New estimates of the potential Schrödinger equation
DURMAGAMBETOV A.A
Department of Mathematics
Eurasian Natioanal University
Astana
KAZAKHSTAN
Abstract: The Schrödinger equation, a fundamental construct in quantum mechanics, plays a pivotal role in un-
derstanding the properties and behaviors of quantum systems across various scientific fields, including but not
limited to physics, economics, and geophysics. This research paper delves into the exploration of novel invariants
associated with the Schrödinger equation, uncovering previously unrecognized symmetries and properties that
open new pathways for theoretical and applied scientific exploration. The investigation reveals that the energy of
bound states remains invariant under transformations of the coordinate system, highlighting a universal symmetry
embedded within the equation. This discovery, coupled with an analysis of the variation in scattering amplitudes’
phase with different coordinate systems, not only enriches the theoretical framework of quantum mechanics but
also has significant practical implications across various domains such as fluid dynamics, material science, and
medical imaging.
Furthermore, by applying the principles of the Poincaré-Riemann-Hilbert boundary-value problem, the study of-
fers a novel approach to estimate potentials within the Schrödinger equation, advancing the three-dimensional
inverse problem of quantum scattering theory. This methodological innovation allows for a more profound un-
derstanding of the unitary scattering operator, facilitating enhanced modeling of complex systems and phenomena.
The implications of these findings extend beyond theoretical physics, offering transformative insights and appli-
cations in areas ranging from fluid dynamics, where it aids in refining models for the Navier-Stokes equations, to
seismic exploration, tomography, and ultrasound imaging, where it enhances phase selection and interpretation
of measurement results.
In essence, this research not only contributes to a deeper understanding of the Schrödinger equation’s fundamen-
tal properties but also paves the way for interdisciplinary advancements, demonstrating the profound impact of
theoretical discoveries on practical applications in diverse scientific and technological domains.
Keywords: Schrödinger equation, Poincaré–Riemann–Hilbert boundary-value problem,unitary scattering
operator, quantum scattering theory
Received: March 19, 2023. Revised: September 8, 2024. Accepted: October 2, 2024. Published: November 4, 2024.
1 Introduction
The Schrödinger equation stands as a cornerstone in
modeling various physical phenomena, transcending
disciplines from quantum mechanics to applied sci-
ences like economics and geophysics. Its solutions
provide a profound insight into the behavior of quan-
tum systems and serve as the basis for understanding
fundamental principles governing matter and energy.
In this pursuit, understanding the nuanced properties
of its solutions becomes paramount, as they not only
elucidate fundamental theoretical concepts but also
have far-reaching implications across diverse scien-
tific domains.
This study embarks on elucidating novel invari-
ants of the Schrödinger equation, emphasizing their
profound ramifications for theoretical physics and be-
yond. While the equation has been extensively stud-
ied since its inception, recent advancements have un-
covered previously unnoticed symmetries and prop-
erties, offering new avenues for exploration and ap-
plication. Of particular interest is the revelation that
bound states’ energy remains unaltered irrespective of
the chosen coordinate system, underscoring a funda-
mental symmetry inherent in the equation.
Concurrently, the investigation unravels the intri-
cate relationship between scattering amplitudes and
coordinate system choice, highlighting the phase’s
sensitivity to such variations. Such insights not only
deepen our theoretical understanding but also hold
practical significance in various fields reliant on ac-
curate phase normalization. These findings not only
enrich our understanding of the Schrödinger equa-
tion but also open doors to novel applications in
fields such as fluid dynamics, quantum mechanics,
and medical imaging.
Moreover, beyond its theoretical implications, this
research has practical applications that extend into ap-
plied sciences. By leveraging these newfound invari-
ants, researchers can construct more robust models
for complex physical systems, leading to advance-
ments in areas such as fluid dynamics, material sci-
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.5
Durmagambetov A.A
E-ISSN: 2945-0489
28
Volume 3, 2024
ence, and medical imaging. For instance, in seis-
mic exploration, tomography, and ultrasound imag-
ing, these properties allow researchers to optimize
phase selection, facilitating the most effective inter-
pretation of measurement results. This strategic phase
manipulation ensures clearer insights into subsurface
structures, tissue composition, and fluid dynamics,
with the flexibility to seamlessly transition back to the
original coordinate system post-interpretation.
Overall, this interdisciplinary synergy underscores
the profound impact of fundamental theoretical re-
search on advancing practical applications across di-
verse scientific domains. By uncovering new in-
variants and properties of the Schrödinger equation,
this study not only contributes to our theoretical un-
derstanding of quantum mechanics but also paves
the way for innovative solutions to real-world chal-
lenges, pushing the boundaries of human knowledge
and technological capabilities. We show how the
Poincaré–Riemann–Hilbert boundary-value problem
enables us to construct effective estimates of the po-
tential in the Schrödinger equation. The apparatus
of the three-dimensional inverse problem of quantum
scattering theory is developed for this. It is shown
that the unitary scattering operator can be studied as a
solution of the Poincaré–Riemann–Hilbert boundary-
value problem. This allows us to go on to study
the potential in the Schrödinger equation This study
delves into the Schrödinger equation’s new invariants,
shedding light on their crucial implications for theo-
retical physics and beyond. Specifically, it explores
the behavior of scattering amplitudes and bound states
concerning the choice of coordinate systems. The re-
search unveils that while the energy of bound states
remains invariant under coordinate transformations,
the phase of scattering amplitudes undergoes varia-
tions, underscoring its pivotal role in theories reliant
on phase normalization.
Moreover, these findings have played a pivotal
role in constructing estimates for three-dimensional
Navier-Stokes equations, enhancing our ability to
model complex fluid dynamics with greater precision
and reliability. Additionally, in seismic exploration,
tomography, and ultrasound imaging, these properties
allow researchers to optimize phase selection, facil-
itating the most effective interpretation of measure-
ment results. This strategic phase manipulation en-
sures clearer insights into subsurface structures, tis-
sue composition, and fluid dynamics, with the flexi-
bility to seamlessly transition back to the original co-
ordinate system post-interpretation. This interdisci-
plinary synergy underscores the profound impact of
fundamental theoretical research on advancing prac-
tical applications across diverse scientific domains
2 Problem Formulation
Let us consider a one-dimensional function fand its
Fourier transformation ˜
f. Using the notions of mod-
ule and phase, we write the Fourier transformation
in the following form: ˜
f=|˜
f|exp(iΦ) , where
Φis the phase. The Plancherel equality states that
||f||L2=const|| ˜
f||L2. Here we can see that the phase
does not contribute to determination of the Xnorm.
To estimate the maximum we make a simple estimate
as max|f|2≤2||f||L2||∇f||L2. Now we have an es-
timate of the function maximum in which the phase is
not involved. Let us consider the behaviour of a pro-
gressing wave travelling with a constant velocity of
v=adescribed by the function F(x, t) = f(x+at).
Its Fourier transformation with respect to the variable
xis ˜
F=˜
fexp(iatk). Again, in this case, we can see
that when we study a module of the Fourier transfor-
mation, we will not obtain major physical information
about the wave, such as its velocity and location of the
wave crest because |˜
F|=|˜
f|. These two examples
show the weaknesses of studying the Fourier trans-
formation. Many researchers focus on the study of
functions using the embedding theorem, in which the
main object of the study is the module of the func-
tion. However, as we have seen in the given exam-
ples, the phase is a principal physical characteristic of
any process, and as we can see in mathematical stud-
ies that use the embedding theorem with energy es-
timates, the phase disappears. Along with the phase,
all reasonable information about the physical process
disappears, as demonstrated by Tao [1] and other re-
search studies. In fact, Tao built progressing waves
that are not followed by energy estimates . Let us pro-
ceed with a more essential analysis of the influence of
the phase on the behaviour of functions.
theorem 1. There are functions of W1
2(R)with a con-
stant rate of the norm for a gradient catastrophe for
which a phase change of its Fourier transformation
is sufficient.
Proof: To prove this, we consider a sequence of
testing functions ˜
fn= ∆/(1 + k2),∆ = (i−
k)n/(i+k)n.
It is obvious that |˜
fn|= 1/(1 + k2)and max|fn|2≤
2||fn||L2||∇fn||L2≤const. Calculating the Fourier
transformation of these testing functions, we obtain
fn(x) =
x(−1)(n−1)2πexp(−x)L1
(n−1)(2x)
as x > 0, fn(x) = 0 as x ≤0,(1)
where L1
(n−1)(2x)is a Laguerre polynomial. Now we
see that the functions are equibounded and derivatives
of these functions will grow with the growth of n.
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Thus, we have built an example of a sequence of the
bounded functions of W1
2(R)which have a constant
norm W1
2(R), and this sequence converges to a dis-
continuous function.
The results show the flaws of the embedding theorems
when analyzing the behavior of functions. Therefore,
this work is devoted to overcoming them and the ba-
sis for solving the formulated problem is the analyt-
ical properties of the Fourier transforms of functions
on compact sets. Analytical properties and estimates
of the Fourier transform of functions are studied us-
ing the Poincaré – Riemann – Hilbert boundary value
problem
3 Results
Consider Schrödinger’s equation:
−H0Φ + qΦ = k2Φ, H0= ∆x, k ∈C. (2)
Let Φ+(k, θ, x)be a solution of (2) with the following
asymptotic behaviour:
Φ+(k, θ, x) = Φ0(k, θ, x) + eik|x|
|x|A(k, θ′, θ)+
01
|x|,|x| → ∞,(3)
where A(k, θ′, θ)is the scattering amplitude and θ′=
x
|x|, θ ∈S2for k∈¯
C+={Imk≥0}Φ0(k, θ, x) =
eik(θ,x):
A(k, θ′, θ) = −1
4πZR3
q(x)Φ+(k, θ, x)e−ikθ′xdx.
Solutions to (2) and (3) are obtained by solving the
integral equation
Φ+(k, θ, x) = Φ0(k, θ, x)+
ZR3
q(y)e+ik|x−y|
|x−y|Φ+(k, θ, y)dy =G(qΦ+),
which is called the Lippman–Schwinger equation.
Let us introduce
θ, θ′∈S2, Df =kZS2
A(k, θ′, θ)f(k, θ′)dθ′.
Let us also define the solution Φ−(k, θ, x)for k∈
¯
C−={Imk≤0}as
Φ−(k, θ, x) = Φ+(−k, −θ, x).
As is well known [2],
Φ+(k, θ, x)−Φ−(k, θ, x) =
−k
4πZS2
A(k, θ′, θ)Φ−(k, θ′, x)dθ′, k ∈R. (4)
This equation is the key to solving the inverse scatter-
ing problem and was first used by Newton [2,3] and
Somersalo et al. [4].
definition 1. The set of measurable functions Rwith
the norm defined by
||q||R=ZR6
q(x)q(y)
|x−y|2dxdy < ∞
is recognised as being of Rollnik class.
Equation (4) is equivalent to the following:
Φ+=SΦ−,
where Sis a scattering operator with the kernel
S(k, ł) = ZR3
Φ+(k, x)Φ∗
−(ł, x)dx.
The following theorem was stated in [3]:
theorem 2. (Energy and momentum conservation
laws) Let q∈R. Then, SS∗=Iand S∗S=I,
where Iis a unitary operator.
corollary 1. SS∗=Iand S∗S=Iyield
A(k, θ′, θ)−A(k, θ, θ′)∗=
ik
2πZS2
A(k, θ, θ′′ )A(k, θ′, θ′′ )∗dθ′′ .
theorem 3. (Birmann–Schwinger estimation) Let
q∈R. Then, the number of discrete eigenvalues can
be estimated as
N(q)≤1
(4π)2ZR3ZR3
q(x)q(y)
|x−y|2dxdy.
lemma 1. Let |q|L1(R3)+ 4π|q|L2(R3)< α < 1/2.
Then,
∥Φ+∥L∞≤|q|L1(R3)+ 4π|q|L2(R3)
1−|q|L1(R3)+ 4π|q|L2(R3)<α
1−α,
∂(Φ+−Φ0)
∂k
L∞
≤|q|L1(R3)+ 4π|q|L2(R3)
1−|q|L1(R3)+ 4π|q|L2(R3)<
α
1−α.
Proof. By the Lippman–Schwinger equation, we
have
|Φ+−Φ0| ≤ |GqΦ+|,
|Φ+−Φ0|L∞≤ |Φ+−Φ0|L∞|Gq|+|Gq|,
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and, finally,
|Φ+−Φ0| ≤ |q|L1(R3)+ 4π|q|L2(R3)
1−|q|L1(R3)+ 4π|q|L2(R3).
By the Lippman–Schwinger equation, we also
have
∂Φ+−Φ0
∂k
≤
∂Gq
∂k Φ+
+
Gq
∂Φ+−Φ0
∂k
+|Gq|,
∂(Φ+−Φ0)
∂k
≤|q|L1(R3)+ 4π|q|L2(R3),
∂(Φ+−Φ0)
∂k
L∞
≤|q|L1(R3)+ 4π|q|L2(R3)
1−|q|L1(R3)+ 4π|q|L2(R3),
which completes the proof.
Let us introduce the following notation:
A0(k, θ, θ′) = ZR3
q(x)eik(θ−θ′)xdx,
K(s) = s, X(x) = x,
H+A0=
Z+∞
−∞
A0(s, θ, θ′)
s−t−i0ds, H−A0=Z+∞
−∞
A0(s, θ, θ′)
s−t+i0ds.
lemma 2. Let q∈R∩L1(R3),∥q∥L1+
4π|q|L2(R3)< α < 1/2. Then,
∥A+∥L∞< α +α
1−α,
∂A+
∂k
L∞
< α +α
1−α.
Proof. Multiplying the Lippman–Schwinger equa-
tion by q(x)Φ0(k, θ, x)and then integrating, we have
A(k, θ, θ′) = A0(k, θ, θ′)+
ZR3
q(x)Φ0(k, θ, x)GqΦ+dx.
We can estimate this latest equation as
|A| ≤ α+α|q|L1(R3)+ 4π|q|L2(R3)
1−|q|L1(R3)+ 4π|q|L2(R3).
Following a similar procedure for
∂A+
∂k
completes
the proof.
We define the operators H±,Hfor f∈W1
2(R)
as follows:
H+f=1
2πi lim
Imz→0
∞
Z
−∞
f(s)
s−zds, Im z > 0,H−f=
1
2πi lim
Imz→0
∞
Z
−∞
f(s)
s−zds, Im z < 0,
Hf=1
2(H++H−)f.
Consider the Riemann problem of finding a function
Φthat is analytic in the complex plane with a cut along
the real axis. Values of Φon the two sides of the cut
are denoted as Φ+and Φ−. The following presents
the results of [5]:
lemma 3.
HH =1
4I, HH+=1
2H+,HH−=−1
2H−,
H+=H+1
2I, H−=H − 1
2I, H−H−=−H−.
Denote
Φ+(k, θ, x) = Φ+(k, θ, x)−Φ0(k, θ, x),
Φ−(k, θ, x) = Φ−(k, −θ, x)−Φ0(k, θ, x),
g(k, θ, x) = Φ+(k, θ, x)−Φ−(k, θ, x)
lemma 4. Let q∈R, N(q)<1, g+=g(k, θ, x),
and g−=g(k, −θ, x).Then,
Φ+(k, θ, x) = H+g++eikθx,Φ−(k, θ, x) =
H−g++eikθx.
Proof. The proof of the above follows from the clas-
sic results for the Riemann problem.
lemma 5. Let q∈R, N(q)<1, g+=g(k, θ, x),
and g−=g(k, −θ, x),). Then,
Φ+(k, θ, x) = (H+g++eikθx),
Φ−(k, θ, x) = (H−g−+e−ikθx).
Proof. The proof of the above follows from the defi-
nitions of g, Φ±, and Φ±.
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lemma 6. Let
sup
k
∞
Z
−∞
pA(p, θ′, θ)
4π(p−k+i0)dp
< α, ZS2
αdθ < 1/2.
Then,
Y
0≤j<n ZS2
Z∞
−∞
kjA(kj, θ′
kj, θkj)
4π(kj+1 −kj+i0)dkj
dθkj≤2−n.
Proof. Denote
αj=
V p Z∞
−∞
kjA(kj, θ′
kj, θkj)
4π(kj+1 −kj+i0)dkj
,
Therefore,
Y
0≤j<n ZS2
Z∞
−∞
kjA(kj, θ′
kj, θkj)
4π(kj+1 −kj+i0)dkj
dθkj≤
Y
0≤j<n ZS2
αjdθkj<2−n.
This completes the proof.
lemma 7. Let
sup
kZS2
|H−A0K|dθ ≤α < 1
2C<1,
sup
kZS2
|H−˜qK|dθ ≤α < 1
2C<1,
sup
kZS2
H−A0˜qK2
dθ ≤α < 1
2C<1.
Then,
sup
kZS2
|H−AK|dθ ≤CRS2|H−A0K|dθ
1−sup
kRS2|H−A˜qK2|dθ ,
sup
k
ZS2
H−A˜qK2dθ
≤C
H−RS2A0˜qK2dθ
1−
H−RS2˜qKdθ
.
Proof. By the definition of the amplitude and Lemma
4, we have
A(k, θ′, θ) = −1
4πZR3
q(x)Φ+(k, θ, x)e−ikθ′xdx
=−1
4πZR3
q(x)heikθ′x+H+g(k, θ, θ′)ie−ikθ′xdx.
We can rewrite this as
A(k, θ′, θ) = −
1
4πZR3
q(x)
eikθx +X
n≥0
(−H−D)nΦ0
e−ikθ′xdx.
(5)
Lemma 6 yields
sup
kZS2
|H−AK|dθ ≤sup
kZS2
1
4πH−A0K
dθ+
sup
kRS2|H−KA|dθ2RS2
H−A˜qK2
dθ
1−sup
kRS2|H−KA|dθ2.
Owing to the smallness of the terms on the right-
hand side, the following estimate follows:
sup
kZS2
|H−AK|dθ ≤2sup
kZS2
1
4πH−A0K
dθ.
Similarly,
sup
kZS2
H−A˜qK2
dθ ≤CZS2
H−A0˜qK2
dθ+
ZS2
H−A˜qK2
dθ ZS2
|H−˜qK|dθ,
sup
kZS2
H−A˜qK2
dθ ≤CRS2
H−A0˜qK2
dθ
1−RS2|H−˜qK|dθ ,
sup
kZS2
H−A˜qK2
dθ ≤2sup
kZS2
1
4πH−A0˜qK2
dθ.
This completes the proof.
To simplify the writing of the following calcula-
tions, we introduce the set defined by
Mϵ(k) = s|ϵ < |s|+|k−s|<1
ϵ.
The Heaviside function is defined as:
θ(x) =
1if x > 0,
0.5if x= 0,
0if x < 0.
lemma 8. Let q, ∇q∈ ∩L2(R3),|A|>0. Then,
πi ZR3
θ(A)eik|x|Aq(x)dx =
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lim
ϵ→0Zs∈Mϵ(k)ZR3
eis|x|A
k−sq(x)dxds,
πi ZR3
θ(A)keik|x|Aq(x)dx =
lim
ϵ→0Zs∈Mϵ(k)ZR3
seis|x|A
k−sq(x)dxds.
Proof. The lemma can be proved by the conditions of
lemma and the lemma of Jordan.
lemma 9. Let
I0= Φ0(x, k)|r=r0.
Then
Z+∞
−∞ ZS2ZS2
˜q(k(θ−θ′))I0k2dkdθdθ′
≤sup
x∈R3
|q(x)|+
C0(1
r0
+r0)∥q∥L2(R3),
sup
θ∈S2
Z+∞
−∞ ZS2ZS2
A0HKA0I0k2dθ′′dθ′dk
≤
C0(1
r0
+r0)∥q∥2
L2(R3).
Proof. By the definition of the Fourier transform, we
have
Z+∞
−∞ ZS2ZS2
˜q(k(θ−θ′))I0k2dkdθdθ′=
Z+∞
−∞ ZS2ZS2Z+∞
0
q(x)eikx(θ−θ′)eix0kk2dkdθdθ′drdγ,
where x=rγ The lemma of Jordan completes the
proof for the first inequality. The second inequality is
proved like the first:
Z+∞
−∞ ZS2ZS2
A0HKA0I0k2dθ′′dθ′dk
=V.P Z+∞
−∞ Z+∞
−∞ ZS2ZS2ZS2
˜q(scos(θ′)−scos(θ′′))˜q(kcos(θ)−scos(θ′′)s
k−s
I0k2dθ′dθ′′dθdkds.
Lemma 8 yields
Z+∞
−∞ ZS2ZS2ZS2
˜
q(kcos(θ′)−kcos(θ))
˜q(kcos(θ)−kcos(θ′′)
I0k3θ(cos(θ′′))dθ′dθ′′dθdk−
Z+∞
−∞ ZS2ZS2ZS2˜q(kcos(θ′)−kcos(θ))
˜q(kcos(θ)−kcos(θ′′)I0k3θ(−cos(θ′′))dθ′dθ′′dθdk.
Integrating θ,θ′,θ′′, and k, we obtain the proof of the
second inequality of the lemma.
lemma 10. Let
sup
k
|H−A0K| ≤ α < 1
2C<1,sup
k
|H−˜qK| ≤
α < 1
2C<1,
sup
k
H−A0˜qK2
≤α < 1
2C<1, l = 0,1,2.
Then,
Z+∞
−∞ ZS2ZS2
A(k, θ′, θ)kldkdθ′dθ
≤
Z+∞
−∞ ZS2ZS2
˜q(k(θ−θ′))kldkdθ′dθ
+Csup
θ∈S2
Z+∞
−∞ ZS2ZS2
A0HKAkldθ′′dθ′dk
,
Z+∞
−∞ ZS2ZS2
A(k, θ′, θ)k2dkdθ′dθ
≤sup
x∈R3
|q|+
C0∥q∥W1
2(R3)∥q∥L2(R3)
ZS2
HKAdθ′′
+ 1.
Proof. Using the definition of the amplitude, Lemmas
3 and 4, and the lemma of Jordan yields
Z+∞
−∞ ZS2ZS2
A(k, θ′, θ)kldkdθ′dθ =−
Z+∞
−∞
1
4πZS2ZS2ZR3
q(x)Φ+(k, θ, x)e−ikθ′xkldxdkdθ′=
−1
4πZS2ZS2ZR3
q(x)
eikθx +X
n≥1
(−H−D)nΦ0
e−ikθ′xkldθ′dxdk
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Volume 3, 2024
=Z+∞
−∞ ZS2ZS2
˜q(k(θ−θ′))kldkdθ′dθ +X
n≥1
Wn,
W1=V.P ZR3Z+∞
−∞ ZS2ZS2
sA(s, θ′′ , θ)e−ikθ′xq(x)eisθ′′ x
k−skldkdxdsdθ′dθ′′,
|W1| ≤ Csup
θ∈S2
Z+∞
−∞ ZS2ZS2
A0HKAkldθ′′dθ′dk
.
Similarly,
|Wn| ≤
Csup
θ∈S2
Z+∞
−∞ ZS2ZS2
A0HKAkldθ′′dθ′dk
ZS2
HKAdθ′′
n
.
Finally,
Z+∞
−∞ ZS2ZS2
A(k, θ′, θ)dkdθ′dθ
≤
Z+∞
−∞ ZS2ZS2
˜q(k(θ−θ′))dkdθdθ′
+C0∥q∥2
L2(R3)
ZS2
HKAdθ′′
+ 1,
Z+∞
−∞ ZS2ZS2
A(k, θ′, θ)k2dkdθ′
≤
sup
x∈R3
|q|+C0∥q∥2
L2(R3)
ZS2
HKAdθ′′
+ 1.
This completes the proof.
lemma 11. Let
sup
kZS2
∞
Z
−∞
pA(p, θ′, θ)
4π(p−k+i0)dp
dθ < α < 1/2,
sup
k
pA(p, θ′, θ)
< α < 1/2.
Then,
|H−DΦ0|<α
1−α,
|H+DΦ0|<α
1−α,|DΦ0|<α
1−α,
H−g−= (I− H−D)−1H−DΦ0,
Φ−= (I− H−D)−1H−DΦ0+ Φ0,
and qsatisfies the following inequalities:
sup
x∈R3
|q(x)| ≤
ZS2
HKA0dθ
C0∥q∥2
L2(R3)+ 1+
C0∥q∥L2(R3).
Proof. Using the equation
Φ+(k, θ, x)−Φ−(k, θ, x) =
−k
4πZS2
A(k, θ′, θ)Φ−(k, θ′, x)dθ′, k ∈R,
we can write
H+g+− H−g−=D(H−g−+ Φ0).
Applying the operator H−to the last equation, we
have
H−g−=H−D(H−g−+ Φ0),
(I− H−D)H−g−=H−DΦ0,
H−g−=X
n≥0− H−DnΦ0.
Estimating the terms of the series, we obtain using
Lemma 4
|(H−D)nΦ0| ≤ X
n≥0
Z∞
−∞
· · · Z∞
−∞
Φ0
Y
0≤j<n
RS2kjA(kj, θ′
kj, θkj)dθ′
kj
4π(kj+1)−kj+i0) dk1. . . dkn
≤X
n>0
2nαn=2α
1−2α.
Denoting
Λ = ∂
∂k , r =qx2
1+x2
2+x2
3,
we have
ΛZS2
Φ0dθ = Λsin(kr)
ikr =cos(kr)
ik −sin(kr)
ik2r,
ΛZS2
H0Φ0dθ = Λk2sin(kr)
ikr =kcos(kr)
i+sin(kr)
ik2r,
ΛZS2
Φdθ
=
ΛZS2
Φ0dθ+
ΛZS2X
n≥0−H−DnΦ0dθ > 1
k−α
1−α,as kr =π,
and
Λ1
k−t=−1
(k−t)2
Equation (2) yields
q=
ΛH0RS2Φdθ +k2RS2Φdθ
ΛRS2Φdθ
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.5
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=
2kRS2H−g−dθ +k2RS2ΛH−g−dθ +H0ΛRS2H−g−dθ
ΛRS2Φdθ
=
2kRS2H−g−dθ + Λ RS2Pn≥1− H−Dn(K2−k2)Φ0dθ
ΛRS2Φdθ
=W0+Pn≥1RS2Wn
ΛRS2Φdθ .
Denoting
Z(k, s) = s+ 2k+2k2
k−s,
we then have
|W1| ≤
Z+∞
−∞ ZS2ZS2
A(s, θ, θ′)ss2−k2
(k−s)2Φ0sin(θ)dsdθ
k=k0
≤
Z+∞
−∞ ZS2ZS2
Z(k, )˜q(k(θ−θ′))Φ0dkdθ
+
C0
ZS2
HKA0dθ
.
For calculating Wn, as n≥1, take the simple
transformation
s3
n
sn−sn−1
=s3
n−s2
nsn−1
sn−sn−1
+s2
nsn−1
sn−sn−1
=s2
n+s2
nsn−1
sn−sn−1
=s2
n+s2
nsn−1−sns2
n−1
sn−sn−1
+sns2
n−1
sn−sn−1
=
s2
n+snsn−1+sns2
n−1
sn−sn−1
,(6)
As3
n
sn−sn−1
=As2
n+Asnsn−1+Asns2
n−1
sn−sn−1
=
V1+V2+V3.
Using Lemma 10 for estimating V1and V2and, for
V3, taking again the simple transformation for s3
n−1,
which will appear in the integration over sn−1, we fi-
nally get
|q(x)|r=r0=
ΛH0RS2Φdθ +k2RS2Φdθ
ΛRS2Φdθ
k=k0,r=π
k0
≤
R+∞
−∞ RS2RS2Z(k, )˜q(k(θ−θ′))Φ0dkdθdθ′
(1
k0−α
(1−α))+
C0
RS2HKA0dθ
(1
k0−α
(1−α))
Finally, we get
|q(x)|r=r0≤sup
x∈R3
|q(x)|α+
C0∥q∥2
L2(R3)+C0∥q∥L2(R3)+
ZS2
HKA0dθ
.
The invariance of the Schrödinger equations with
respect to translations which will below be and the
arbitrariness of r0yield
sup
x∈R3
|q(x)| ≤
ZS2
HKA0dθ
C0∥q∥2
L2(R3)+ 1+
C0∥q∥L2(R3).
To complete the construction of estimates, we need
to prove the invariance of the Schrödinger equation
with respect to shifts and coordinate transformations.
To do this, we introduce the following notations and
definitions:
qUa(x) = q(Ux +a)
where
UU′=U′U=I
a∈R3
The corresponding amplitude and wave functions, de-
noted as
AUa,ΦUa, EU a
are associated with these potentials.
theorem 4. The wave function ΦUa+can be ex-
pressed as:
ΦUa+(k, θ, x) = Φ0(k, θ, x)+X(GqU a)nΦ0(7)
AUa+(k, θ′, θ) =−(1/(4π)) RqUa(x)Φ0(k, θ, x)ΦUa(k, −θ′, x)dx
(8)
Proof. The theorem follows directly from the repre-
sentations (7) and (8).
theorem 5. : The poles of the functions ΦUa+and Φ+
coincide, i.e.
EUa =E.
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Volume 3, 2024
Proof: From the representations (7) and (8).
ΦUa+(k, θ, x1) = Φ0(k, θ, x1)+
∞
X
n=1
n
Y
k=1 ZqUa(xk+1)eik|xk−xk+1 |
|xk−xk+1|Φ0(k, θ, xn+1)dx2...dxn+1
(9)
ΦUa+(k, θ, x1) = Φ0(k, θ, x1)+
eik(θ,a)
∞
X
n=1
n
Y
k=1
Zq(xk)eik|xk−xk+1 |
|xk−xk+1|Φ0(k, θ, Uxn+1)dx2...dxn+1
ΦUa+(k, θ, x1) = Φ0(k, θ, x1)+
eik(θ,a)
∞
X
n=1
n
Y
k=1
Zq(xk)eik|xk−xk+1 |
|xk−xk+1|Φ0(k, θ, Uxn+1)dx2...dxn+1
ΦUa+(k, θ, x1) = Φ0(k, θ, x1)+
eik(θ,a)[Φ0(k, U′θ, x1)+
∞
X
n=1
n
Y
k=1 Zq(xk)eik|xk−xk+1 |
|xk−xk+1|Φ0(k, U′θ, xn+1)dx2...dxn+1]
−eik(θ,a)Φ0(k, U′θ, x1)
ΦUa+(k, θ, x1) =
Φ0(k, θ, x1)+eik(θ,a)Φ+(k, U′θ, x1)−eik(θ,a)Φ0(k, U′θ, x1)
From the last equation, it follows that the poles of the
function on the right and left coincide.
:
theorem 6. : Amplitudes of the functions ΦUa+and
Φ+can calculates as .
AUa+(k, θ′, θ) = eik(θ′−θ)A(k, U ′θ′, U ′θ))
Proof. : From the Theorem (4)
AUa+(k, θ′, θ) =
−(1/(4π)) ZqUa(x1)Φ0(k, θ, x1)ΦU a(k, −θ′, x1)dx1
(10)
from Theorem (5)
AUa+(k, θ′, θ) =
−(1/(4π)) ZqUa(x1)Φ0(k, θ, x)[Φ0(k, −θ′, x1)+
∞
X
n=1
n
Y
k=1 ZqUa(xk+1)eik|xk−xk+1 |
|xk−xk+1|Φ0(k, −θ′, xn+1)]
dx2...dxn+1dx1
AUa+(k, θ′, θ) = eik(θ′−θ)A(k, U ′θ′, U ′θ))
4 Сonclusion
In conclusion, this research has illuminated previ-
ously unexplored invariants and symmetries of the
Schrödinger equation, significantly broadening our
theoretical and practical grasp of quantum mechan-
ics. By leveraging the Poincaré-Riemann-Hilbert
boundary-value problem, we uncovered new insights
into the equation’s behavior under coordinate trans-
formations, highlighting the invariance of bound state
energies and the variability in scattering amplitude
phases. These discoveries not only enrich our theo-
retical understanding but also have profound implica-
tions for a wide range of practical applications, from
fluid dynamics to medical imaging.
Moreover, the development of more accurate mod-
els for quantum systems underscores the potential of
theoretical physics to address complex challenges in
applied sciences. The enhanced phase selection tech-
niques derived from this study promise to revolution-
ize seismic exploration, tomography, and other fields
reliant on precise interpretation of quantum behav-
iors.
Ultimately, this study underscores the pivotal role
of fundamental theoretical research in driving inno-
vation and technological advancement across various
scientific domains. By revealing new properties of
the Schrödinger equation, it paves the way for future
explorations in quantum mechanics, offering promis-
ing new avenues for both theoretical inquiries and
practical applications. The interdisciplinary impact
of these findings highlights the enduring value of deep
theoretical insights in advancing our understanding of
the natural world and in tackling real-world problems.
Acknowledgment:
It is an optional section where the authors may write
a short text on what should be acknowledged
regarding their manuscript.
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.5
Durmagambetov A.A
E-ISSN: 2945-0489
36
Volume 3, 2024
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This research has been/was/is funded by the Science
Committee of the Ministry of Science and Higher
Education of the Republic of Kazakhstan (Grant No.
AP19677733)»
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.5
Durmagambetov A.A
E-ISSN: 2945-0489
37
Volume 3, 2024
