# An efficient approximation for accelerating convergence of the numerical
power series. Results for the $1D-$Schrödinger’s equation

A. Bağcı<EMAIL_ADDRESS>Computational Physics Laboratory, Department of
Physics, Faculty of Arts and Sciences, Pamukkale University, Çamlaraltı,
Kınıklı Campus, Denizli, Turkey  Z. Güneş  Department of Chemical Engineering,
Faculty of Engineering, Pamukkale University, Çamlaraltı, Kınıklı Campus,
Denizli, Turkey

###### Abstract

The numerical matrix Numerov algorithm is used to solve the stationary
Schrödinger equation for central Coulomb potentials. An efficient
approximation for accelerating the convergence is proposed. The Numerov method
is error$-$prone if the magnitude of grid$-$size is not chosen properly. A
number of rules so far, have been devised. The effectiveness of these rules
decrease for more complicated equations. Efficiency of the technique used for
accelerating the convergence is tested by allowing the grid$-$sizes to have
variationally optimum values. The method presented in this study eliminates
the increased margin of error while calculating the excited states. The
results obtained for energy eigenvalues are compared with the literature. It
is observed that, once the values of grid$-$sizes for hydrogen energy
eigenvalues are obtained, they can simply be determined for the hydrogen
iso$-$electronic series as, $h_{\varepsilon}(Z)=h_{\varepsilon}(1)/Z$.

Keywords

Numerov method, Screened Columb potential, Schrödinger equation

PACS numbers

… .

##  I Introduction

The wave$-$function that holds all the required information of a system is
obtained from analytical or numerical solution of the quantum mechanical
wave$-$equation such as Schrödinger or Dirac equation. Characteristics of the
potential determines an exact analytical solution for the wave$-$equation
whether available or not. If the potential function $V(r,r_{0})$ is separable
depending on its variables $r,r_{0}$ into two factors 1_Nichols_1965  may be
reduced to soluble form. Such simplicity however, appears only for a few
idealized systems. Complexity increases and the potential function may not be
separable for more realistic representations. In this case, an analytical
solution based on the approximation methods is required. Among these
approximations, the perturbation, variation and Wentzel-Kramers-Brillouin
(WKB) methods have generally been used in the literature 2_Ishikawa_2002 ;
3_Landau_1977 . Although these analytical approaches are preferred for use in
atomic and molecular calculations, they are limited in terms of applications
2_Ishikawa_2002 . A solution free from symmetry properties of a potential
inevitably leads to numerical approaches 4_Dijk_2016 .
Numerical integration formula for integrating the time$-$independent
Schrödinger equation 5_Uria_1996

$\displaystyle-\frac{\hbar}{2m}\psi^{\prime\prime}\left(x\right)+V\left(x\right)\psi\left(x\right)\approx
E\psi\left(x\right)\hskip 5.69054ptx\in\left[a,b\right]$ (1)

which was suggested in the early days of quantum mechanics by Numerov
6_Numerov_1924 ; 7_Numerov_1927 . The Numerov’s method takes advantage of
specific structure of the Eq. (1). It involves no first derivative and it is
linear in $\psi$. The Eq. (1) may be transformed to dimensionless form as,

$\displaystyle\psi^{\prime\prime}\left(x\right)-f\left(x\right)\psi\left(x\right)=0,$
(2)

where, $f\left(x\right)=-2m\left(E-V\left(x\right)\right)/\hbar^{2}$.
The solution $\psi\left(x\right)$ at the nodes of $N$ uniform grid points,
namely, $x_{0},x_{1},x_{2},...,x_{N}$ with $h_{\varepsilon}=x_{i+1}-x_{i}$,
where $h_{\varepsilon}$ grid$-$size for $\varepsilon^{th}$ energy state. The
Numerov’s algorithm includes higher order terms than the well$-$known
finite$-$difference with an error term that is of order
$\mathcal{O}\left(h^{2}\right)$. It is derived by considering conjointly the
forward, backward Taylor expansion of $\psi\left(x\right)$ up to order
$\mathcal{O}\left(h^{6}\right)$ 8_Graen_2016 . The following relation is
finally obtained $\left[\psi_{i}\equiv\psi\left(x_{i}\right),f_{i}\equiv
f_{i}\left(x\right)\right]$,

$\psi_{i+1}-2\psi_{i}+\psi_{i-1}=\\\
-\frac{h^{2}}{12}\left(f_{i+1}\psi_{i+1}+10f_{i}\psi_{i}+f_{i-1}\psi_{i-1}\right)+\mathcal{O}\left(h^{6}\right).$
(3)

The Eq. (3) is a numerical one$-$dimensional pattern 9_Chow_1972 . The
wavefunction $\psi\left(x\right)$ is calculated iteratively 10_Gonzales_1997 .
An initial guess for $E$ is required. The interval $\left[a,b\right]$ is
finite and implicit the Dirichlet boundary conditions at
$\psi_{0}=\psi\left(a\right)=0$, $\psi_{N}=\psi\left(b\right)=0$ 8_Graen_2016
. Richardson extrapolation 5_Uria_1996  or Cooley’s correction formula
11_Cooley_1961 ; 12_Guest_1974 ; 13_Du_1990 ; 14_Izaac_2018  which are based
on perturbation theory, may be used to improve the approximation for the
energy with each iteration.
Several other variants in order to increase both stability and accuracy while
calculating the resonance problems and high$-$lying bound states were
proposed. Generalization of the algorithm to an error of arbitrary order
15_Hajj_1974 ; 16_Fack_1987 ; 17_Berghe_1989 ; 18_Ixaru_1980 ; 19_Ixaru_1985 ;
20_Allison_1991 ; 21_Killingbeck_1999 ; 22_Wang_2004 ; 23_Yang_2017 ;
24_Obaidat_2021 ; 25_Medvedeva_2021  and its extension for solution of
differential equations with more than one$-$dimension 8_Graen_2016 ;
26_Kobeisse_1974 ; 27_Hajj_1975 ; 28_Hajj_1982 ; 29_Hajj_1985 ; 30_Eckert_1989
; 31_Avdelas_2000 ; 32_Kalogiratou_2005 ; 33_Kuenzer_2019  form the main
framework of the studies. A Numerov$-$type exponentially fitted method was
suggested 18_Ixaru_1980 ; 19_Ixaru_1985 ; 20_Allison_1991 ;
32_Kalogiratou_2005 ; 34_Raptis_1978 ; 35_Berghe_1990 ; 36_Berghe_1990 ;
37_Simos_1996 ; 38_Simos_1998 ; 39_Simos_1999 ; 40_Konguetsof_2003 ;
41_Aguiar_2005 ; 42_Berghe_2007 ; 43_Tsitouras_2018 , accordingly. Here, the
coefficients in the Eq. (3) are replaced by some arbitrary parameters. It is
assumed that the solution is a linear combination of the exponential
functions. The parameters are then obtained from the resulting system of
differential equations. Another method derived from the Eq. (3) is so called
re-normalized Numerov method. It was obtained by making two transformations
44_Johnson_1977 ; 45_Johnson_1978 ; 46_Leroy_1985 ; 47_Karman_2014 ;
48_Zhao_2016 . The first transformation is used to decrease the number of
steps required for calculation and the second one is used to replace the
three$-$point recurrence relation with two$-$point relations by defining a
ratio. If the iteration is stopped at any point, the last two elements of the
ratio are immediately available. They are used to calculate the
wave$-$function $\left(\psi\right)$ around that point within a normalized
factor 44_Johnson_1977 . Non$-$uniform grid$-$size $h_{i}$,
$h_{i+1}=x_{i+1}-x_{i}$ was also considered to be use 49_Kobeissi_1988 ;
50_Bieniasz_2004 ; 51_Aguiar_2005 ; 52_Ramos_2005 ; 53_Speciale_2020 ;
54_Brunetti_2021 . It is derived due to the need to solve the transport model
for semiconductor devices where, the standard Numerov procedure with uniform
grid$-$size is not applicable 53_Speciale_2020 ; 54_Brunetti_2021 . The range
of application of the Numerov’s algorithm is indubitably not limited to the
aforementioned examples 47_Karman_2014 ; 48_Zhao_2016 ; 50_Bieniasz_2004 ;
53_Speciale_2020 ; 54_Brunetti_2021 . It is used to determine presence of the
energetics for short intramolecular O$-$H$\cdots$O in enzyme and photo centers
by nuclear motion of the involved hydrogen atom 8_Graen_2016 , to solve the
Schrödinger equation with complex potential boundaries for open multi$-$layer
heterojunction systems 55_Lin_2018  and to study energy spectra of mesons and
hadrons 56_Ali_2020 . Two$-$dimensional quantum dot eigen$-$functions for a
much larger spectrum of external harmonic frequencies are calculated as well
57_Caruso_2019 .

As an alternative to the iterative procedure the Numerov’s algorithm can be
transformed to a matrix form. This matrix method was suggested in 5_Uria_1996
and was revisited in 58_Mohandas_2012 . It leads to renewed interest in
solution of the one$-$dimensional Schrödinger equation. Convergence properties
for the Numerov’s algorithm has been recently investigated for various forms
of central potentials 59_Xie_2021  where, the relativistic effects have also
been taken into account. In this matrix method, the problem is reduced to the
following form of the algebraic generalized$-$eigenvalue equation,

$\displaystyle Ax=\lambda Bx.$ (4)

Computation of the energy levels and the wave$-$functions in the Eq. (1) is
now reduced to the computation of eigenvalues and eigen$-$functions of
triangular symmetric matrices $A$, $B$. This leads to the ground and the
excited states of the Eq. (1) to be calculated simultaneously. No initial
guess is required 5_Uria_1996 . Efficient computation of the Eq. (4) for
matrices with large numbers of non$-$zero elements has attracted considerable
research interest because it is encountered in many applications in physics,
chemistry and engineering. Gaussian elimination, $LU-$factorization, inverse
iteration methods are the ones were studied in detail 60_Stewart_1973 .

The proposed approach in this work can be considered within the scope of
matrix Numerov method. The number of grid points is reduced while the accuracy
is sufficiently increased. This is achieved by manipulating the potential in
the Schrödinger equation. Due to singularity of the Coulomb potential,
hydrogen-like eigen$-$functions have non$-$continuous derivatives at $r=0$
61_Fattal_1996 . Correctly representation of the radial wave$-$function
require use of tremendous grid points such that, one has to consider how to
compute Eq. (4) effectively. The solution of the Schrödinger equation for the
central Coulomb potential is therefore examined as a sample. The error term is
embedded in a certain weight to the matrix that represents the effective
potential. Such weight leads to approximately determine the grid$-$size.
Variational stability is tested via the optimization procedure. The results
are presented for hydrogen$-$like atoms.

Table 1:  Variationally optimum values of grid$-$sized to be used in computation of electronic energy spectrum of hydrogen atom. The calculations are performed in atomic units (a. u.) for the pure Coulomb potential where, $\nu=0$ and $\mu\left(N,0,0\right)=0$. $N$ | $h_{1s}$ | $h_{2s}$ | $h_{3s}$ | $h_{4s}$ | $h_{5s}$ | $h_{10s}$
---|---|---|---|---|---|---
$500$ | $0.016764$ | $0.041090$ | $0.073143$ | $0.112906$ | $0.160429$ | $0.518061$
$1000$ | $0.009129$ | $0.022209$ | $0.039261$ | $0.060262$ | $0.085171$ | $0.268797$
$1500$ | $0.006378$ | $0.015458$ | $0.027225$ | $0.041659$ | $0.058729$ | $0.183493$
$2000$ | $0.004941$ | $0.011920$ | $0.020973$ | $0.032028$ | $0.045083$ | $0.140004$
$2500$ | $0.004048$ | $0.009749$ | $0.017134$ | $0.026108$ | $0.036703$ | $0.113499$
$5000$ | $0.002171$ | $0.005199$ | $0.009070$ | $0.013798$ | $0.019333$ | $0.059092$
$7500$ | $0.001498$ | $0.003590$ | $0.006244$ | $0.009485$ | $0.013268$ | $0.040302$
$10000$ | $0.001160$ | $0.002755$ | $0.004798$ | $0.007261$ | $0.010146$ | $0.030702$
$15000$ | $0.000804$ | $0.001902$ | $0.003286$ | $0.004979$ | $0.006947$ | $0.020917$
Table 2:  Variationally minimum values for electronic energy spectrum of hydrogen atom with the Coulomb potential in atomic units (a. u.) where, $\nu=0$ and $\mu\left(N,0,0\right)=0$. $N$ | $E_{1s}\left(N,h_{1s},0\right)$ | $E_{2s}\left(N,h_{2s},0\right)$ | $E_{3s}\left(N,h_{3s},0\right)$ | $E_{4s}\left(N,h_{4s},0\right)$ | $E_{5s}\left(N,h_{5s},0\right)$ | $E_{10s}\left(N,h_{10s},0\right)$
---|---|---|---|---|---|---
$500$ |  0.49989 54423 |  0.12492 43874 |  0.05548 76675 |  0.03118 52044 |  0.01993 68695 |  0.00494 40578
$1000$ |  0.49996 90692 |  0.12497 76755 |  0.05553 54185 |  0.03123 05773 |  0.01998 07819 |  0.00498 07719
$1500$ |  0.49998 49515 |  0.12498 91750 |  0.05554 57959 |  0.03124 05672 |  0.01999 06272 |  0.00499 01987
$2000$ |  0.49999 10010 |  0.12499 35460 |  0.05554 97440 |  0.03124 43816 |  0.01999 44088 |  0.00499 40255
$2500$ |  0.49999 39700 |  0.12499 56864 |  0.05555 16764 |  0.03124 62510 |  0.01999 62672 |  0.00499 59622
$5000$ |  0.49999 82742 |  0.12499 87763 |  0.05555 44612 |  0.03124 89454 |  0.01999 89507 |  0.00499 88427
$7500$ |  0.49999 91736 |  0.12499 94173 |  0.05555 50364 |  0.03124 95010 |  0.01999 95043 |  0.00499 94518
$10000$ |  0.49999 95110 |  0.12499 96564 |  0.05555 52504 |  0.03124 97072 |  0.01999 97095 |  0.00499 96789
$15000$ |  0.49999 97669 |  0.12499 98372 |  0.05555 54115 |  0.03124 98622 |  0.01999 98636 |  0.00499 98498
|  |  0.49999 99167111Ref. 69_Purevkhuu_2021  $\left(N\approx 25000\right)$
---
0.49999 0222Ref. 5_Uria_1996
|  0.12499 99896111Ref. 69_Purevkhuu_2021  $\left(N\approx 25000\right)$
---
0.124999222Ref. 5_Uria_1996
0.05555 4222Ref. 5_Uria_1996  |  |  |
Table 3:  Results of computation for electronic energy spectrum of hydrogen
atom with screened Coulomb potential in atomic units (a. u.) where, $\nu\neq
0$ and $\mu\left(N,0,\nu\right)\neq 0$.

$N$ | $E_{1s}\left(N,h_{1s},\nu\right)$ | $E_{2s}\left(N,h_{2s},\nu\right)$ | $E_{3s}\left(N,h_{3s},\nu\right)$ | $E_{4s}\left(N,h_{4s},0\right)$ | $E_{5s}\left(N,h_{5s},\nu\right)$ | $E_{10s}\left(N,h_{10s},\nu\right)$
---|---|---|---|---|---|---
$500$ |  0.49999 81387 |  0.12499 35304 |  0.05555 00986 |  0.03124 43978 |  0.01999 43970 |  0.00499 46365
$1000$ |  0.49999 95666 |  0.12499 81450 |  0.05555 39025 |  0.03124 81011 |  0.01999 78112 |  0.00499 57720
$1500$ |  0.49999 98415 |  0.12499 91616 |  0.05555 48140 |  0.03124 91164 |  0.01999 89347 |  0.00499 95057
$2000$ |  0.49999 99394 |  0.12499 95392 |  0.05555 51559 |  0.03124 95120 |  0.01999 93941 |  0.00499 96099
$2500$ |  0.49999 99699 |  0.12499 97087 |  0.05555 53086 |  0.03124 96943 |  0.01999 96129 |  0.00499 97098
$5000$ |  0.50000 00010 |  0.12499 99336 |  0.05555 55062 |  0.03124 99361 |  0.01999 99133 |  0.00499 99207
$7500$ |  0.49999 99960 |  0.12499 99684 |  0.05555 55341 |  0.03124 99728 |  0.01999 99626 |  0.00499 99651
$10000$ |  0.50000 00037 |  0.12499 99866 |  0.05555 55485 |  0.03124 99899 |  0.01999 99842 |  0.00499 99865
$15000$ |  0.50000 00036 |  0.12499 99959 |  0.05555 55548 |  0.03124 99981 |  0.01999 99956 |  0.00499 99976
| $i=3$ | $i=1$ | $i=2$ | $i=4$ | $i=6$ | $i=15$

00footnotetext: $i$ is the number of iterations

##  II Modified Matrix Numerov Representation for The Hamiltonian

Taking into account the Eqs. (1,2) the exact second derivative for
$\psi\left(x\right)$ is given as,

$\displaystyle\psi^{\prime\prime}\left(x\right)=-\frac{2m}{\hbar^{2}}\left[E-V\left(x\right)\right]\psi\left(x\right).$
(5)

The Eq. (5) from the perspective of numerical integration, may be represented
by the following statement;

$\displaystyle Exact=Numerical+\mathcal{O}\left(h^{6}\right).$

By inserting the Eq. (5) in the above statement and assuming that the error
term is of order $\mathcal{O}\left(h^{6}\right)$ we have the following
property,

$\displaystyle\hat{\beta}\left(h^{6}\right)\psi\left(x\right)=\mathcal{O}\left(h^{6}\right)\psi\left(x\right),$
(6) $\displaystyle
Numerical=-\frac{2m}{\hbar^{2}}\left[E-V\left(x\right)\pm\beta\left(h^{6}\right)\right]\psi\left(x\right).$
(7)

If the potential term defined as,

$\displaystyle
V^{\prime}\left(x\right)=V\left(x\right)\pm\beta\left(h^{6}\right),$ (8)

the Eq. (7) becomes analogous to Eq. (2). It is unlikely to find a definition
for $\beta\left(h^{6}\right)$. Instead the potential $V\left(x\right)$ may be
replaced by the one involving a screening parameter $\mu$ (more precisely a
function). Thus,

$\displaystyle V^{\prime}\left(x\right)=V_{\mu}\left(x\right).$ (9)

In general we can say that the function $\mu$ is a function of number of
grid$-$points. It must be a function such that it satisfies the conditions
given for the screened potential as below,

$\displaystyle\lim_{\mu\rightarrow
0}V^{\prime}_{\mu}\left(x\right)=V\left(x\right),$ (10)
$\displaystyle\lim_{N\rightarrow\infty}V^{\prime}_{\mu}\left(x\right)=V\left(x\right).$
(11)

We have derived two variants of $\mu$ in order to solve the radial stationary
Schrödinger equation of an electron moving through the Coulomb potential.

##  III A method to determine the function $\mu\left(N\right)$

Depending on the number of grid$-$points $N$ and angular momentum quantum
number $l$, the two versions of the $\mu$ function in this study are defined
as,

$\displaystyle\mu_{1}\left(N,l,\nu_{1}\right)=Exp\left[{\frac{1}{N^{\nu_{1}(l+1)}}}\right]-1,$
(12)
$\displaystyle\mu_{2}\left(N,l,\nu_{2}\right)=Erf\left[{\frac{1}{N^{\nu_{2}(l+1)}}}\right],$
(13)

where, $\left\\{\nu_{1},\nu_{2}\right\\}\in\mathcal{R}^{+}$,
$1\leq\left\\{\nu_{1},\nu_{2}\right\\}<\infty$.
Minimum and maximum values to be found for energy eigenvalues correspond to
the $\nu=1$ and $\nu\rightarrow\infty$ ($\mu\rightarrow 0$, represents the
pure Coulomb potential), respectively. According to the value of $\nu$ the
interval for numerically obtained energy eigenvalues are determined as,

$\displaystyle E_{1}\leq E_{\nu}\leq E_{\infty}=E_{Coulomb}.$

The Eqs. (12, 13) for any value of angular momentum quantum number $l$ satisfy
the following property,

$\displaystyle\lim_{N\rightarrow\infty}\mu\left(N,l,\nu\right)=0.$ (14)

The Eq. (4) is solved iteratively since it is now depends to the values of
$\nu$. The number of iterations are sensitive to the choice of initial values
of $\nu$. Large number of grid$-$points approximate both values of the $\mu$
and the grid$-$size $h$ to zero. A simple approach as
$\mu_{1}\left(N,l,\nu_{1}\right)=h_{1s}^{2}$,
$\mu_{2}\left(N,l,\nu_{2}\right)=h_{1s}^{2}$ or visa versa (see the Table 1)
that promise better initial values for $\nu$ (or if $\nu$ is known for $h$) is
used, accordingly. Alternatively, one can chose a large value of $\nu$ as an
initial input. Two upper limit of summations (number of grid$-$points) are
selected. The calculations are performed for these two upper limit of
summations using the $\nu_{1}$. If the energy eigenvalues are increasing, then
the $\nu_{2}$ is used. The iteration is terminated. Else, $\nu_{1}$ is used in
the Eq. (13) to derive a new $\nu_{1}$. This new value of $\nu_{1}$ is used in
the Eq. (12). The calculations are performed until the energy eigenvalues
start to increasing. An algorithm for the above procedure may be summarized
as,

Step 1. Choose two upper limit of summation $\left(N_{1},N_{2}\right)$.
Step 2. Choose a grid$-$size for each upper limit of summation or use the ones
given in the Table 1.
Step 3. Obtain initial values of $\nu_{1}$ and $\nu_{2}$ for each $N$ by
solving the equations $\mu_{1}\left(N,l,\nu_{1}\right)=h^{2}$,
$\mu_{2}\left(N,l,\nu_{2}\right)=h^{2}$. $i=0$ (iteration number).
Step 4. Calculate the energy eigenvalues
$E_{N_{1}}\left(h_{1},\nu_{1}\right)$, $E_{N_{2}}\left(h_{2},\nu_{1}\right)$.
Step 5. If
$E_{N_{1}}\left(h_{1},\nu_{1}\right)<E_{N_{2}}\left(h_{2},\nu_{1}\right)$ and
$i=0$
Print $E_{N_{2}}\left(h_{2},\nu_{2}\right)$. Break.
Step 6. Else if
$E_{N_{1}}\left(h_{1},\nu_{1}\right)>E_{N_{2}}\left(h_{2},\nu_{1}\right)$
Insert the $\nu_{1}$ in the Eq. (13). Obtain new values for $\mu_{2}$. Solve
the equation $\mu_{1}\left(N,l,\nu_{1}\right)=\mu_{2}$. This derives a new
value for $\nu_{1}$. $i=i+1$. Go to step 4.
Step 7. Write,

$\displaystyle\frac{\nu^{2,i-1}_{1}E^{i-1}_{N_{2}}+\nu^{2,i}_{1}E^{i}_{N_{2}}}{\nu^{2,i-1}_{1}+\nu^{2,i}_{1}}.$

Note that $E^{i}_{N}\equiv E^{i}_{N}\left(h,\nu\right)\equiv
E_{nl}\left[h,\mu\left(N,l,\nu\right)\right]$ of $i^{th}$ iteration.
$n=1,2,3,...$, $l=n-1$, represent the corresponding energy state.
$\nu_{1}^{pi}$, $\nu_{2}^{pi}$ represent $i^{th}$ values of $\nu_{1}$ and
$\nu_{2}$ for grid$-$points $N_{1},N_{2}$ with $p$, $p=1$ or $p=2$,
respectively. $h_{1}$ and $h_{2}$ are the grid$-$sizes for the corresponding
energy eigenvalues in $N_{1}$, $N_{2}$. The convergence properties and the
algorithm of the procedure with an explicit application are given in the
Figures 1, 2.

##  IV Example: Hydrogen$-$like atoms

The radial Schrödinger equation for an electron in a spherically symmetric
potential has the following form 3_Landau_1977 ,

$\displaystyle\hat{H}_{r}\Psi\left(r\right)=E_{r}\Psi\left(r\right),$ (15)

where,

$\displaystyle\hat{H}_{r}=-\frac{1}{2}\Bigg{(}\frac{\partial^{2}}{\partial
r^{2}}-\frac{l(l+1)}{r^{2}}\Bigg{)}+\frac{1}{r},$ (16)

is the one$-$electron Hamilton operator [in atomic units (a. u.);
$\hbar=1$,$m=1$and $e^{2}/4\pi\epsilon_{0}=1$]. Note that, the Eq. (16)
depends upon $l$ yet, the exact solution of the Eq. (15) with $\hat{H}_{r}$
for hydrogen atom is $n^{2}-$fold degenerate. It is referred to as accidental
degeneracy. The discussion on the origin of such degeneracy for hydrogen atom
is treated rigorously in 62_Deshmukh_2015 . The results obtained from
numerical solution of the Schrödinger equation using the standard Numerov’s
method for $\hat{H}_{r}$ operator do not posses $n^{2}-$fold degeneracy (see
the following section for more detail). The Numerov’s method with a modified
effective potential on the other hand, reveals the accidental degeneracy.

In order to apply the Numerov method with a modified effective potential, the
Eq. (15) is transformed into a linear finite$-$difference equation. The
effective potential is defined as,

$\displaystyle V^{\prime}\equiv
V^{\prime}_{eff}\left(r\right)=\frac{1}{2}\frac{\left[l(l+1)\right]}{r^{2}}+\frac{1}{r^{1-\mu}}.$
(17)

The standard Numerov algorithm given in the Eq. (3) is used in following. The
resulting equation can be rearranged into the following form
$\left(\Psi\equiv\Psi\left(r\right)\right)$ (5_Uria_1996, ; 58_Mohandas_2012,
),

$-\frac{1}{2}\frac{\left(\Psi_{i-1}-2\Psi_{i}+\Psi_{i+1}\right)}{h^{2}}\\\
+\frac{\left(V^{\prime}_{i-1}\Psi_{i-1}+10V^{\prime}_{i}\Psi_{i}+V^{\prime}_{i+1}\Psi_{i+1}\right)}{12}\\\
=E\frac{\left(\Psi_{i-1}+10\Psi_{i}+\Psi_{i+1}\right)}{12}.$ (18)

$N$ equations of the linear system given above in the matrix form are written
as,

$\displaystyle-\frac{1}{2}A\Psi+BV^{\prime}\Psi=EB\Psi,$ (19)

where, $\Psi$ as the column vector
$\left(...\Psi_{i-1},\Psi_{i},\Psi_{i+1}...\right)$,
$A=\left(\mathbb{I}_{-1}-2\mathbb{I}_{0}+\mathbb{I}_{1}\right)/h^{2}$,
$B=\left(\mathbb{I}_{-1}-2\mathbb{I}_{0}+\mathbb{I}_{1}\right)/12$,
$V^{\prime}=diag\left(...V^{\prime}_{i-1},V^{\prime}_{i},V^{\prime}_{i+1}...\right)$,
$\mathbb{I}_{p}$ is a matrix of $1s$ along the $pth$ diagonal and zeros
elsewhere. The boundary conditions $\Psi\left(0\right)=\Psi\left(R\right)=0$
for some large $R$ are implemented by taking $N\times N$ sub$-$matrices of $A$
and $B$. The optimized values of $h$ is obtained by minimizing the energy
eigenvalues. Then, the solution for $\Psi$ is found at the interval
$\left[0,R\right]$, $R=N\times h$.

##  V Results and Discussions

The modified Numerov algorithm proposed in the present paper is based on the
potential function has a screening parameter, $\mu$. Such a potential is first
suggested to calculate the matrix elements of molecular properties such as
quadrupole coupling tensor, electron$-$photon hyperfine interaction, chemical
shift and spin$-$orbit coupling 63_Silverstone_1971 . Later on, an exponential
function was embedded into this potential function 64_Guseinov_1988 . It was
referred to as non$-$central interaction potential 65_Guseinov_2002 . It is
defined as,

$\displaystyle
f_{\mu\kappa\sigma}\left(\zeta,\vec{r}\right)=r^{\mu-1}e^{-\zeta
r}\left(\frac{4\pi}{2\kappa+1}\right)^{1/2}S_{\kappa\sigma}\left(\theta,\varphi\right),$
(20)

where $\mu>=-\left(\kappa-1\right)$ and $\kappa>=0$,
$S_{\kappa\sigma}\left(\theta,\varphi\right)$ are the real or complex
spherical harmonics. The case $\kappa=\sigma=0$ corresponds to the screened
central potential. It was also considered that $\mu$ can take non$-$integer
values at the range of $-1\leq\mu\leq 0$. In this case the potential referred
to as correlated interaction potential in which it was claimed that the
electron correlation effects are directly involved 66_Guseinov_2008 . This
type of potential was used to improve Hartree$-$Fock self$-$consistent field
calculations 67_Guseinov_2008 . It is on the other hand, obvious that the real
values of $\mu$ may cause variational instability.

Based on the perspective of numerical integration, the aforementioned method
in this study regards $\mu$ as a function enables the exact value for the
second order derivative of $\psi\left(x\right)$ to be acquired at a finite
number of grid$-$points. The function $\mu$ is chosen depending on the
conditions given in the Eq. (10, 11) and the Eq. (14). The values of $\mu$
functions for certain number of grid$-$point are determined through the Eqs.
(12, 13). The Eqs. (12, 13) depend on the angular momentum quantum number $l$
and their values significantly decrease by increasing the $l$.

The energy eigenvalues are usually calculated by requiring the
eigen$-$functions to satisfy certain boundary conditions 68_Chin_2019 . The
method presented in this study avoids dependence of energy eigenvalues to the
eigen$-$functions. It suggests first to calculate the eigenvalues. The
corresponding eigen$-$functions are then calculated easily. In this case all
the energy eigenvalues should be tried up to a certain precision. This
approximation was used previously in 68_Chin_2019 . An infinite potential
barrier at some radius $C$ added to a potential. The correct eigenvalue is
then obtained in the limit $C\rightarrow\infty$. It is referred to as hardwall
method. The number of significant digits provided by this method is limited to
a few. In this work, an explicit function that satisfy the conditions given in
the Eqs. (10, 11, 14) is alternatively added to the potential. The presented
algorithm provides arbitrary number of significant digit because the range of
values of $\mu$ indicates where the exact energy eigenvalue is (see Figure 2).
This yields the variational instability to be eliminated as well. The
parameter $\mu$ in the non$-$central interaction potential free from
restriction and the energy eigenvalues are unbounded from below unless the
Eqs. (10, 11, 14) are taken into account.

Figure 1: Results for the $4s$ electronic energy state depending on the upper
limit of summation $\left(N\right)$ in atomic units (a. u.). The dotted,
dashed, dot$-$dashed and black lines represents $E\left(h,\nu_{1}\right)$ with
values of $\nu_{1}$ obtained from $2^{th}$ $\left(i=2\right)$, $3^{th}$
$\left(i=3\right)$, $4^{th}$ $\left(i=4\right)$, $5^{th}$ $\left(i=5\right)$
iterations, respectively. $\Delta E=|E_{exact}|-|E\left(h,\nu_{1}\right)|$.
$N^{\prime}=N/100$. $E_{N}\left(h,\nu_{1}\right)\equiv
E_{4s}\left[h,\mu\left(N,l,\nu_{1}\right)\right]$. The results are multiplied
by $10^{6}$.

The time$-$independent one$-$dimensional Schrödinger equation given in the
previous section is solved as a sample. The method of solution has been
incorporated into a computer program written in the $Mathematica$ programming
language. Variationally optimum values for the grid$-$size $h$ are used in
calculations. The optimization procedure is implemented in order to test
stability of the suggested method.
Results for the $4s$ electronic energy state of hydrogen atom depending on the
upper limit of summation $\left(N\right)$ are given in the Figure 1. This
figure demonstrates how the correct energy eigenvalues to be found by using
the procedure presented in the Section III. It can be seen from this Figure
that, the first four iterations convergence from above while the fifth
convergences from below. This convergence property exposes at which iteration
the algorithm should decide to stop. The algorithm of the procedure with an
application for ground state energy calculation is given in the Figure 2.

Figure 2:  The algorithm of the procedure given in the Section III with an
example for ground state energy calculation of hydrogen atom. $N_{1},N_{2}$
are the grid$-$points. $h_{1},h_{2}$ are the grid$-$sizes for $N_{1}$,
$N_{2}$. $i$ represents the number of iteration. $\nu_{1}^{pi}$,
$\nu_{2}^{pi}$ represent $i^{th}$ of $\nu_{1}$ and $\nu_{2}$ for grid$-$points
$N_{1},N_{2}$ with $p$, $p=1$ or $p=2$, respectively.Start $N_{1}=7500$
---
$N_{2}=10000$
$h_{1}=0.001498$
$h_{2}=0.001160$
(Table 1)
$i=0$ (iteration number)
$\nu_{1}^{10}=1.457709$
---
$\nu_{1}^{20}=1.467792$
$0^{th}$ iteration, $i=0$
---
$E^{0}_{N_{1}}\left(h,\nu_{1}^{10}\right)=-0.49999\hskip 2.84526pt97807$
$E^{0}_{N_{2}}\left(h,\nu_{1}^{20}\right)=-0.49999\hskip 2.84526pt98747$
$1^{th}$ iteration, $i=1$
$E^{1}_{N_{1}}\left(h,\nu_{1}^{11}\right)=-0.49999\hskip 2.84526pt98586$
$E^{1}_{N_{2}}\left(h,\nu_{1}^{21}\right)=-0.49999\hskip 2.84526pt99214$
$2^{th}$ iteration, $i=2$
$E^{2}_{N_{1}}\left(h,\nu_{1}^{12}\right)=-0.49999\hskip 2.84526pt99465$
$E^{2}_{N_{2}}\left(h,\nu_{1}^{22}\right)=-0.49999\hskip 2.84526pt99741$
$3^{th}$ iteration, $i=3$
$E^{3}_{N_{1}}\left(h,\nu_{1}^{13}\right)=-0.50000\hskip 2.84526pt00458$
$E^{3}_{N_{2}}\left(h,\nu_{1}^{23}\right)=-0.50000\hskip 2.84526pt00336$
$E^{i}_{N_{1}}<E^{i}_{N_{2}}$
---
$i=0$
$E^{0}_{N_{2}}\left(h,\nu_{2}^{20}\right)$
---
$E^{i}_{N_{1}}>E^{i}_{N_{2}}$ $i=i+1$
---
$1^{th}$ iteration, $i=1$
$\nu_{1}^{11}=1.444173$
$\nu_{1}^{21}=1.454678$
$2^{th}$ iteration, $i=2$
$\nu_{1}^{12}=1.430636$
$\nu_{1}^{22}=1.441564$
$3^{th}$ iteration, $i=3$
$\nu_{1}^{13}=1.417100$
$\nu_{1}^{23}=1.428451$
Print
---
$E=-0.50000\hskip 2.84526pt00037$
Print
---
$E^{0}_{N_{2}}\left(h,\nu_{2}^{20}\right)$
yesnoyesnoStop

The energy eigenvalues of hydrogen atom, up to a principal quantum number $n$,
$n=10$, are investigated. The results are presented in the Tables 1, 2, 3.
Variationally optimum values for the grid$-$sizes depending on the upper limit
of summation $N$ are given in the Table 1. They are obtained by minimizing the
energy eigenvalues via the Powell’s optimization procedure 70_Powell_1964 . In
the Tables 2, 3, the results of computation for ground and excited states are
presented. The matrix form approach is used to solve the Eq. (4) for Coulomb
and screened Coulomb potentials, respectively. They are compared with the
results given in 5_Uria_1996  obtained via Richardson’s extrapolation and with
the results obtained via band matrix technique that recently have been
published in 69_Purevkhuu_2021 . The most accurate results that the standard
Numerov method with error term is of order $\mathcal{O}\left(h^{4}\right)$ can
provide, were presented in 69_Purevkhuu_2021 . The eigenvalue problem for
large matrices must be overcome for such work. The modified subroutines from
the LINPACK library: dgbsl.f, dgbfa.f were accordingly, performed in
69_Purevkhuu_2021 . Fairly large number of grid$-$points (i.e., $\approx
25000$) were used. It was reported that the Numerov’s method for $s$ state
with error term only is of order $\mathcal{O}\left(h^{2}\right)$ due to
singularity of Coulomb potential. Note that, for the $p$ states is
$\mathcal{O}\left(h^{3}\right)$ and for the $d$ states is
$\mathcal{O}\left(h^{4}\right)$.
The results given in the Tables 2, 3 are presented for $s$ states,
accordingly. Although less number of grid$-$points are used, more accurate
than the ones presented in both Table 2 and 69_Purevkhuu_2021  are given the
Table 3. In this table, instead of using large number of grid$-$points as in
69_Purevkhuu_2021 , the range of values of $\nu$ are reduced. The following
conclusions are achieved consequently:

  * •

The method suggested in this study accelerating the convergence of numerical
power series related with the Numerov method.

  * •

Difficulty of obtaining the eigenvalues for large matrices is eliminated.

  * •

The standart Numerov method is limited to error term is of order
$\mathcal{O}\left(h^{4}\right)$. Table 3 show that, the technique developed
here, improves the accuracy for any energy state. It permits to find the
energy eigenvalues with more then $\mathcal{O}\left(h^{4}\right)$ even for
potential with a singularity.

  * •

Extension of the Numerov method to error term is of arbitrary order is
problematic because $B^{-1}A$ in the Eq. (19) is not sparse and not symmetric
33_Kuenzer_2019 . Such necessity may also be eliminated. Instead, the range of
values of $\nu$ may be further constrained.

Solution of Hartree$-$Fock equations for atoms and molecules provides
essential input information in investigating the internal structures and
reaction dynamics of complex systems in fields like condensed matter physics,
quantum chemistry, and plasma physics. 71_Froese_1963 ; 72_Fischer_1978 ;
73_Jiao_2019 . The Hartree$-$Fock equations are solved by self$-$consistent
field method. It is an iterative solution obtained by rewriting each
Hartree$-$Fock equation in the form of the Schrödinger equation with the
non$-$local potential. In addition to field of studies referred in the Section
I, the convergence acceleration method for numerical power series suggested in
this work is also useful to improve the numerical solutions for Hartree$-$Fock
equations.
The computational aspect of the formulae given in the present work for
numerical calculation of the two$-$ and three$-$dimensional Schrödinger
equations will be the subject of the next work.

## Acknowledgement

In this study, the authors were supported by the Scientific Research
Coordination Unit of Pamukkale University under the project number 2020BSP011.
One of the author Z. G. is an undergraduate student works under the
supervision of A. B. She would like to express her gratitude to the Pamukkale
University, Department of Physics for the support during her minor program in
physics.

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# Federated Learning for Water Consumption Forecasting in Smart Cities

Mohammed El Hanjri3, Hibatallah Kabbaj3, Abdellatif Kobbane3, Amine Abouaomar1
3ENSIAS, Mohammed V University in Rabat, Morocco.  1School of Science and
Engineering, Al Akhawayn University in Ifrane, Morocco.

###### Abstract

Water consumption remains a major concern among the world’s future challenges.
For applications like load monitoring and demand response, deep learning
models are trained using enormous volumes of consumption data in smart cities.
On the one hand, the information used is private. For instance, the precise
information gathered by a smart meter that is a part of the system’s IoT
architecture at a consumer’s residence may give details about the appliances
and, consequently, the consumer’s behavior at home. On the other hand,
enormous data volumes with sufficient variation are needed for the deep
learning models to be trained properly. This paper introduces a novel model
for water consumption prediction in smart cities while preserving privacy
regarding monthly consumption. The proposed approach leverages federated
learning (FL) as a machine learning paradigm designed to train a machine
learning model in a distributed manner while avoiding sharing the users data
with a central training facility. In addition, this approach is promising to
reduce the overhead utilization through decreasing the frequency of data
transmission between the users and the central entity. Extensive simulation
illustrate that the proposed approach shows an enhancement in predicting water
consumption for different households.

###### Index Terms:

Federated learning, edge learning, water consumption prediction

##  I Introduction

In order to develop smart cities, load forecasting is a crucial step. Long-
term consumption forecasts of various vital resources are essential for
infrastructure management and planning [1, 2]. In addition, short and medium
term consumption forecasts are critical for system performance. The daily
operational efficiency of water distribution, in particular, requires accurate
short-term forecasting, which relies on the collection and analysis of large
volumes of data from the various domains[3]. However, the forecasting of
individual short-term loads is a challenging task to accomplish due to the
high volatility of user profiles. Indeed, the water demand of a household or
building is strongly influenced by the behavior of its residents, which is too
stochastic and therefore difficult to anticipate.

To predict load and consumption, different approaches have been proposed [4],
including linear and dynamic programming, heuristic methods, game theory, and
fuzzy methods [5]. The problem is that existing centralized and decentralized
approaches do not consider online solutions for large-scale real-world
databases, which is usually the case for residential consumption [6, 7]. Due
to large amounts of data from water meters, these approaches generate notable
slowdowns, which are not optimal for real-time applications. In the era of big
data, many machine learning methods are emerging as suitable for overcoming
this limitation by automatically exploiting, controlling, and optimizing load
prediction models [8]. This can be achieved by applying successive
transformations to historical data to train robust machine learning models to
accommodate the high uncertainty of water patterns, and control demand based
on load patterns and market cost signals.

By utilizing the caching capabilities coupled with the Fog characteristics,
the authors provide a green method to enhance performance of the energy
efficiency in a fog computing environment and Internet of Everything (IoE)
devices. To handle distributed networking, processing, and storage resources,
they make use of fog computing. As content requesters with limited resources,
IoT devices will make use of cutting-edge caching techniques to deliver high-
quality services that demand a lot of computational power and a high
throughput [9, 10, 11].

Recently, the machine learning community proposed Federated Learning (FL) as a
novel on-device method to address privacy issues while also growing the number
and variety of data sets [12]. A decentralized machine learning method called
federated learning allows any device to train a central model without
delivering any data [1, 7]. The model is initially initialized by the server
either randomly or using data that is readily accessible to the public [13].
The model is then given to a group of clients that have been chosen at random,
where it will be trained locally using their data. Each client updates the
model’s weights, which are then communicated to the server, averaged, and used
to update the overall model. This process will be continued until the global
model stabilizes [14].

This paper’s primary goal is to assess how well the Federated Learning
approach worked for the STLF challenge on home water use. For time series
forecasting, we use a deep neural network called Long Short Term Memory (LSTM)
[15], which makes predictions about future measurements of the water
consumption of the home based on past ones. We investigate a set of structures
with comparable characteristics (geographical location, type of building).
Federated learning is carried out using home grid Edge technology [16, 17].
Edge equipment, such as smart meters or more sophisticated devices, is
frequently installed at the end of the water distribution system to serve as a
smart interface between the customer and the water supply. We made the
following contributions to this effort, in brief: Using Edge technology in
smart cities, We first suggest an enabling architecture for FL, then we
simulate the potential accuracy gain of FL and last we use numerical findings
to simulate the potential network load gain. We also include in these
contributions the improvement in privacy made available by decentralization
and Edge computing.

The rest of this paper is organized as follows, with section II discussing
related works with a focus on load forecasting and privacy. The system model
is described in Section III, while Section IV presents the simulation and
numerical findings. Section V serves as the paper’s conclusion.

##  II Related Work

Deep neural networks, especially in particular Long-short term memory (LSTM),
have been employed extensively in recent research to address the difficulty of
short-term load forecasting. However, the results fall short of the required
level of precision in terms of Mean Average Percentage Error (MAPE) and Root
Mean Square Error (RMSE), despite benchmarks demonstrating LSTM’s capability
in comparison to other methods [18]. The authors of [19] suggest using a
sequence-to-sequence LSTM version of LSTM to increase forecasting accuracy.
This variant performs better for data with a one-minute resolution but does
not significantly outperform regular LSTM for data with an hour resolution.
Additionally, some writers [20] employ the evolutionary algorithm to solve the
challenge of selecting the optimal LSTM network because they view it as a
hyperparameter tuning problem.

According to some other studies, the issue is actually with data-driven
forecasting models capacity to generalize, not just with neural network
architecture. In fact, when applied to fresh datasets, several of the
presented models accuracy decreases [18].

Grouping data from several clients is another method for enhancing the
training data. By grouping people with comparable profiles, the authors of
[21] are able to lower the variation of uncertainty within groups. To solve
the overfitting issue, authors in [22] suggest a pooling strategy that
broadens the diversity of the data. However, these approaches are very
centralized and prone to privacy problems.

In the proposed study, we recommend using the federated learning strategy to
carry out client selection and neural network training at the edge. This will
allow the use of data to build a reference model without sacrificing resident
privacy. In order to do this, the Home Area Network (HAN) Edge Equipment is
utilized.

##  III System Model

We suggest the network architecture shown in Figure 1, which has a Multi-
access Edge Computing (MEC) server and users as its two primary parts. The
users are buildings having Edge equipment made up primarily of smart meters
and other HAN devices. For STLF, FL is utilized to create a global model based
on LSTM. The MEC server orchestrates the training rounds, which are carried
out by the users using information about their local water consumption. We go
into greater detail regarding LSTM and its use in forecasting, as well as FL
and its use in our system model, in this part.

Figure 1: Network architecture and components

###  III-A Time series forecasting using LSTM

In this work, the time series forecasting technique with LSTM is used to
predict the future water load. A time series is an organized collection of
equally spaced data sets that shows how a certain variable has changed over
time. By modeling the connections between the points of the current data and
the points of the past, time series forecasting is made possible, but the
precision of the forecasts is greatly influenced by the model that is used and
the caliber of the historical data points.
Recurrent neural networks (RNNs) like LSTM are more effective than typical
RNNs and fundamentally different from conventional feedforward RNNs. The
strength of LSTM is sequence learning. It can resolve vanishing and exploding
gradient difficulties, which are frequent in RNNs, and establish temporal
correlations between earlier data points and the current situation. Gradient
exploding relates to the opposite process, whereas gradient vanishing
describes how the norm of the gradient for long-term components gets less,
leading weights to never change at lower levels. This is made possible by two
of its essential elements: the memory cell, which is utilized to recall
significant prior events, and the gates, which control the information flow.
The input gate, output gate, and forget gate are the three gates that make up
an LSTM. During the learning process, they acquire the skill of clearing the
memory of irrelevant data. Utilizing LSTM and its derivatives, nearly all
state-of-the-art outcomes in sequence learning are accomplished. For
residential STLF, it is anticipated that the LSTM network will be able to
track the states of the inhabitants, abstract some states from the provided
consumption profile, and forecast future consumption using the learned data.

###  III-B Federated Learning

A type of machine learning called federated learning involves distributing the
majority of the training process across various clients, or computer devices.
On mobile device keyboards for next word prediction, Google first put out and
executed the idea [23]. For many situations, this strategy is ideal: 1) In
highly distributed systems, when there are orders of magnitude more devices
than data center nodes, 2) When data privacy is important, and 3) When data is
huge in comparison to model updates. In cases when datasets are uneven or not
dispersed evenly, federated learning has been shown to be quite effective.
Following is an example of federated learning iteration: First, the current
model is given to each client in a selected subset. In this instance, clients
are housed using Edge equipment (e.g. smart meters). On locally stored data, a
few clients choose to compute Stochastic Gradient Descent (SGD) updates. A
server then integrates the client updates to produce a new global model. A
different group of clients is sent the updated model. Until the necessary
forecast accuracy is attained, this process is repeated. Algorithm 1 describes
the procedures in detail.
The server employs the Federated Averaging algorithm to integrate the client
updates [12]. The starting global model is first randomly initialized or
trained beforehand using publicly accessible data. A subset $S$ of clients
with sufficient data records and diverse enough consumption loads to augment
the training data are sent a global model $\omega_{t}$ by the server in each
training round $t$. This requirement was introduced to make sure we have
enough variety among the data points to accurately depict the occupants’
typical consumption. Each client $s$ in the subset then employs $m_{s}$
samples from its own local data. In our case, the volume is based on the
amount of data that is locally stored and the length of time that the smart
meter has been providing data. The dataset used consists of sliding windows
with a predetermined number of look-back steps. The average gradient $g_{s}$
is then calculated by each client $s$ using SGD, with a learning rate $\eta$.
The server receives the updated models $\omega_{s}$ and aggregates them.

Initialize the model in training round $t=0$;

while  _$t\leq t_{max}$_ do

Select subset $S$ of clients;

for  _client $s$ in $S$_ do

if  _$\Delta(monthlyload) >threshold$_ then

$s$ obtains model $\omega_{t}$;

$s$ calculates average gradient $g_{s}$ with SGD;

$s$ updates local model $\omega_{t+1}^{s}\leftarrow\omega_{t}^{s}-\eta g_{s}$;

$s$ communicates updated model to server;

end if

end for

server updates global model using the formula in eq (1); begin the following
round $t\leftarrow t+1$.

end while

Algorithm 1 Federated Averaging Algorithm. $t_{max}$ is the maximum number of
rounds and $M=\sum_{s}m_{s}$.

$\omega_{t+1}=\sum_{s=0}^{S}\frac{m_{s}}{M}\omega_{t+1}^{s}$ (1)

Figure 2: The distribution of water consumption data in building 04 (BLD 04).
(HCF; Hundred-Cubic Foot)

The centralized strategy might not, however, account for every user’s water
consumption. Personalization is one suggestion for resolving this issue. Many
applications, which call for analyzing and responding to user behavior, are
focused on personalization. To create a customized model for each user, the
centralized model is retrained using user-specific data. This can be
accomplished by retraining the model locally over a limited number of epochs
using only the customer’s data.
Since user identities are still at risk and can be ascertained through reverse
engineering, federated learning poses less of a privacy threat than
centralized server storage. Weight changes are processed in memory and are
deleted after aggregation, whereas model updates received by each client are
transient and never retained on the server. Individual weight uploads must not
be reviewed or assessed according to the federated learning approach. Because
the network and server cannot be trusted with precise user data, this is
nevertheless more secure than server training. For billing purposes, some data
must still be supplied in an aggregated form, but these data are not
particularly revealing.

###  III-C Reward in the network load

First, we establish the network load $Q_{C}$ for a server in centralized
training in Eq. 1 and the network load in FL $Q_{F}$ in Eq. 2, in order to
compare the reward in network load in FL to centralized training.
$\ell_{G}$ is the length of the global model, and $\ell_{s}$ is the length of
the data that the client $s$ supplied. The number of hops between client $s$
and the server in the centralized training is denoted by $h_{s}$.

$Q_{C}=\sum_{s=1}^{M}\ell_{s}\times h_{s}$ (2)

and

$Q_{F}=\ell_{G}\times\sum_{t=1}^{t_{max}}\sum_{s=1}^{S}h_{s,t}$ (3)

where $S$ is the total number of users in each batch, and $h_{s,t}$ is the
number of hops between the client $s$ chosen in round $t$ and the server.
We define the reward in networking load using Eq.1 and Eq.2, as follows:

$R=1-\frac{Q_{F}}{Q_{C}}$ (4)

##  IV Simulation and Results

###  IV-A Dataset Pre-Processing

This study made use of data from the Kaggle website. Data about New York’s
water consumption and costs include Consumption and expense data broken down
by borough and development monthly. Utility vendor and meter information is
included in the data set. For over 300 buildings in the New York city, New
York Water data provides water consumption and cost statistics at ”monthly”
intervals between 2013 and 2020.

###  IV-B Numerical Results

For deep learning models to perform the best in forecasting, hyper-parameter
adjustment is crucial. However, we only evaluate the federated learning
approach in this paper.
This section examines how personalization affects the efficiency of the
models. First, we examine if localizing the model’s training for the
participating clients improves outcomes.

We chose a client (building; BLD 04) randomly from the sample of participants
to show how personalization can improve predictions. The distribution of water
consumption data in building 04 (BLD 04) between 2013 and 2020 is given as
shown in Figure 2.
We have used the matching personalized models as well as the global model.
Figure 3 illustrates the actual and anticipated consumption profiles between
2018 and 2020.

Figure 3: predictions for the client’s (BLD 04) consumption for the following
month for those who took part in training the global model.
(HCF; Hundred-Cubic Foot)

Using just a portion of the users that make up the population, we come to the
conclusion that it is possible to train robust models for the consumption
profiles of that population. Retraining the model can produce a customized
model that better follows the profile’s characteristics and produces
predictions with higher accuracy for applications with high accuracy needs.
Even said, for brand-new customers lacking sufficient data for customisation,
the predictions made using the global model can be a useful starting point.
The edge devices ability to provide local training is a necessity for the
suggested approach’s feasibility. The performance of new IoT devices will very
certainly be compromised by training a neural network, even though they have
the CPU power to execute complex machine learning models.

##  V Conclusion

Given the stochastic nature of consumption profiles, projecting individual
short-term loads is a difficult challenge. For the purpose of short-term load
forecasting in smart cities, we proposed a system model in this research that
uses federated learning to address issues with privacy and data diversity.
Contrary to centralized techniques, federated learning in the proposed system
trains models on edge devices, limiting security concerns to those specific to
the device. We ran tests to assess how well centralized and customized models
performed in federated contexts. The simulation results show that it is a
promising approach for developing high-performing models with a significantly
reduced networking load than a centralised model, while preserving the privacy
of consumption data.

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# Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean
and Hyperbolic spaces

Vicent Gimeno# Departament de Matemàtiques-INIT, Universitat Jaume I,
Castelló, Spain<EMAIL_ADDRESS>and  Vicente Palmer  Departament de
Matemàtiques-INIT, Universitat Jaume I, Castelló, Spain<EMAIL_ADDRESS>
###### Abstract.

We study the topology of (properly) immersed complete minimal surfaces $P^{2}$
in Hyperbolic and Euclidean spaces which have finite total extrinsic
curvature, using some isoperimetric inequalities satisfied by the extrinsic
balls in these surfaces, (see [12]). We present an alternative and partially
unified proof of the Chern-Osserman inequality satisfied by these minimal
surfaces, (in $\mathbb{R}^{n}$ and in $\mathbb{H}^{n}(b)$), based in the
isoperimetric analysis above alluded. Finally, we show a Chern-Osserman type
equality attained by complete minimal surfaces in the Hyperbolic space with
finite total extrinsic curvature.

###### Key words and phrases:

Area growth, minimal surfaces, Chern-Osserman inequality, finite topological
type, compactification, Euler characteristic.

###### 2000 Mathematics Subject Classification:

Primary 53C20 ; Secondary 53C42, 49Q05

# Supported by the Fundació Caixa Castelló-Bancaixa Grants P1.1B2006-34 and
P1.1B2009-14
* Supported by MICINN grant No. MTM2010-21206-C02-02.

##  1\. Introduction

Let us consider $P^{2}$ be a complete and minimal surface immersed in
$\mathbb{R}^{n}$ and with finite total curvature
$\int_{P}K^{P}d\sigma<\infty$, being $K^{P}$ the Gauss curvature of the
surface. Then we have the following equality (resp. inequality), known as the
Chern-Osserman formula, (see [1], [3] and [8]):

(1.1)
$-\chi(P)=\frac{1}{4\pi}\int_{P}\|B^{P}\|^{2}d\sigma-\operatorname{Sup}_{r}\frac{\operatorname{Vol}(P^{2}\cap
B^{0,n}_{r})}{\operatorname{Vol}(B^{0,2}_{r})}\leq\frac{1}{4\pi}\int_{P}\|B^{P}\|^{2}d\sigma-k(P)$

where $\chi(P)$ is the Euler characterisitic of $P$, $k$ is its number of
ends, $B^{P}$ is the second fundamental foorm of $P$ in $\mathbb{R}^{n}$ and
$B^{b,n}_{r}$ denotes the geodesic $r$-ball in the simply connected real space
form $\mathbb{K}^{n}(b)$.

To have finite total scalar (extrinsic) curvature
$\int_{P}\|B^{P}\|^{2}d\sigma<\infty$ is equivalent to the finiteness of the
total Gaussian curvature (the original assumption in [3]) when the surface is
minimal and immersed in $\mathbb{R}^{n}$. From this point of view, it is
natural to wonder if it is possible to stablish a Chern-Osserman inequality
(or equality) for complete minimal surfaces with finite total extrinsic
curvature (properly) immersed in the hyperbolic space. This question has been
addressed by Q. Chen and Y. Cheng in the papers [4] and [5]. They proved, for
a complete minimal surface $P^{2}$ (properly) immersed in $\mathbb{H}^{n}(b)$
and such that $\int_{P}\|B^{P}\|d\sigma<\infty$, that
$\operatorname{Sup}_{r}\frac{\operatorname{Vol}(P^{2}\cap
B^{-1,n}_{r})}{\operatorname{Vol}(B^{-1,2}_{r})}<\infty$ and the following
version of the Chern-Osserman Inequality, in terms of the volume growth of the
extrinsic balls:

(1.2)
$-\chi(P)\leq\frac{1}{4\pi}\int_{P}\|B^{P}\|^{2}d\sigma-\operatorname{Sup}_{r}\frac{\operatorname{Vol}(P^{2}\cap
B^{-1,n}_{r})}{\operatorname{Vol}(B^{-1,2}_{r})}$

The proofs given by these authors are different for those for the Euclidean
case, and rely heavily on the properties of the hyperbolic functions.

We present in this paper a partial unification of the proof of the Chern-
Osserman inequality (in terms of the volume growth) for complete minimal
surfaces with finite total extrinsic curvature immersed in Euclidean or
Hyperbolic spaces. This partial unification is based in obtaining estimates
for the Euler characteristic of the extrinsic balls (given in Lemma 3.1, and
Proposition 3.2) and in the isoperimetric inequality for the extrinsic balls
given in Theorem 1.1 in [12]. These results are based, in its turn, on the
divergence Theorem and the Hessian and Laplacian comparison theory of
restricted distance function, (see [6], [7] and [13]) which involves bounds on
the mean curvature of the submanifold.

We have proved the following Chern-Osserman inequality, which encompasses
inequalities (1.1) and (1.2):

######  Theorem A.

Let $P^{2}$ be an complete minimal surface immersed in a simply connected real
space form with constant sectional curvature $b\leq 0$, $\mathbb{K}^{n}(b)$.
Let us suppose that $\int_{P}\|B^{P}\|^{2}d\sigma<\infty$. Then

  1. (1)

$P$ has finite topological type.

  2. (2)

$\operatorname{Sup}_{t>0}(\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B^{b,2}_{t})})<\infty$

  3. (3)

$-\chi(P)\leq\frac{\int_{P}\|B^{P}\|^{2}}{4\pi}-\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}$

where $\chi(P)$ is the Euler characteristic of $P$.

Although with this approach we are not able to state equality (1.1) in the
Euclidean setting, we shall prove in Theorem B the following Chern-Osserman
type equality for cmi surfaces in the Hyperbolic space:

######  Theorem B.

Let $P^{2}$ be a complete immersed minimal surface in $\mathbb{H}^{n}(b)$. Let
us suppose that $\int_{P}\|B^{P}\|^{2}d\sigma<\infty$. Then

(1.3)
$-\chi(P)=\frac{1}{4\pi}\int_{P}\|B^{P}\|^{2}d\sigma-\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}-\frac{1}{2\pi}G_{b}(P)$

where $G_{b}(P)$ is a nonnegative and finite quantity which do not depends on
the exhaustion by extrinsic balls $\\{D_{t}\\}_{t>0}$ of $P$ and is given by

(1.4) $\displaystyle G_{b}(P)$
$\displaystyle:=\lim_{t\to\infty}\left(h_{b}(t)\operatorname{Vol}(B^{b,2}_{t})(\frac{(\operatorname{Vol}(D_{t}))}{\operatorname{Vol}(B^{b,2}_{t})})^{\prime}\right.$
$\displaystyle\left.+\int_{\partial D_{t}}\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\|\operatorname{\nabla}^{P}r\|}\rangle
d\sigma_{t}\right)$

###  1.1. Outline

The outline of the paper is following. In Section §.2 we present the basic
facts about the Hessian comparison theory of restricted distance function we
are going to use, obtaining as a corollary the compactification of cmi
surfaces in $\mathbb{K}^{n}(b)$ with finite total extrinsic curvature,
(Corollary 2.3). Section §.3 is devoted to the unified proof of the Chern-
Osserman inequality for complete minimal surfaces with finite total extrinsic
curvature immersed in Euclidean and Hyperbolic spaces (Theorem A), and in
Section §.4 it is proved a Chern-Osserman type equality satisfied by the cmi
surfaces in $\mathbb{H}^{n}(b)$ (Theorem B).

##  2\. Preliminaires

###  2.1. The extrinsic distance

We assume throughout the paper that $P^{2}$ is a complete, non-compact,
immersed, $2$-dimensional submanifold in a simply connected real space form of
non-positive constant sectional curvature $\mathbb{K}^{n}(b)$,
($\mathbb{K}^{n}(b)=\mathbb{R}^{n}$ when $b=0$ and
$\mathbb{K}^{n}(b)=\mathbb{H}^{n}(b)$ when $b<0$) . All the points in these
manifolds are poles. Recall that a pole is a point $o$ such that the
exponential map

$\exp_{o}\colon T_{o}N^{n}\to N^{n}$

is a diffeomorphism. For every $x\in N^{n}\setminus\\{o\\}$ we define
$r_{o}(x)=\operatorname{dist}_{N}(o,x)$, and this distance is realized by the
length of a unique geodesic from $o$ to $x$, which is the radial geodesic from
$o$. We also denote by $r$ the restriction
$r|_{P}:P\to\mathbb{R}_{+}\cup\\{0\\}$. This restriction is called the
extrinsic distance function from $o$ in $P^{m}$. The gradients of $r$ in $N$
and $P$ are denoted by $\operatorname{\nabla}^{N}r$ and
$\operatorname{\nabla}^{P}r$, respectively. Let us remark that
$\operatorname{\nabla}^{P}r(x)$ is just the tangential component in $P$ of
$\operatorname{\nabla}^{N}r(x)$, for all $x\in S$. Then we have the following
basic relation:

(2.1)
$\nabla^{N}r=\operatorname{\nabla}^{P}r+(\operatorname{\nabla}^{N}r)^{\bot},$

where $(\operatorname{\nabla}^{N}r)^{\bot}(x)=\nabla^{\bot}r(x)$ is
perpendicular to $T_{x}P$ for all $x\in P$.

On the other hand, we should recall that all immersed surfaces $P$ in the real
space forms of non-positive constant sectional curvature
$N^{n}=\mathbb{K}^{n}(b)$ which satisfies
$\int_{P}\|B^{P}\|^{2}d\sigma<\infty$ are properly immersed (see [1], [10] and
[11]). Therefore, we can omit the hypothesis about the properness of the
immersion when we assume that $\int_{P}\|B^{P}\|^{2}d\sigma<\infty$.

######  Definition 2.1.

Given a connected and complete surface $P^{2}$ properly immersed in a manifold
$N^{n}$ with a pole $o\in N$, we denote the extrinsic metric balls of radius
$t>0$ and center $o\in N$ by $D_{t}(o)$. They are defined as the intersection

$D_{t}(o)=B^{N}_{t}(o)\cap P=\\{x\in P\colon r(x)<t\\},$

where $B^{N}_{t}(o)$ denotes the open geodesic ball of radius $R$ centered at
the pole $o$ in $N^{n}$.

######  Remark a.

We want to point out that the extrinsic domains $D_{t}(o)$ are precompact
sets, (because we assume in the definition above that the submanifold $P$ is
properly immersed), with boundary $\partial D_{t}(o)$ being a immersed curve
in $P$. The generical smoothness of $\partial D_{t}(o)$ follows from the
following considerations: the distance function $r$ is smooth in
$\mathbb{K}^{n}(b)\setminus\\{o\\}$ since $\mathbb{K}^{n}(b)$ to possess a
pole $o\in\mathbb{K}^{n}(b)$, ($b\leq 0$). Hence the restriction $r|_{P}$ is
smooth in $P$ and consequently the radii $t$ that produce smooth boundaries
$\partial D_{t}(o)$ are dense in $\mathbb{R}$ by Sard’s theorem and the
Regular Level Set Theorem.

######  Remark b.

When the submanifold considered is totally geodesic, namely, when $P$ is a
Hyperbolic or an Euclidean subespace of the ambient real space form, the
extrinsic balls become geodesic balls, and its boundary is the distance
sphere. We recall here that the mean curvature of the geodesic sphere in the
real space form $\mathbb{K}^{n}(b)$, ’pointed inward’ is (see [12]):

$h_{b}(t)=\left\\{\begin{array}[]{l}\sqrt{b}\cot\sqrt{b}t\,\,\text{ if
}\,\,b>0\\\ 1/t\,\,\text{ if }\,\,b=0\\\ \sqrt{-b}\coth\sqrt{-b}t\,\,\text{ if
}\,\,b<0\end{array}\right.$

###  2.2. Hessian comparison analysis of the extrinsic distance

Let us consider now $D_{t}$ an extrinsic ball in a complete and properly
immersed minimal surface $P$ in the real space form $\mathbb{K}^{n}(b)$ with
$b\leq 0$. We are going to apply Gauss-Bonnet formula to the curve $\partial
D_{t}$. To do that, we need to compute its geodesic curvature in the following

######  Proposition 2.2.

Given $\partial D_{t}$ the smooth closed curves in $P$,

(2.2) $k_{g}^{\partial
D_{t}}=\frac{h_{b}(t)}{\|\operatorname{\nabla}^{P}r\|}+\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\|\operatorname{\nabla}^{P}r\|}\rangle$

###### Proof.

Let $\\{e,\nu\\}\subset TP$ be an orthonormal frame along the curve $\partial
D_{t}$, where $e$ is the unit tangent vector to $\partial D_{t}$ and
$\nu=\frac{\operatorname{\nabla}^{P}r}{\|\operatorname{\nabla}^{P}\|}$ is the
unit normal to $\partial D_{t}$ in $P$, pointed outward.

From the definition of geodesic curvature of the extrinsic boundaries
$\partial D_{t}$, we have

(2.3)
$k_{g}^{t}=-\langle\nabla_{e}^{P}e,\frac{\nabla^{P}r}{\|\nabla^{P}r\|}\rangle$

Then, having on account the definition of Hessian

$Hess^{P}r(e,e)=\langle\nabla^{P}\nabla^{P}r,e\rangle$

and the fact that $\nabla^{P}r$ and $e$ are orthogonal,

(2.4) $k_{g}^{t}=\frac{1}{\|\nabla^{P}r\|}Hess^{P}r(e,e)$

But, given $X\in T_{q}P$ unitary, (see [7] and [13] for detailed
computations):

(2.5)
$\operatorname{Hess}^{P}(r)(X,X)\,=\,h_{b}(r)\left(\,1-\langle\,X,\nabla^{\mathbb{K}^{n}(b)}r\,\rangle^{2}\,\right)+\langle\,\nabla^{\mathbb{K}^{n}(b)}r,\,B^{P}(X,X)\,\rangle\,$

where $B^{P}$ is the second fundamental form of $P$ in $N$.

Applying at this point equation (2.5):

(2.6)
$k_{g}^{t}=\frac{1}{\|\nabla^{P}r\|}\\{h_{b}(r)+\langle\nabla^{\perp}r,B^{P}(e,e)\rangle\\}$

∎

Now, we consider $\\{D_{t}\\}_{t>0}$ an exhaustion of $P$ by extrinsic balls.
Recall than an exhaustion of the submanifold $P$ is a sequence of subsets
$\\{D_{t}\subseteq P\\}_{t>0}$ such that:

  * •

$D_{t}\subseteq D_{s}$ when $s\geq t$

  * •

$\cup_{t>0}D_{t}=P$

Using the equality (2.2) for the geodesic curvature of the extrinsic curves we
have the following result

######  Theorem 2.3.

Let $P^{2}$ be an complete minimal surface immersed in a simply connected real
space form with constant sectional curvature $b\leq 0$, $\mathbb{K}^{n}(b)$.
Let us suppose that $\int_{P}\|B^{P}\|^{2}d\sigma<\infty$. Then

(i) $P$ is diffeomorphic to a compact surface $P^{*}$ punctured at a finite
number of points.

(ii) For all sufficiently large $t>R_{0}>0$, $\chi(P)=\chi(D_{t})$ and hence,
given $\\{D_{t}\\}_{t>0}$ an exhaustion of $P$ by extrinsic balls,

$\chi(P)=\lim_{t\to\infty}\chi(D_{t})$

###### Proof.

Let us consider $\\{D_{t}\\}_{t>0}$ an exhaustion of $P$ by extrinsic balls,
centered at the pole $o\in\mathbb{K}^{n}(b)$. We apply Lemma 2.2 to the smooth
curves $\partial D_{t}$: As

$-\|B^{P}\|\leq\langle
B^{P}(e,e),\operatorname{\nabla}^{\bot}r\rangle\leq\|B^{P}\|$

we have, on the points of the curve $q\in\partial D_{t}$,

(2.7) $\displaystyle\|\operatorname{\nabla}^{P}r\|(q)\cdot k_{g}^{\partial
D_{t}}(q)$ $\displaystyle=h_{b}(r_{o}(q))+\langle
B^{P}(e,e),\operatorname{\nabla}^{\bot}r\rangle(q)\,$ $\displaystyle\geq
h_{b}(r_{o}(q))-\|B^{P}\|(q)$

Using now Proposition 2.2 in [1], when $P^{2}$ is a cmi in $\mathbb{R}^{n}$ or
Lemma 3.1 in [11], when $P^{2}$ is a cmi in $\mathbb{H}^{n}(b)$, we know that
$\|B^{P}\|(q)$ goes uniformly to $0$ as $t=r_{o}(q)\to\infty$. Hence, for all
the points $q\in\partial D_{t}$ and for sufficiently large $t$,

(2.8) $\|\operatorname{\nabla}^{P}r\|(q)\cdot k_{g}^{\partial D_{t}}(q)\ >0$

Hence, $\|\operatorname{\nabla}^{P}r\|>0$ in $\partial D_{t}$, for all
sufficiently large $t$. Fixing a sufficienty large radius $R_{0}$, we can
conclude that the extrinsic distance $r_{o}$ has no critical points in
$P\setminus\operatorname{D}_{R_{0}}$.

The above inequality implies that for this sufficienty large fixed radius
$R_{0}$, there is a diffeomorphism

$\Phi:P\setminus\operatorname{D}_{R_{0}}\to\partial D_{R_{0}}\times[0,\infty[$

In particular, $P$ has only finitely many ends, each of finite topological
type.

To proof this we apply Theorem 3.1 in [9], concluding that, as the extrinsic
annuli $A_{R_{0},R}(o)=D_{R}(o)\setminus D_{R_{0}}(o)$ contains no critical
points of the extrinsic distance function
$r_{o}:P\longrightarrow\mathbb{R}^{+}$ because inequality (2.8), then
$D_{R}(o)$ is diffeomorphic to $D_{R_{0}}(o)$ for all $R\geq R_{0}$.

The above diffeomorfism implies that we can construct $P$ from $D_{R_{0}}$
($R_{0}$ big enough) attaching annulis and that $\chi(P\setminus D_{t})=0$
when $t\geq R_{0}$. Then, for all $t>R_{0}$,

$\chi(P)=\chi(D_{t}\cup(P\setminus D_{t}))=\chi(D_{t})$

∎

##  3\. Proof of Theorem A

We begin with the following results which are the common ingredient of the
proof, both for the Euclidean and Hyperbolic cases :

######  Lemma 3.1.

Let $P^{2}\subset\mathbb{K}^{n}(b)$ be a surface properly immersed in a real
space form with curvature $b\leq 0$, let $D_{t}$ be an extrinsic disc in $P$
of radius $t>0$ and let $\partial D_{t}$ the extrinsic circle. Then:

(3.1) $\int_{\partial
D_{t}}\frac{||\nabla^{\bot}r||^{2}}{||\nabla^{P}r||}d\sigma_{t}\leq\int_{\partial
D_{t}}\frac{1}{||\nabla^{P}r||}-h_{b}(t)\operatorname{Vol}(D_{t})d\sigma_{t}$

###### Proof.

Tracing equality (2.5) we obtain the following expression for the Laplacian of
the extrinsic distance in this context:

(3.2)
$\Delta^{P}(r)\,=\,(m-\|\nabla^{P}r\|^{2})h_{b}(r)+m\langle\,\nabla^{N}r,\,H_{P}\,\rangle\quad,$

where $H_{P}$ denotes the mean curvature vector of $P$ in $N$ and $h_{b}(r)$
is the mean curvature of the geodesic $r$-spheres in $\mathbb{K}^{n}(b)$.
Applying divergence theorem we have

(3.3) $\displaystyle\int_{\partial
D_{t}}\frac{||\nabla^{\bot}r||^{2}}{||\nabla^{P}r||}d\sigma_{t}=\int_{\partial
D_{t}}\frac{1}{||\nabla^{P}r||}d\sigma_{t}-\int_{\partial
D_{t}}||\nabla^{P}r||d\sigma_{t}=\int_{\partial
D_{t}}\frac{1}{||\nabla^{P}r||}d\sigma_{t}$
$\displaystyle-\int_{D_{t}}\Delta^{P}rd\sigma=\int_{\partial
D_{t}}\frac{1}{||\nabla^{P}r||}d\sigma_{t}-\int_{D_{t}}(2-||\nabla^{P}r||^{2})h_{b}(r)d\sigma$
$\displaystyle\leq\int_{\partial
D_{t}}\frac{1}{||\nabla^{P}r||}d\sigma_{t}-\int_{D_{t}}h_{b}(r)d\sigma\leq\int_{\partial
D_{t}}\frac{1}{||\nabla^{P}r||}d\sigma_{t}-h_{b}(t)\operatorname{Vol}(D_{t})$

∎

######  Proposition 3.2.

Let $P^{2}\subset\mathbb{K}^{n}(b)$ be a complete minimal surface properly
immersed in a real space form with curvature $b\leq 0$, let $D_{t}$ be an
extrinsic disc in $P$ of radius $t>0$ and let $\partial D_{t}$ be its
boundary. Then:

(3.4) $\displaystyle-2\pi\chi$
$\displaystyle(D_{t})+(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})\operatorname{Vol}(D_{t})$
$\displaystyle+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\int_{\partial
D_{t}}\frac{1}{\|\nabla^{P}r\|}d\sigma_{t}\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$

where $R(t)=\int_{D_{t}}\|B^{P}\|^{2}d\sigma$, $\|B^{P}\|$ is the norm of the
second fundamental form of $P$ in $\mathbb{K}^{n}(b)$, $\chi(D_{t})$ is the
Euler’s characterisc of $D_{t}$ and, given $\alpha\in]0,2[$ ,

$f_{b,\alpha}^{2}(t)=\alpha h_{b}(t)$

###### Proof.

Integrating along $\partial D_{t}$ equation (2.2) and using Gauss-Bonnet
theorem and co-area formula, (see [14]), we obtain

(3.5) $\displaystyle 2$
$\displaystyle\pi\chi(D_{t})-\int_{D_{t}}K^{P}d\sigma=$ $\displaystyle
h_{b}(t)\int_{\partial
D_{t}}\frac{1}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}+\int_{\partial
D_{t}}\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\|\operatorname{\nabla}^{P}r\|}\rangle
d\sigma_{t}$

where we denote as $K^{P}$ the Gauss curvature of $P$.

But , on $\partial D_{t}$,

$-\|B^{P}\|\frac{\|\operatorname{\nabla}^{\bot}r\|}{\|\operatorname{\nabla}^{P}r\|}\leq\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\|\operatorname{\nabla}^{P}r\|}\rangle\leq\|B^{P}\|\frac{\|\operatorname{\nabla}^{\bot}r\|}{\|\operatorname{\nabla}^{P}r\|}$

so, as $f_{b,\alpha}(t)\geq 0\,\forall t>0$, having into account the
inequality among the arithmetic and geometric mean and applying co-area
formula:

(3.6) $\displaystyle 2$
$\displaystyle\pi\chi(D_{t})-\int_{D_{t}}K^{P}d\sigma=h_{b}(t)\int_{\partial
D_{t}}\frac{1}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}$
$\displaystyle+\int_{\partial D_{t}}\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\|\operatorname{\nabla}^{P}r\|}\rangle
d\sigma_{t}\,\geq\,h_{b}(t)\int_{\partial
D_{t}}\frac{1}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}$
$\displaystyle-\frac{1}{2}\int_{\partial
D_{t}}\frac{\|B^{P}\|^{2}}{f_{b,\alpha}^{2}(r)\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}-\frac{1}{2}\int_{\partial
D_{t}}\frac{f_{b,\alpha}^{2}(r)\|\operatorname{\nabla}^{\bot}r\|^{2}}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}$
$\displaystyle\geq h_{b}(t)\int_{\partial
D_{t}}\frac{1}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}-\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)-\frac{f_{b,\alpha}^{2}(t)}{2}\int_{\partial
D_{t}}\frac{\|\operatorname{\nabla}^{\bot}r\|^{2}}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}$

Then, using inequality (3.1) of Lemma 3.1 in the last member of the
inequalities (3.6) and applying Gauss equation for minimal surfaces in the
real space forms $\mathbb{K}^{n}(b)$, we have

(3.7) $\displaystyle 2\pi\chi$
$\displaystyle(D_{t})-b\operatorname{Vol}(D_{t})+\frac{1}{2}R(t)\geq(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\int_{\partial
D_{t}}\frac{1}{\|\operatorname{\nabla}^{P}r\|}d\sigma_{t}$
$\displaystyle-\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2}\operatorname{Vol}(D_{t})$

and hence

(3.8) $\displaystyle-2\pi\chi$
$\displaystyle(D_{t})+(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})\operatorname{Vol}(D_{t})$
$\displaystyle+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\int_{\partial
D_{t}}\frac{1}{\|\nabla^{P}r\|}\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$

∎

We are going to divide the proof in two cases: the Case I, where the ambient
space is the Hyperbolic space $\mathbb{H}^{n}(b)$, and the Case II where the
ambient space is the Euclidean space $\mathbb{R}^{n}$.

### Case I

Let us consider $P$ (properly) immersed in $\mathbb{H}^{n}(b)$. Let
$\\{D_{t}\\}_{t>0}$ be an exhaustion of $P$ by extrinsic balls. Using co-area
formula, we know that

(3.9) $\frac{d}{dt}\operatorname{Vol}(D_{t})=\int_{\partial
D_{t}}\frac{1}{\|\nabla^{P}r\|}d\sigma_{t}$

Hence, applying Proposition 3.2 we have

(3.10) $\displaystyle-2$
$\displaystyle\pi\chi(D_{t})+(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})\operatorname{Vol}(D_{t})$
$\displaystyle+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\frac{d}{dt}\operatorname{Vol}(D_{t})\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$

On the other hand, from 3.9,
$\frac{d}{dt}\operatorname{Vol}(D_{t})\geq\operatorname{Vol}(\partial D_{t})$.
Therefore, using inequality (3.10) we obtain

(3.11) $\displaystyle-2\pi\chi(D_{t})$
$\displaystyle+\operatorname{Vol}(D_{t})\left[(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\frac{\operatorname{Vol}(\partial
D_{t})}{\operatorname{Vol}(D_{t})}\right]$
$\displaystyle\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$

Applying isoperimetric inequality in [12], (Theorem 1.1), we have

(3.12) $\displaystyle-2\pi\chi(D_{t})$
$\displaystyle+\operatorname{Vol}(D_{t})\left[(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\frac{\operatorname{Vol}(S_{t}^{b,1})}{\operatorname{Vol}(B_{t}^{b,2})}\right]$
$\displaystyle\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$

Hence, using the fact that

$b\operatorname{Vol}(B^{b,2}_{t})+h_{b}(t)\operatorname{Vol}(S^{b,1}_{t}=2\pi\,\,\,\forall
t>0$

we obtain, with some computations

(3.13)
$\begin{split}-2\pi\chi(D_{t})+\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}&\left[2\pi-2\pi\frac{f_{b,\alpha}^{2}(t)}{2}\frac{\operatorname{Vol}(B^{b,2}_{t})}{\operatorname{Vol}(S_{t}^{b,1})}\right]\\\
\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)\end{split}$

Therefore, for all $t>0$,

(3.14)
$\begin{split}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}\left(1-\frac{\alpha
h_{b}(t)}{2}\frac{\operatorname{Vol}(B_{t}^{b,2})}{\operatorname{Vol}(S_{t}^{b,1})}\right)&-\chi(D_{t})\\\
\leq\frac{R(t)}{4\pi}+\frac{R^{\prime}(t)}{4\pi\alpha h_{b}(t)}\end{split}$

As $\frac{||B^{P}||^{2}}{h_{b}(t)}\leq\frac{1}{\sqrt{-b}}||B^{P}||^{2}$, then
$\int_{P}||B^{P}||^{2}d\sigma<\infty$ implies
$\int_{P}\frac{||B^{P}||^{2}}{h_{b}(t)}d\sigma<\infty$. Hence, by co-area
formula:

(3.15) $\int_{0}^{\infty}\left(\int_{\partial
D_{t}}\frac{||B^{P}||^{2}}{||\operatorname{\nabla}^{P}r||h_{b}(r)}\right)dt=\int_{0}^{\infty}\left(\frac{R^{\prime}(t)}{h_{b}(t)}\right)dt<\infty$

Therefore, there is a monotone increasing (sub)sequence
$\\{t_{i}\\}_{i=1}^{\infty}$ tending to infinity, (namely, $t_{i}\to\infty$
when $i\to\infty$), such that
$\frac{R^{\prime}(t_{i})}{h_{b}(t_{i})}\rightarrow 0$ when $i\to\infty$.

Let us consider the exhaustion of $P$ by these extrinsic balls, namely,
$\\{D_{t_{i}}\\}_{i=1}^{\infty}$. Then we have, replacing $t$ for $t_{i}$ and
taking limits when $i\to\infty$ in inequality (3.14) and applying Theorem 2.3
(ii),

(3.16)
$\begin{split}\operatorname{Sup}_{i}&\frac{\operatorname{Vol}(D_{t_{i}})}{\operatorname{Vol}(B_{t_{i}}^{b,2})}\left(1-\frac{\alpha}{2}\right)-\chi(P)\\\
&\leq\lim_{i\to\infty}\frac{R(t_{i})}{4\pi}=\frac{1}{4\pi}\int_{P}\|B^{P}\|^{2}d\sigma<\infty\end{split}$

for all $\alpha$ such that $0<\alpha<2$.

Hence, as $\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}$
is a continuous non decreasing function of $t$, we can conclude that
$\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}<\infty$
and $-\chi(P)<\infty$.

Then, letting $\alpha$ tend to $0$ in (3.16), we get, for all $t>0$:

(3.17)
$\displaystyle\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}-\chi(P)\leq\frac{\int_{P}\|B^{P}\|^{2}}{4\pi}$

### Case II

Let us consider $P$ immersed in $\mathbb{R}^{n}$. We consider, as in the proof
above, an exhaustion of $P$ by extrinsic balls, $\\{D_{t}\\}_{t>0}$, but now,
and following [1], these extrinsic balls will be centered at the origin
$0\in\mathbb{R}^{n}$, which we assume, without loss of generality, that
belongs to the surface $P$. Applying Proposition 3.2 we have

(3.18) $\displaystyle-2\pi\chi$
$\displaystyle(D_{t})+(\frac{\alpha}{2t^{2}})\operatorname{Vol}(D_{t})$
$\displaystyle+(\frac{1}{t}-\frac{\alpha}{2t})\int_{\partial
D_{t}}\frac{1}{\|\nabla^{P}r\|}\leq\frac{1}{2}R(t)+\frac{t}{2\alpha}R^{\prime}(t)$

Now, as $\int_{P}||B^{P}||^{2}d\sigma<\infty$, we can apply Proposition 2.2 in
[1], so we have, for $\alpha\in]0,2[$,

(3.19) $\frac{t}{2\alpha}R^{\prime}(t)=\frac{t}{2\alpha}\int_{\partial
D_{t}}\frac{\|B^{P}\|^{2}}{\|\nabla^{P}r\|}d\sigma\leq\frac{\mu(t)}{2\alpha
t}\int_{\partial D_{t}}\frac{1}{\|\nabla^{P}r\|}d\sigma$

being $\mu(t)$ such that $\lim_{t\to\infty}\mu(t)=0$ and therefore, from
(3.18),

(3.20)
$\displaystyle-2\pi\chi(D_{t})+\operatorname{Vol}(D_{t})(\frac{\alpha}{2t^{2}})$
$\displaystyle+(\frac{1}{t}-\frac{\alpha}{2t}-\frac{\mu(t)}{2\alpha
t})\int_{\partial
D_{t}}\frac{1}{\|\nabla^{P}r\|}d\sigma_{t}\,\leq\,\frac{1}{2}R(t)$

On the other hand, $\frac{1}{t}-\frac{\alpha}{2t}-\frac{\mu(t)}{2\alpha t}\geq
0$ if and only if $\mu(t)\leq\alpha(2-\alpha)$, which it is true for $t$ big
enough, namely, for $t>t_{\alpha}$ because $\lim_{t\to\infty}\mu(t)=0$. Hence,
as $\operatorname{Vol}(\partial D_{t})\leq\int_{\partial
D_{t}}\frac{1}{\|\nabla^{P}r\|}d\sigma_{t}$, and applying Theorem 1.1 in [12],
we have that inequality (3.20) becomes, for all $t>t_{\alpha}$

(3.21) $\displaystyle-2\pi\chi(D_{t})$
$\displaystyle+\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}\left[2\pi(1-\frac{\alpha}{2}-\frac{\mu(t)}{2\alpha})+\frac{\pi\alpha}{2}\right]\leq\,\frac{1}{2}R(t)$

Then, taking limits when $t\to\infty$ in inequality (3.21) and applying
Theorem 2.3, we have that $\lim_{t\to\infty}\mu(t)=0$ and
$\chi(P)=\lim_{t\to\infty}\chi(D_{t})$, so we obtain, for all $\alpha$ such
that $0<\alpha<2$:

(3.22) $\displaystyle
2\pi\operatorname{Sup}_{t}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}$
$\displaystyle\left(1-\frac{\alpha}{2}+\frac{\pi\alpha}{2}\right)$
$\displaystyle-2\pi\chi(P)\leq\frac{\int_{P}\|B^{P}\|^{2}}{2}<\infty$

Therefore we obtain
$\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}<\infty$
and $-\chi(P)<\infty$.

Then, letting $\alpha$ tend to $0$ we obtain, for all $t>0$:

(3.23)
$\displaystyle\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}-\chi(P)\leq\frac{\int_{P}\|B^{P}\|^{2}}{4\pi}$

##  4\. Proof of Theorem B

In Corollary 2.3, it was obtained a sufficienty large radius $R_{0}$, such
that the extrinsic distance $r_{p}$ has no critical points in
$P\setminus\operatorname{D}_{R_{0}}$.

Hence for this sufficienty large fixed radius $R_{0}$, there is a
diffeomorphism

$\Phi:P\setminus\operatorname{D}_{R_{0}}\to\partial D_{R_{0}}\times[0,\infty[$

so, in particular, $P$ has only finitely many ends, each of finite topological
type.

The above diffeomorfism implied that we could construct $P$ from $D_{R_{0}}$
($R_{0}$ big enough) attaching annulis and that $\chi(P\setminus D_{t})=0$
when $t\geq R_{0}$, and hence for all $t>R_{0}$, $\chi(P)=\chi(D_{t})$.

Let us consider now an exhaustion by extrinsic balls $\\{D_{t}\\}_{t>0}$ of
$P$ such that the extrinsic distance $r_{o}$ has no critical points in
$P\setminus\operatorname{D}_{R_{0}}$.

Applying now Gauss-Bonnet Theorem to the extrinsic balls $D_{t}$

(4.1) $2\pi\chi(P)=\int_{D_{t}}K^{P}d\sigma+\int_{\partial
D_{t}}k_{g}d\sigma_{t}$

Having in to account equation (2.2) and the Gauss formula, we have, for all
sufficiently large radius $t>R_{0}$

(4.2) $\displaystyle 2\pi\chi(P)$
$\displaystyle=-\frac{1}{2}\int_{D_{t}}\|B^{P}\|^{2}+b\operatorname{Vol}(D_{t})+h_{b}(t)\left(\operatorname{Vol}(D_{t})\right)^{\prime}$
$\displaystyle+\int_{\partial D_{t}}\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\|\operatorname{\nabla}^{P}r\|}\rangle
d\sigma_{t}=-\frac{1}{2}\int_{D_{t}}\|B^{P}\|^{2}d\sigma$
$\displaystyle+\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}\left(b\cdot\operatorname{Vol}(B_{t}^{b,2})+h_{b}(t)(\operatorname{Vol}(D_{t}))^{\prime}\frac{\operatorname{Vol}(B_{t}^{b,2})}{\operatorname{Vol}(D_{t})}\right.$
$\displaystyle\left.+\frac{\operatorname{Vol}(B_{t}^{b,2})}{\operatorname{Vol}(D_{t})}\int_{\partial
D_{t}}\langle
B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\|\operatorname{\nabla}^{P}r\|}\rangle
d\sigma_{t}\right)$

But
$2\pi=b\cdot\operatorname{Vol}(B_{t}^{b,2})+h_{b}(t)\operatorname{Vol}(S^{b,1}_{t})\,\,\,\forall
t>0$, so, for all sufficiently large radius $t>R_{0}$ and after some
computations:

(4.3) $\displaystyle 2\pi\chi(P)=-\frac{1}{2}\int_{D_{t}}\|B^{P}\|^{2}d\sigma$
$\displaystyle+2\pi\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}+h_{b}(t)\operatorname{Vol}(B^{b,2}_{t})(\frac{(\operatorname{Vol}(D_{t}))}{\operatorname{Vol}(B^{b,2}_{t})})^{\prime}$
$\displaystyle+\int_{\partial
D_{t}}<B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\|\operatorname{\nabla}^{P}r\|}>d\sigma_{t}$

The above equation is valid for all $t>R_{0}$, so, taking limits when
$t\to\infty$, we can define

(4.4) $\displaystyle G_{b}(P)$
$\displaystyle:=\lim_{t\to\infty}\left(h_{b}(t)\operatorname{Vol}(B^{b,2}_{t})(\frac{(\operatorname{Vol}(D_{t}))}{\operatorname{Vol}(B^{b,2}_{t})})^{\prime}\right.$
$\displaystyle\left.+\int_{\partial
D_{t}}<B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\|\operatorname{\nabla}^{P}r\|}>d\sigma_{t})\right)$

Using equalities (4.3), we have that

(4.5)
$G_{b}(P)=2\pi\chi(P)+\frac{1}{2}\int_{D_{t}}\|B^{P}\|^{2}d\sigma-2\pi\operatorname{Sup}_{t}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}<\infty$

and hence, $G_{b}(P)$ do not depends on the exhaustion $\\{D_{t}\\}_{t>0}$.

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