arXiv subjects

Joseph Amal Nathan

Publications and source records attributed to Joseph Amal Nathan.

11 recordsLinked to original sources

arXiv2021

$π$ and Arc-Length

We use the classical definitions (i) $π$ is the ratio of area to the square of the radius of a circle; (ii) $π$ is the ratio of circumference to the diameter of a circle, to prove $π$'s existence within the purview of Euclidean geometry. Next we show that the "arc-length" (Definition 1) is deducible from Euclidean geometry. Then we prove the Non-Euclidean-Axioms(NEA) of Archimedes (Corollary 4 and 5) and that the arc-length integral converges to the arc-length. We justify why `Euclidean Metric' (Definition 5) is a correct metric for arc-length; derive expressions for area, circumference of a circle and finally prove the equivalence of definitions (i) and (ii).

math.HO↗

arXiv2018

Divergence of $\langle p^6\rangle$ in discontinuous potential wells

The surprising divergence of the expectation value $<\!p^6\!>$ for the square well potential is known. Here, we prove and demonstrate the divergence of $<\!p^6\!>$ in potential wells which have a finite jump discontinuity; apart from the square-well two-piece half-potentials wells are examples. These half-potential wells can be expressed as $V(x)=-U(x) Θ(x)$, where $Θ(x)$ is the Heaviside step function. $U(x)$ are continuous and differentiable functions with minimum at $x=0$ and which may or not vanish as $x\sim \infty$.

quant-ph↗

arXiv2018

Expectation values of $p^2$ and $p^4$ in the square well potential

Position and momentum representations of a wavefunction $ψ(x)$ and $ϕ(p)$, respectively are physically equivalent yet mathematically in a given case one may be easier or more transparent than the other. This disparity may be so much so that one has to device a special strategy to get the quantity of interest in one of them. We revisit finite square well (FSW) in this regard. Circumventing the the problems of discontinuity of second and higher derivatives of $ψ(x)$ we obtain simple analytic expressions of $<\!p^2\!>$ and $<\!p^4\!>$. But it is the surprising fall-off of $ϕ(p)$ as $p^{-6}$ that reveals and restricts $<\!p^s\!>$ to be finite and non-zero only for $s=2,4$. In finding $<\!p^s\!>(s=2,4)$ from $ϕ(p)$, $p$-integrals are improper which for time-being, have been evaluated numerically to show the agreement between two representations.

quant-ph↗

arXiv2016

Transparency of the PT-symmetric complex potentials for coherent injection

Two port s-matrix for a complex PT-symmetric potential may have uni-modular eigenvalues. If this happens for all energies, there occurs a perfect emission of waves at both ends. We call this phenomenon transparency which is distinctly different from coherent perfect absorption with or without lasing. Using the versatile PT-symmetric complex Scarf II (scattering) potential, we demonstrate analytically that the transparency can occur regardless of whether PT-symmetry is unbroken or broken or if there are only scattering states. In these three cases, for a given value of the strength of the real part; the strength of the imaginary part $|V_2|$ of the potential lies in $(0, V_α), (V_α, V_β)$ and $(0,V_β)$ respectively. Several other numerically solved potentials also support our findings.

quant-ph↗

arXiv2015

Real discrete spectrum of complex PT-symmetric scattering potentials

We investigate the parametric evolution of the real discrete spectrum of several complex PT symmetric scattering potentials of the type $V(x)=-V_1 F_e(x) + i V_2 F_o(x), V_1>0, F_e(x)>0$ by varying $V_2$ slowly. Here $e,o$ stand for even and odd parity and $F_{e,o}(\pm \infty)=0$. Unlike the case of Scarf II potential, we find a general absence of the recently explored accidental (real to real) crossings of eigenvalues in these scattering potentials. On the other hand, we find a general presence of coalescing of real pairs of eigenvalues to the complex conjugate pairs at a finite number of exceptional points. We attribute such coalescings of eigenvalues to the presence of a finite barrier (on the either side of $x=0$ ) which has been linked to a recent study of stokes phenomenon in the complex PT-symmetric potentials.

quant-ph↗

arXiv2015

Accidental crossings of eigenvalues in one-dimensional complex PT-symmetric Scarf-II potential

So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: $E^n_{\pm}(λ)$ with $n=0,1,2...$. Then if strength $(λ)$ of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at $λ=λ_{*}$; this is unlike one dimensional Hermitian potentials. However, we show that the corresponding eigenstates at $λ=λ_{*}$ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at $λ=λ_{c}$. To re-emphasize, sharply at $λ=λ_{*}$ and $λ_{c}$, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.

quant-ph↗

arXiv2015

A new solvable complex PT-symmetric potential

We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of $H=-d^2/dx^2+V(x)$. For smaller values of $a,g <1$, there is a finite number of real discrete eigenvalues. As $a$ and $g$ increase, there exist exceptional points (EPs), $g_n$ (for fixed values of $a$) causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates $ψ_n(x)$ satisfy (1): PT$ψ_n(x)=1 ψ_n(x)$ and (2): PT$ψ_{E_n}(x)=ψ_{E^*_n}(x)$, for real and complex energy eigenvalues, respectively.

quant-ph↗

arXiv2015

Real discrete spectrum in the complex non-PT-symmetric Scarf II potential

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, $V(x)$, where the spectrum is again discrete and real. Showing that the Hamiltonian, $p^2/2m+V(x)$, in the new cases is pseudo-Hermitian could be challenging, if possible.

quant-ph↗

arXiv2013

A unique method to evaluate the general integral $\int_0^\infty dx\frac{\sin ^a px \cos ^c qx}{x^b} $

All integrals available in literature and books, that are related to Sinc(=sin x/x) function, are special cases of the general form of the integral given in the title. The evaluation of the integral is divided into two cases (i) $a$ and $b$ of same parity, which is easier to evaluate and (ii) $a$ and $b$ of different parity, a difficult case. Amazingly and may be for the first time, a divergent integral is used in evaluating this difficult case with the help of a simple but a special combinatorial expression. The combinatorial identity is derived from the power reduction formula of the Sines and Cosines. The method adopted in this paper is unique and makes it relatively easy to evaluate this integral.

math.HO↗

arXiv2003

The diophantine equation x^3/3+y^3+z^3-2xyz=0

We will be presenting two theorems in this paper. The first theorem, which is a new result, is about the non-existence of integer solutions of the cubic diophantine equation. In the proof of this theorem we have used some known results from theory of binary cubic forms and the method of infinite descent, which are well understood in the purview of Elementary Number Theory(ENT). In the second theorem, we show, that the famous Fermat's Last Theorem(FLT) for exponent 3 and the first theorem are equivalent. So Theorem1 and 2 constitute an alternate proof for the non-existence of integer solutions of this famous cubic Fermat's equation. It is well known, that from L.Euler(1770) to F.J.Duarte(1944) many had given proof of FLT for exponent 3. But all proofs uses concepts, which are beyond the scope of ENT. Hence unlike other proofs the proof given here is as an ENT proof of FLT for exponent 3.

math.GM↗

arXiv2003

First Case of Fermat's Last Theorem

In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be proved for all primes greater than 3. With a computer Conjecture1 was verified to be true for primes $\leq 2437$ and Conjecture2 for primes $\leq 100003$.

math.HO↗