wykresx.eps


Acta Polytechnica Vol. 51 No. 1/2011

Propagators of Generalized Schrödinger Equations Related by
First-order Supersymmetry

A. Schulze-Halberg

Abstract

Weconstruct an explicit relation betweenpropagators of generalized Schrödinger equations that are linkedbyafirst-order
supersymmetric transformation. Our findings extend and complement recent results on the conventional case [1].

Keywords: generalized Schrödinger equation, propagator, supersymmetry.

1 Introduction
The formalism of supersymmetry (SUSY) is a well-
known tool for identifying integrable cases of the
Schrödinger equation and for generating the corre-
sponding solutions. The principal idea is to relate
two Schrödinger equations and their solutions via a
supersymmetrical (or Darboux [3]) transformation,
such that known solutions of one equation can be
mapped onto solutions of the second equation. There
is a vast amount of literature on the SUSY formalism
and its applications, for details we refer the reader
to [5, 2] and references therein. While the concept
of the SUSY formalism as a method for generat-
ing solutions is popular, it is less known that two
Schrödinger equations related by a SUSY transfor-
mation (SUSY partners) share more properties than
the link between their solutions. In particular, SUSY
establishes a connectionbetween thepropagatorsand
theGreen’s functions of the underlyingpartner equa-
tions: the propagators are linked by means of an in-
tegral expression [1], while theGreen’s functions sat-
isfy a simple trace formula [9, 6]. It is interesting to
note that this trace formulapersistsundergeneraliza-
tion of the Schrödinger equation to the effectivemass
case [8] or to a generalized Sturm-Liouville prob-
lem [7]. Due to the close relation between Green’s
function and the propagator, one would expect that
the propagator relation found in [1] also extends to
generalized cases of the Schrödinger equation. This
is in fact true, as will be shown in this note. We re-
strict ourselves to first-order SUSY transformations
of generalized Schrödinger equations, a brief review
of which and of related theory is given in section 2.
Subsequently, the explicit integral formula that links
the propagators of our SUSY partner equations is
done in section 3.

2 Preliminaries
In the following we briefly summarize basic facts
about generalized Schrödinger equations, their

SUSY formalism, propagators and Green’s func-
tions.

Generalized Schrödinger equation. We con-
sider the following generalized Sturm-Liouville prob-
lemonthe real interval (a, b), equippedwithDirichlet
boundary conditions:

f(x)ψ′′(x)+ f ′(x)ψ′(x)+

[Eh(x)− V (x)]ψ(x) = 0, x ∈ (a, b) (1)
ψ(a)= ψ(b) = 0. (2)

Here f, h, V are smooth, real functions, with f, h pos-
itive and bounded in a and b. The constant E will
be referred to as energy, and in solutions of (1), (2)
that belong to the discrete spectrum, E stands for
the spectral value. Any solution ψ of (1), (2) be-
longing to a value E from the discrete spectrum, is
located in the weighted Hilbert space L2h(a, b) with
weight function h [4]. The lowest valueof the discrete
spectrumwill be called the ground state anddenoted
by E0 with corresponding solution ψ0. The interval
(a, b) can be unbounded, that is, a or b can represent
minus infinity or infinity, respectively (however, if a
and/or b are finite, thenwe require f, h, V to be con-
tinuous there). We see that the problem (1), (2) can
be singular, which means that its spectrum can ad-
mit a continuous part. Equation (1) will be referred
to as the generalized Schrödinger equation, since its
special cases are frequently encountered in Quantum
Mechanics, such as the Schrödinger equation for ef-
fective mass or with a linearly energy-dependent po-
tential. In the quantum-mechanical context, E de-
notes the energy associatedwith a solution ψ, and V
stands for the potential.

Generalized SUSY formalism. We summarize
results from [10]. The boundary-value problem (1),
(2) can be linked to another problem of the same
kind by means of the SUSY transformation method.
Consider

f(x)φ′′(x)+ f ′(x)φ′(x)+

63



Acta Polytechnica Vol. 51 No. 1/2011

[Eh(x) − U(x)]φ(x) = 0, x ∈ (a, b) (3)
φ(a)= φ(b) = 0, (4)

where the same settings imposed for (1), (2) ap-
ply. Clearly, a solution φ = φ(x) and the potential
U = U(x) are in general different from their respec-
tive counterparts ψ and V . Now, suppose that ψ and
u are solutions of the boundary-value problem (1),
(2) and of equation (1) at real energies E and λ ≤ E,
respectively. Define the SUSY transformation of ψ
as

Du,xψ(x) =

√
f(x)
h(x)

W(u, ψ)(x)
u(x)

=√
f(x)
h(x)

[
−

u′(x)
u(x)

ψ(x)+ ψ′(x)

]
, (5)

where W(u, ψ) stands for the Wronskian of the func-
tions u, ψ and the second indexof D denotes the vari-
ablewhich the derivatives in theWronskian apply to.
The function φ = Du,xψ as defined in (5) solves the
boundary-value problem (3), (4), if the potential U
is given in terms of its counterpart V , as follows:

U(x) = V (x) −2f(x)
d2

dx2

{
log[u(x)]

}
+[

f(x)h′(x)
h(x)

− f ′(x)
]

u′(x)
u(x)

−
f ′′(x)
2
+

[f ′(x)]2

4f(x)
+
3f(x)[h′(x)]2

4h2(x)
−

f(x)h′′(x)
h(x)

. (6)

Note that (5) remains valid when multiplied by a
constant, which can be used for normalization. Now,
depending on the choice of the auxiliary solution u
in (5), the discrete spectrum of problem (3), (4) can
be affected in three possible ways: if λ = E0 and
u = ψ0, then E0 is removed from the spectrum of
(3), (4). The opposite case, creation of a new spec-
tral value λ < E0, happens if the auxiliary solution u
does not fulfill the boundary conditions (4). Finally,
the spectra of both problems (1), (2) and (3), (4) are
the same, if we pick λ < E0 and u that fulfills only
one of the boundary conditions (2).

Propagator and Green’s function. Thepropaga-
tor governs a quantum system’s time evolution. For
a stationary Schrödinger equation, the propagator K
has the defining property

Ψ(x, t) = exp(−iEt)ψ(x)=∫
(a,b)

K(x, y, t)ψ(y)dy, (7)

as the solution Ψ of the time-dependent Schrödinger
equation is related to its stationary counterpart ψ
by the exponential factor used for separating time

and spatial variable. Suppose problem (1), (2) ad-
mits a complete set of eigenfunctions (ψn), n =
0,1,2, . . . , M ∈ N0, where M can stand for infinity,
and (ψk), k ∈ R, belonging to the discrete and the
continuous part of the spectrum, respectively. Then
the propagator K has the representation

K(x, y, t) = h(y)

[
M∑

n=0

ψn(x)exp(−iEnt)ψn(y)+∫
R

ψk(x)exp(−ik2t)ψk(y)dk
]

, (8)

where En and k
2 stand for the spectral values be-

longing to the discrete and continuous spectrum, re-
spectively. The Green’s function G of problem (1),
(2) has two equivalent representations [4], both of
which we will use here. In order to state the first
representation, let ψ0,l and ψ0,r be solutions of equa-
tion (1) that fulfill theunilateralboundaryconditions
ψ0,l(a) = ψ0,r(b) = 0. The Wronskian W(ψ0,l, ψ0,r)
of these funtions is given by

W(ψ0,l, ψ0,r)(x)=
c0

f(x)
, (9)

where c0 is a constant that depends on the explicit
formof ψ0,l and ψ0,r. Nowwecangive thefirst repre-
sentationof theGreen’s function G0 forourboundary
value problem (1), (2):

G(x, y) = −
1
c0

[
ψ0,l(y)ψ0,r(x)θ(x − y)+

ψ0,l(x)ψ0,r(y)θ(y − x)
]
, (10)

where c0 is the constant from (9) and θ stands for the
Heaviside distribution. The second representation of
the Green’s function G can be obtained as follows,
provided problem (1), (2) admits a complete set of
solutions:

G(x, y) =
M∑

n=0

ψn(x)ψn(y)
En − E

+

∫
R

ψk(x)ψk(y)
k2 − E

dk, (11)

where the notation is the same as in (8).

3 Propagators related by
generalized SUSY

In order to obtain a relationbetween the propagators
of the two boundary-value problems (1), (2) and (3),
(4),we take thepropagator K1 of the secondproblem
and express it through quantities related to the first
problem. For the sake of simplicity we assume for
now that the two boundary-value problems have the

64



Acta Polytechnica Vol. 51 No. 1/2011

same discrete spectrumand that both of them admit
a complete setof solutionsbelonging to adiscreteand
a continuous part of the spectrum. Furthermore, we
assume that the solutionsof problem(1), (2) are real-
valued functions. This is no restriction, as equation
(1) involves only real functions.

3.1 General case

The construction of our propagator K1 is similar to
the way it was done in [1]. According to representa-
tion (8) we have

K1(x, y, t) = h(y)

[
M∑

n=0

φn(x)exp(−iEnt)φn(y)+∫
R

φk(x)exp(−ik2t)φk(y)dk
]

. (12)

The notation is the same as in (8), only φn and φk
must be replaced by ψn and ψk, respectively. Now,
since the solutions φn, φk are obtained by means
of a SUSY transformation (5) from ψn, ψk, we can
rewrite (12) as follows, taking into account normal-
ization constants Ln and Lk, respectively:

K1(x, y, t) = h(y)Du,xDu,y ·[
M∑

n=0

L2nψn(x)exp(−iEnt)ψn(y)+

+
∫
R

L2kψk(x)exp(−ik
2t)ψk(y)dk

]
.

In the next step we apply the defining property (7)
to the previously obtained expression:

K1(x, y, t)= h(y)Du,xDu,y ·[
M∑

n=0

L2n

∫
(a,b)

K0(x, z, t)ψn(z)dzψn(y)+

∫
R

L2k

∫
(a,b)

K0(x, z, t)ψk(z)dzψk(y)dk

]
. (13)

We choose the free constants as Ln = 1/(En − λ)
1
2 ,

n = 1, . . . , M, and Lk = 1/(k
2 − λ)

1
2 , k ∈ R, where

λ is the discrete spectral value associated with the
auxiliary function u. After regrouping terms, these
settings render (13) in the form

K1(x, y, t)= h(y)Du,xDu,y

{∫
(a,b)

K0(x, z, t) ·[
M∑

n=0

ψn(z)ψn(y)
En − λ

+
∫
R

ψk(z)ψk(y)
k2 − λ

dk

]
dz

}
= (14)

h(y)Du,xDu,y

∫
(a,b)

K0(x, z, t)G0(z, y)dz, (15)

where G0 is the Green’s function of our boundary-
value problem (1), (2) in its form (11), taken at
energy λ. Relation (15) gives the final connection
between the propagators of our two boundary-value
problems, provided they admit the same discrete
spectrum. In the case that problem (3), (4) admits
an additional discrete spectral value λ with the cor-
responding solution φ−1, formula (12) must be mod-
ified as follows:

K1(x, y, t) = h(y)

[
M∑

n=0

φn(x)exp(−iEnt)φn(y)+

φ−1(x)exp(−iλt)φ−1(y)+∫
R

φk(x)exp(−ik2t)φk(y)dk
]
. (16)

From this point, the additional term is maintained
until the final relation between the propagators K0
and K1 results as

K1(x, y, t)=

h(y)

{
Du,xDu,y

[∫
(a,b)

K0(x, z, t)G0(z, y)dz

]
+

φ−1(x)exp(−iλt)φ−1(y)
}

,

where the Green’s function G0 is to be taken at en-
ergy λ. Finally, if problem (3), (4) admits one dis-
crete spectral value less than its initial counterpart,
formula (12) remains the same except that the sum
starts at one instead of at zero. This is maintained
until formula (14), where summation now starts at
one. Expression (15) then turns into

K1(x, y, t) = h(y)Du,xDu,y

∫
(a,b)

K0(x, z, t) ·

lim
E→E0

[
G0(z, y)−

ψ0(z)ψ0(y)
E0 − E

]
dz,(17)

where the Green’s function G0 is to be taken at en-
ergy E. In summary, the last three expressions stand
for the final relations between the propagators of
our two boundary-value problems. Clearly, in the
conventional case h = 1, the above expressions re-
duce correctly to the known relations [1]. In general,
our expressions cannot be simplified anymore, un-
less more information on the auxiliary solution u is
known. We will now study such a case.

3.2 Special case: ground state as
auxiliary solution

Let us assume that the auxiliary solution u is chosen
to be the ground state ψ0, associated with the spec-
tral value E0, of problem (1), (2). According to our

65



Acta Polytechnica Vol. 51 No. 1/2011

brief SUSY review in section 2, this choice implies
that the discrete spectrum of problem (3), (4) will
not contain the value E0 anymore. Wewill now show
that the corresponding relationbetween thepropaga-
tors (17),where u is replacedby ψ0, canbe simplified
considerably. While the general procedure of simpli-
fication follows a similar way as in the conventional
case [1], one must keep track of the nonconstant fac-
tor in front of (5). Before we start simplifying (17),
we observe that the limit and the operator Dψ0,y in
(17) commute, because

Dψ0,y lim
E→E0

[
G0(z, y)−

ψ0(z)ψ0(y)
E0 − E

]
=

lim
E→E0

[
M∑

n=1

ψn(z)Dψ0,yψn(y)
En − E

+

∫
R

ψk(z)Dψ0,yψk(y)
k2 − E

dk

]
.

The first term in the sum vanishes, as Dψ0,yψ0(y)=
0, so we obtain

Dψ0,y lim
E→E0

[
G0(z, y)−

ψ0(z)ψ0(y)
E0 − E

]
=

lim
E→E0

[Dψ0,yG0(z, y)] .

This property will be useful for rewriting (17). Note
that for the sake of simplicitywe divide by the factor
h:

1
h(y)

K1(x, y, t)=

Dψ0,x

∫
(a,b)

K0(x, z, t) lim
E→E0

[Dψ0,yG0(z, y)] dz =

Dψ0,x

∫
(a,y)

K0(x, z, t) ·

lim
E→E0

[
−
1
c0

Dψ0,yψ0,l(y)ψ0,r(z)

]
dz +

Dψ0,x

∫
(y,b)

K0(x, z, t) ·

lim
E→E0

[
−
1
c0

ψ0,l(z)Dψ0,yψ0,r(y)

]
dz. (18)

We will now determine the limits that the integrals
contain. To this end, first note that according to (5)
the Wronskian W(ψ0, ψ0,r) is involved in the limits,
which we will now find by means of the differential
equation that it obeys. We have

W(ψ0, ψ0,r)
′(y)=

d
dy

[
ψ0(y)ψ

′
0,r(y)− ψ0,r(y)ψ

′
0(y)
]
=

ψ0(y)ψ
′′
0,r(y)− ψ

′′
0(y)ψ0,r(y)= (19)

h(y)
f(y)

ψ0(y)ψ0,r(y)(E0 − E)−
f ′(y)
f(y)

W(ψ0, ψ0,r)(y).

Note that in the third line we replaced the second
derivatives by means of the generalized Schrödinger
equation (1). Equation (19) can be solved with re-
spect to the Wronskian, giving

W(ψ0, ψ0,r)(y)=
E0 − E
f(y)

∫
(y,b)

h(z)ψ0(z)ψ0,r(z)dz,

where a constant of integration has been set to zero.
Thus, we have

Dψ0,yψ0,r(y) =

√
1

f(y)h(y)
E0 − E
ψ0(y)

·∫
(y,b)

h(z)ψ0(z)ψ0,r(z)dz.

Thus, the term inside the limit in (18) reads

−
1
c0

ψ0,l(z)Dψ0,yψ0,r(y)=

−
E0 − E

c0

√
1

f(y)h(y)
ψ0,l(z)
ψ0(y)

·∫
(y,b)

h(z)ψ0(z)ψ0,r(z)dz. (20)

According to (9), we have c0 = f(b)Wψ0,l,ψ0,r(b),
where the right hand side could have been evalu-
ated at any point of [a, b]; the choice b will prove
convenient in subsequent calculations. Thus, tak-
ing into account the fact that ψ0,r(b) = 0, we have
c0 = −f(b)ψ0,l(b)ψ′0,r(b), which we will now express
by means of an integral. To this end, consider our
generalized Schrödinger equation (1) and its deriva-
tive with respect to E, each multiplied by a ψ0,l and
its derivative with respect to E, respectively:

{
f(z)ψ′′0,l(z)+ f

′(z)ψ′0,l(z)+

[Eh(z)− V (z)]ψ0,l(z)
} ∂

∂E
ψ0,l(z)= 0[

f(z)
∂

∂E
ψ′′0,l(z)+ f

′(z)
∂

∂E
ψ′0,l(z)+

h(z)ψ0,l(z)+ Eh(z)
∂

∂E
ψ0,l − V (z)

∂

∂E
ψ0,l

]
×

ψ0,l(z)= 0

Taking the difference of these two equations yields
the following result:

f ′(z)ψ′0,l(z)
∂

∂E
ψ0,l(z)− f ′(z)ψ0,l(z)

∂

∂E
ψ′0,l(z)−

h(z)ψ20,l(z)+

f(z)ψ′′0,l(z)
∂

∂E
ψ0,l(z)− f(z)ψ0,l(z)

∂

∂E
ψ′′0,l(z)= 0.

66



Acta Polytechnica Vol. 51 No. 1/2011

Rewriting and integrating this equation gives∫
(a,b)

h(z)ψ20,l(z)dz =∫
(a,b)

d
dz

[
f(z)ψ′0,l(z)

] ∂
∂E

ψ0,l(z)dz −∫
(a,b)

ψ0,l(z)
d
dz

[
f(z)

∂

∂E
ψ′0,l(z)

]
dz.

The two integrals on the right hand side can each be
reformulated using integration by parts.∫

(a,b)

d
dz

[
f(z)ψ′0,l(z)

] ∂
∂E

ψ0,l(z)dz =

∂

∂E
ψ0,l(z)f(z)ψ0,l(z)

∣∣∣∣∣
b

a

−
∫
(a,b)

f(z)ψ0,l(z)
∂

∂E
ψ′0,l(z)dz∫

(a,b)
ψ0,l(z)

d
dz

[
f(z)

∂

∂E
ψ′0,l(z)

]
dz =

f(z)ψ0,l(z)
∂

∂E
ψ′0,l(z)

∣∣∣∣∣
b

a

−
∫
(a,b)

f(z)ψ0,l(z)
∂

∂E
ψ′0,l(z)dz.

Observe that both expressions involve the same inte-
gral term on their right hand sides. Therefore, if we
substitute into their difference above, we obtain∫

(a,b)
h(z)ψ20,l(z)dz =

f(b)ψ′0,l(b)
∂

∂E
ψ0,l(b)− f(b)ψ0,l(b)

∂

∂E
ψ′0,l(b). (21)

Note that in the last stepwe substituted the limits of
integration andmadeuse of the fact that ψ0,l(a)=0.
Now we are ready to compute the limits in (18). Ac-
cording to (20), the first limit reads

lim
E→E0

[
−
1
c0

Dψ0,yψ0,l(y)ψ0,r(z)

]
=

lim
E→E0

[
E0 − E

f(b)ψ0,l(b)ψ′0,r(b)

√
1

f(y)h(y)
ψ0,l(z)
ψ0(y)

·

∫
(y,b)

h(z)ψ0(z)ψ0,r(z)dz

]
=

lim
E→E0

[
E0 − E
ψ0,l(b)

]
·

lim
E→E0

[
1

f(b)ψ′0,r(b)

√
1

f(y)h(y)
ψ0,l(z)
ψ0(y)

·

∫
(y,b)

h(z)ψ0(z)ψ0,r(z)dz

]
=

−
1

∂
∂E

ψ0,l(b)

∣∣∣∣∣
E=E0

·

lim
E→E0

[
1

f(b)ψ′0,r(b)

√
1

f(y)h(y)
ψ0,l(z)
ψ0(y)

·

∫
(y,b)

h(z)ψ0(z)ψ0,r(z)dz

]
. (22)

In the next step we substitute the first factor us-
ing our relation (21), which we need to evaluate at
E = E0. Since our generalized Schrödinger equa-
tion (1) can only have two linearly independent so-
lutions at E = E0, the three solutions ψ0,l, ψ0,r and
ψ0 become linearly dependent there. In particular,
they must all fulfill the same boundary conditions
(2), which implies for the right hand side of (21) that[

f(b)ψ′0,l(b)
∂

∂E
ψ0,l(b)−

f(b)ψ0,l(b)
∂

∂E
ψ′0,l(b)

]∣∣∣∣∣
E=E0

=

f(b)ψ′0,l(b)
∂

∂E
ψ0,l(b)

∣∣∣∣∣
E=E0

.

Now, (21) can be solved for
∂

∂E
ψ0,l(b):

∂

∂E
ψ0,l(b)

∣∣∣∣∣
E=E0

=

[
1

f(b)ψ′0,l(b)

∫
(a,b)

h(z)ψ20,l(z)dz

]∣∣∣∣∣
E=E0

.

We use this to replace the first factor of (22) and get

lim
E→E0

[
−
1
c0

Dψ0,yψ0,l(y)ψ0,r(z)

]
=

− lim
E→E0

[
ψ′0,l(b)

ψ′0,r(b)

√
1

f(y)h(y)
ψ0,l(z)
ψ0(y)

·∫
(y,b) h(z)ψ0(z)ψ0,r(z)dz∫
(a,b) h(z)ψ

2
0,l(z)dz

]
.

For taking the limit we recall that at E = E0 the
functions ψ0,l, ψ0,r and ψ0 become linearly depen-
dent. The respective proportionality constants can-
cel out and we obtain

lim
E→E0

[
−
1
c0

Dψ0,yψ0,l(y)ψ0,r(z)

]
=

−
√

1
f(y)h(y)

ψ0(z)
ψ0(y)

∫
(y,b)

h(z)ψ20(z)dz∫
(a,b)

h(z)ψ20(z)dz
.

67



Acta Polytechnica Vol. 51 No. 1/2011

The second limit in (18) is found in a similar fashion,
yielding

lim
E→E0

[
−
1
c0

ψ0,l(z)Dψ0,yψ0,r(y)

]
=√

1
f(y)h(y)

ψ0(z)
ψ0(y)

∫
(a,y) h(z)ψ

2
0(z)dz∫

(a,b) h(z)ψ
2
0(z)dz

,

note that there is no negative sign in front. Now our
two limits can be plugged into the propagator (18):

1
h(y)

K1(x, y, t)= Dψ0,x

∫
(a,y)

K0(x, z, t) ·[
−
√

1
f(y)h(y)

ψ0(z)
ψ0(y)

∫
(y,b) h(w)ψ

2
0(w)dw∫

(a,b)
h(w)ψ20(w)dw

]
dz +

Dψ0,x

∫
(y,b)

K0(x, z, t) ·[√
1

f(y)h(y)
ψ0(z)
ψ0(y)

∫
(a,y) h(w)ψ

2
0(w)dw∫

(a,b) h(w)ψ
2
0(w)dw

]
dz. (23)

In order to join the two terms, we rewrite the inner
integral over (y, b) as a difference of integrals over
(a, b) and (a, y), respectively:

1
h(y)

K1(x, y, t)=

Dψ0,x

∫
(a,y)

K0(x, z, t) ·[
−
√

1
f(y)h(y)

ψ0(z)
ψ0(y)

]
dz +

Dψ0,x

∫
(a,y)

K0(x, z, t) ·[√
1

f(y)h(y)
ψ0(z)
ψ0(y)

∫
(a,y)

h(w)ψ20(w)dw∫
(a,b)

h(w)ψ20(w)dw

]
dz +

Dψ0,x

∫
(y,b)

K0(x, z, t) ·[√
1

f(y)h(y)
ψ0(z)
ψ0(y)

∫
(a,y)

h(w)ψ20(w)dw∫
(a,b)

h(w)ψ20(w)dw

]
dz =

−
√

1
f(y)h(y)

1
ψ0(y)

Dψ0,x ·∫
(a,y)

K0(x, z, t)ψ0(z)dz +√
1

f(y)h(y)

∫
(a,y) h(w)ψ

2
0(w)dw

ψ0(y)
∫
(a,b) h(w)ψ

2
0(w)dw

Dψ0,x ·∫
(a,b)

K0(x, z, t)ψ0(z)dz. (24)

Since according to (5) we have

Dψ0,x

∫
(a,b)

K0(x, z, t)ψ0(z)dz =

exp(−iE0t)Dψ0,xψ0(x) = 0,

relation (24) turns after multiplication by h into its
final form

K1(x, y, t) = −
√

h(y)
f(y)

1
ψ0(y)

Dψ0,x ·∫
(a,y)

K0(x, z, t)ψ0(z)dz. (25)

Alternatively, in (23) one can write the inner inte-
gral over (a, y) as a difference of integrals over (a, b)
and (y, b), respectively. This gives a result slightly
different from (25):

K1(x, y, t) =

√
h(y)
f(y)

1
ψ0(y)

Dψ0,x ·∫
(y,b)

K0(x, z, t)ψ0(z)dz.

It can be seen immediately that for a conventional
Schrödinger equation (1) with f = 1 and h = 1, our
results reduce correctly to the known findings [1].

4 Concluding remarks

We have obtained a relation between propagators of
generalized Sturm-Liouville problems that are con-
nected by means of SUSY transformations. Our re-
sults complement and generalize former findings for
the conventional Schrödinger equation [1]. While in
the latter reference propagators related by higher-
order SUSY transformations are also found to satisfy
simple interrelations, the corresponding situation in
the generalized case is subject to ongoing research.

References

[1] Pupasov, A. M., Samsonov, B. F., Günther, U.:
Exact propagators for SUSY partners,
J. Phys. A 40 (2007), 10557–10589.

[2] Cooper, F., Khare, A., Sukhatme, U.: Super-
symmetry and Quantum Mechanics, Phys. Rep.
251 (1995), 267–388.

[3] Darboux, M. G.: Sur une proposition relative
aux équations linéaires, Comptes Rendus Acad.
Sci. Paris 94 (1882), 1456–1459.

[4] Duffy, D. G.: Green’s functions with applica-
tions, Chapman and Hall, New York, 2001.

[5] Fernandez, D. J. C.: Supersymmetric Quantum
Mechanics, quant-ph/0910.0192.

68



Acta Polytechnica Vol. 51 No. 1/2011

[6] Samsonov, B. F., Sukumar, C. V., Pupa-
sov, A. M.: SUSY transformation of the Green
function and a trace formula, J. Phys. A 38
(2005), 7557–7565.

[7] Schulze-Halberg, A.: Green’s functions and
trace formulas for generalized Sturm-Liouville
problems related by Darboux transformations,
J. Math. Phys. 51 (2010), 053501 (13pp).

[8] Pozdeeva, E., Schulze-Halberg, A.: Trace for-
mula for Green’s functions of effective mass
Schrödinger equations and N-th order Darboux
transformations, Internat. J. Modern Phys. A
23 (2008), 2635–2647.

[9] Sukumar, C. V.: Green’s functions, sum
rules and matrix elements for SUSY partners,
J. Phys. A 37 (2004), 10287–10295.

[10] Suzko, A. A., Schulze-Halberg, A.: Darboux
transformationsand supersymmetry for the gen-
eralized Schrödinger equations in (1+1) dimen-
sions, J. Phys. A 42 (2009), 295203–295217.

Axel Schulze-Halberg
E-mail: xbataxel@gmail.com
Department of Mathematics and Actuarial Science
Indiana University Northwest
3400 Broadway, Gary, IN 46408, USA

69