and mass terms. If some (or all) of these parameters
are trivial, the corresponding PDM Schrödinger equation can have more
extended set of symmetries. For example, it is the case for the potential
and PDM presented in item 3 of the table, compare the list of symmetries
presented in column 4 with (24). The completed list of non-equivalent
symmetries can be found in [13] which generalizes the Boyer results [3] to
the case of PDM Schrödinger equations. As other extensions of results
of [3] we can mention the group classification of the nonlinear Schrödinger
equations [15] and the analysis of its conditional symmetries [6].
Thanks to their extended symmetries the majority of the presented
systems is exactly solvable. In Sections 3 and 4 we present the cor-
responding solutions explicitly and discuss supersymmetric aspects of
some of them. However, two of the presented systems (whose mass and
potential are presented in items 4 and 5 of Table 1) are not separable, if
both arbitrary parameters κ and λ are nonzero. And just these systems
have the most small symmetry. On the other hand, all systems admit-
ting six- or higher-dimensional Lie symmetry algebras are separable and
exactly solvable.
In addition to the symmetry under the six parameter Lie group, equa-
tion (32) (which we call deformed 3d isotropic harmonic oscillator) pos-
sesses a hidden dynamical symmetry with respect to group SO(1, 2).
The effective radial Hamiltonian is shape invariant, and its eigenvalues
can be found algebraically. In spite on the qualitative difference of its
spectra (37) and (38) of the standard 3d oscillator, it keeps the main
supersymmetric properties of the latter. We show that the shape invari-
ance of PDM problems usually attends their extended symmetries.
[1] Anderson R.L., Kumei S., Wulfman C.E., Invariants of the equations of wave
mechanics. I, Rev. Mex. Fis. 21(1972), 1–33.
[2] Barut A.O., Ra̧czka R., Theory of group representations and applications,
World Scientific Publishing Co., Singapore, 1986.
[3] Boyer C.P., The maximal kinematical invariance group for an arbitrary poten-
tial, Helv. Phys. Acta 47 (1974), 450–605.
[4] Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics,
Phys. Rep. 251 (1995), 267–385.
[5] Cruz y Cruz S., Rosas-Ortiz O,, Dynamical equations, invariants and spec-
trum generating algebras of mechanical systems with position-dependent mass,
SIGMA 9 (2013), 004, 21 pp.
130 A.G. Nikitin
[6] Fushchych W.I., Nikitin A.G., Higher symmetries and exact solutions of linear
and nonlinear Schrödinger equation, J. Math. Phys. 38 (1997), 5944–5959.
[7] Hagen C.R., Scale and conformal transformations in Galilean-invariant confor-
mal field theory, Phys. Rev. D 5 (1972), 377–388.
[8] Koç R., Koca M., A systematic study on the exact solution of the position
dependent mass Schrödinger equation, J. Phys. A: Math. Gen. 36 (2003), 8105–
8112.
[9] Naimark M.A., Linear representations of the Lorentz group, Macmillan, New
York, 1964.
[10] Niederer U., The maximal kinematical invariance group of the free Schrödinger
equations, Helv. Phys. Acta 45 (1972), 802–810.
[11] Nikitin A.G., Group classification of systems of nonlinear reaction-diffusion
equations with general diffusion matrix. I. Generalized Ginsburg–Landau equa-
tions, J. Math. Anal. Appl. 324 (2006), 615–628.
[12] Nikitin A.G., Superintegrable and shape invariant systems with position depen-
dent mass, J. Phys. A: Math. Theor. 48 (2015), 335201, 24 pp.
[13] Nikitin A.G., Kinematical invariance groups of the 3d Schrödinger equations
with position dependent masses, J. Math. Phys. 58 (2017), 083508, 16 pp.
[14] Nikitin A.G., The maximal “kinematical” invariance group for an arbitrary
potential revised, J. Math. Phys. Anal. Geom. 14 (2018), 519–531.
[15] Nikitin A.G., Popovych R.O., Group classification of nonlinear Schrödinger
equations, Ukr. Math. J. 53 (2001), 1255–1265.
[16] Nikitin A.G., Wiltshire R.J., Symmetries of systems of nonlinear reaction-dif-
fusion equations, in Symmetry in Nonlinear Mathematical Physics (Kyiv, 1999),
Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 30, Part 1,
Institute of Mathematics, Kyiv, 2000, 47–59.
[17] Nikitin A.G., Wiltshire R.J., Systems of reaction diffusion equations and their
symmetry properties, J. Math. Phys. 42 (2001), 1667–1688.
[18] Nikitin A.G., Zasadko T.M., Superintegrable systems with position dependent
mass, J. Math. Phys. 56 (2015), 042101, 13 pp.
[19] Nikitin A.G., Zasadko T.M., Group classification of Schrödinger equations with
position dependent mass, J. Phys. A: Math. Theor. 49 (2016), 365204, 17 pp.
[20] Quesne C., Quadratic algebra approach to an exactly solvable position-depen-
dent mass Schrödinger equation in two dimensions, SIGMA 3 (2007), 067, 14 pp.
[21] Quesne C., Tkachuk V.M., Deformed algebras, position-dependent effective
masses and curved spaces: an exactly solvable Coulomb problem, J. Phys. A:
Math. Gen. 37 (2004), 4267–4281.
|
| id | oai:trim.imath.kiev.ua:article-361 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:04Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/8a/c8a67d73f7a956f0462349290f23608a.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3612020-08-13T08:52:53Z Exact solvability of PDM systems with extended Lie symmetries Точна розв'язність рівняння Шрьодінгера зі змінним параметром маси з розширеними симетріями Лі Nikitin, A. Нікітін, А. It is shown that all PDM Schrodinger equations admitting more than five-dimensional Lie symmetry algebras (whose completed list can be found in paper [J.~Math. Phys. 58&nbsp; (2017), 083508, 16 pp.] are exactly solvable. The corresponding exact solutions are presented. The supersymmetric aspects of the exactly solvable systems are discussed. Показано, що усі рівняння Шрьодінгера зі змінним параметром маси, які допускають алгебри інваріантності розмірностей більших за п'ять (повний список таких рівнянь наведено у роботі [Math. Phys.&nbsp; 58&nbsp; (2017), 083508, 16 pp.]), є точно розв'язними. Знайдено у явному вигляді відповідні розв'язки та показано їх суперсиметричну природу. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/361 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 113-130 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 113-130 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 113-130 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/361/367 Авторське право (c) 2019 А. Нікітін |
| spellingShingle | Nikitin, A. Нікітін, А. Exact solvability of PDM systems with extended Lie symmetries |
| title | Exact solvability of PDM systems with extended Lie symmetries |
| title_alt | Точна розв'язність рівняння Шрьодінгера зі змінним параметром маси з розширеними симетріями Лі |
| title_full | Exact solvability of PDM systems with extended Lie symmetries |
| title_fullStr | Exact solvability of PDM systems with extended Lie symmetries |
| title_full_unstemmed | Exact solvability of PDM systems with extended Lie symmetries |
| title_short | Exact solvability of PDM systems with extended Lie symmetries |
| title_sort | exact solvability of pdm systems with extended lie symmetries |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/361 |
| work_keys_str_mv | AT nikitina exactsolvabilityofpdmsystemswithextendedliesymmetries AT níkítína exactsolvabilityofpdmsystemswithextendedliesymmetries AT nikitina točnarozv039âznístʹrívnânnâšrʹodíngerazízmínnimparametrommasizrozširenimisimetríâmilí AT níkítína točnarozv039âznístʹrívnânnâšrʹodíngerazízmínnimparametrommasizrozširenimisimetríâmilí |