arXiv subjects

Zafar Ahmed

Publications and source records attributed to Zafar Ahmed.

At least 37 records · Page 2Linked to original sources

arXiv2017

Dirichlet spectrum of the paradigm model of complex PT-symmetric potential: $V(x)=-(ix)^N$

So far the spectra $E_n(N)$ of the paradigm model of complex PT(Parity-Time)-symmetric potential $V_{BB}(x,N)=-(ix)^N$ is known to be analytically continued for $N > 4$. Consequently, the well known eigenvalues of the Hermitian cases ($N=6,10$) cannot be recovered. Here, we illustrate Kato's theorem that even if a Hamiltonian $H(λ)$ is an analytic function of a real parameter $λ$, its eigenvalues $E_n(λ)$ may not be analytic at finite number of Isolated Points (IPs). In this light, we present the Dirichlet spectra $E_n(N)$ of $V_{BB}(x,N)$ for $2\le N<12$ using the numerical integration of Schr{ö}dinger equation with $ψ(x=\pm \infty)=0$ and the diagonalization of $H=p^2/2μ+V_{BB}(x,N)$ in the harmonic oscillator basis. We show that these real discrete spectra are consistent with the most simple two-turning point CWKB (C refers to complex turning points) method provided we choose the maximal turning points (MxTP) [$-a+ib,a+ib, a, b \in {\cal R}$] such that $|a|$ is the largest for a given energy among all (multiple) turning points. We find that $E_n(N)$ are continuous function of $N$ but non-analytic (their first derivative is discontinuous) at IPs $N=4,8$; where the Dirichlet spectrum is null (as $V_{BB}$ becomes a Hermitian flat-top potential barrier). At $N=6$ and $10$, $V_{BB}(x,N)$ becomes a Hermitian well and we recover its well known eigenvalues.

quant-ph↗

arXiv2017

Solvable models of an open well and a bottomless barrier in 1-D

We present one dimensional potentials $V(x)= V_0[e^{2|x|/a}-1]$ as solvable models of a well $(V_0>0)$ and a barrier ($V_0<0$). Apart from being new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of Schr{ö}dinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission $(R/T)$ co-efficients for the barrier have been derived. Interestingly, the crossover energy $E_c$ where $R(E_c)=1/2=T(E_c)$ may occur below/above or at the barrier-top. A connection between poles of these co-efficients and bound state eigenvalues of the well has also been demonstrated.

quant-ph↗

arXiv2017

Spectral statistics for ensembles of various real random matrices

We investigate spacing statistics $p(s)$ and distribution of eigenvalues $D(ε)$ for ensembles of various real random matrices (of order $n \times n, n=2$ and $n>>2$) where the matrix-elements have various Probability Distribution Function (PDF: $f(x)$) including Gaussian. We construct ensembles of $1000$, $100 \times 100$ real random matrices $R$, $C$ (cyclic) and $T$ (tridiagonal) and real symmetric matrices: ${\cal R}'$, ${\cal R}=R+R^t$, ${\cal Q}=RR^t$, ${\cal C}$ (cyclic), ${\cal T}$ (tridiagonal), $T'$ (pseudo-symmetric Tridiagonal), $Θ$ (Toeplitz) , ${\cal D}=CC^t$ and ${\cal S}=TT^t$. We find that the spacing distribution of the adjacent levels of matrices ${\cal R}$ and ${\cal R}'$ under any symmetric PDF of matrix elements is $p_{AB}(s)=A s e^{-Bs^2}$ which approximately conforms to the Wigner surmise as $A/2 \approx B \approx π/4$. But under asymmetric PDFs we observe $A/2 \approx B >>π/4$, where $A,B$ are also sensitive to the choice of the matrix and the PDF. More interestingly, the real symmetric matrices ${\cal C}, {\cal T}, {\cal Q}$, $Θ$ (excepting ${\cal D}$ and ${\cal S}$) and $T'$ (pseudo-symmetric tridiagonal) all conform to the Poisson distribution $p_μ(s) =μe^{-μs}$, where $μ$ depends upon the choice of the matrix and PDF. Let complex eigenvalues of $R$, $C$ and $T$ be $E^c_n$. We show that all $p(s)$ arising due to $\Re(E^c_n)$, $\Im(E^c_n)$ and $|E^c_n|$ of $R$, $C$ and $T$ are also of Poisson type: $μe^{-μs}$. We observe $p(s)$ as half-Gaussian for two real eigenvalues of $C$. For real matrices $R, C, T$, we associate new types of $p(s)$ with them. Lastly, we study the distribution $D(ε)$ of eigenvalues of symmetric matrices (of large order) discussed above.

quant-ph↗

arXiv2017

Coalescing versus merging of energy levels in one-dimensional potentials

The sub-barrier pairs of energy levels of a Hermitian one-dimensional symmetric double well potential are known to merge into one, if the inter-well distance ($a$) is increased slowly. The energy at which the doublets merge are the ground state eigenvalues of independent wells ($ε_0$). We show that if the double well is perturbed mildly by a complex PT-symmetric potential the merging of levels turns into the coalescing of two levels at an exceptional point $a=a_*$. For $a>a_*$, the real part of complex-conjugate eigenvalues coincides with $ε_0$ again. This is an interesting and rare connection between the two phenomena in two domains: Hermiticity and complex PT-symmetry.

quant-ph↗

arXiv2017

Supersymmetric partner potentials arising from nodeless half bound states

A Half Bound State (HBS) $ψ_*(x)$ can be defined as a single, conditional, zero-energy, continuous solution of the one dimensional Schr{ö}dinger equation for a scattering potential well $V(x)$ ($s.t ~ V(\pm \infty)=0$). The non-normalizable and solitary HBS of a potential satisfies Neumann boundary condition that $ψ'_*(\pm \infty)=0$ and it can have $n$ (= 0,1,2,...) number of nodes indicating $n$ number of bound states in $V(x)$ below $E=0$. Here we show that starting with a nodeless HBS, we can construct a (supersymmetric) pair of finite potentials (well, double wells, well-barrier): $V_{\pm}(x)$ having no bound state and they enclose positive area on $x$-axis. On the contrary their negative counterparts $(-cV_{\pm}(x)),c>0$ do have at least one bound state for any arbitrary positive value of $c$. Furthermore, $c V_{\pm}(x),~ c >0$ which binds positive area on x-axis in conformity with Simon's theorem can have at least one bound state only conditionally for instance when $c>1$ or $c>>1$.

quant-ph↗

arXiv2017

The paradoxical zero reflection at zero energy

Usually, the reflection probability $R(E)$ of a particle of zero energy incident on a potential which converges to zero asymptotically is found to be 1: $R(0)=1$. But earlier, a paradoxical phenomenon of zero reflection at zero energy ($R(0)=0$) has been revealed as a threshold anomaly. Extending the concept of Half Bound State (HBS) of 3D, here we show that in 1D when a symmetric (asymmetric) attractive potential well possesses a zero-energy HBS, $R(0)=0$ $(R(0)<<1)$. This can happen only at some critical values $q_c$ of an effective parameter $q$ of the potential well in the limit $E \rightarrow 0^+$. We demonstrate this critical phenomenon in two simple analytically solvable models which are square and exponential wells. However, in numerical calculations even for these two models $R(0)=0$ is observed only as extrapolation to zero energy from low energies, close to a precise critical value $q_c$. By numerical investigation of a variety of potential wells, we conclude that for a given potential well (symmetric or asymmetric), we can adjust the effective parameter $q$ to have a low reflection at a low energy.

quant-ph↗

arXiv2016

Hermitian Hamiltonians: Matrix versus Schr${ö}$dinger's

We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schr{ö}dinger Hamiltonian: $H=p^2/2μ+V(x)$, may not be so. For instance when $V(x)=x, x^3, -x^2$, $H$ does not have even one real discrete eigenvalue. Textbooks do not highlight this distinction. However, if $H$ has real discrete spectrum, by virtue of the expansion theorem, one can convert the eigenvalue problem $Hψ_n=E_n ψ_n$ into a matrix and get eigenvalues $E_n$ by diagonalizing the matrix. We show, that the thus obtained $E_n$ could be accurate, provided $H$ is devoid of scattering states. We suggest that this could be a simple and apt way to introduce the method of Linear Combination of Atomic Orbitals (LCAO) for finding the spectra of molecules. In textbooks, usually the method of matrix-diagonalization appears meagerly as a degenerate perturbation theory for more than one dimensions.

physics.gen-ph↗

arXiv2016

Revisiting double Dirac delta potential

We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ($T=1$). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, $R(E)$] at energy $E=0$ is non-intuitive. However, earlier this has been found and called "threshold anomaly". Here we show that it is a critical phenomena and we can have $ 0 \le R(0)<1$ when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate ($ψ(x)=Ax+B$) of delta well between two symmetric rigid walls for $R(0)=0$. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.

quant-ph↗

arXiv2016

Bound state eigenfunctions need to vanish faster than $|x|^{-3/2}$

In quantum mechanics students are taught to practice that eigenfunction of a physical bound state must be continuous and vanishing asymptotically so that it is normalizable in $x\in (-\infty, \infty)$. Here we caution that such states may also give rise to infinite uncertainty in position $(Δx=\infty)$, whereas $Δp$ remains finite. Such states may be called loosely bound and spatially extended states that may be avoided by an additional condition that the eigenfunction vanishes asymptotically faster than $|x|^{-3/2}$.

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

Avoided crossings of energy levels in one-dimension

Due to the absence of degeneracy in one dimension, when a parameter, $λ$, of a potential is varied slowly the discrete energy eigenvalue curves, $E_n(λ)$, cannot cross but they are allowed to come quite close and diverge from each other. This phenomena is called avoided crossing of energy levels. The parametric evolution of eigenvalues of the generally known one dimensional potentials do not display avoided crossings and on the other hand some complicated and analytically unsolvable models do exhibit this. Here, we show that this interesting spectral property can be found in simple one-dimensional double-wells when width of one of them is varied slowly.

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

New shape resonances in one dimension

Hitherto, a finitely thick barrier next to a well or a rigid wall has been considered the potential of simplest shape giving rise to resonances (metastable states) in one dimension: $x \in(-\infty, \infty)$. In such a potential, there are three real turning points at an energy below the barrier. Resonances are Gamow's (time-wise) decaying states with discrete complex energies $({\cal E}_n = E_n -iΓ_n/2)$. These are also spatially catastrophic states that manifest as peaks/wiggles in Wigner's reflection time-delay at $E = ε_n \approx E_n$. Here we explore potentials with simpler shapes giving rise to resonances - two-piece rising potentials having just one-turning point. We demonstrate our point by using rising exponential profile in various ways.

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↗

arXiv2014

Ahmed's Integral: the maiden solution

In 2001-2002, I happened to have proposed a new definite integral in the American Mthematical Monthly (AMM),which later came to be known in my name (Ahmed). In the meantime, this integral has been mentioned in mathematical encyclopedias and dictionaries and further it has also been cited and discussed in several books and journals. In particular, a google search with the key word "Ahmed's Integral" throws up more than 60 listings. Here I present the maiden solution for this integral.

math.HO↗

arXiv2014

PT-symmetry and Transparency

It is known that the perfect absorption of two identical waves incident on a complex potential from left and right can occur at a fixed real energy and that the time-reversed setting of this system would act as a laser at threshold at the same energy. Here, we argue and show that PT-symmetric potentials are exceptional in this regard which do not allow Coherent Perfect Absorption without lasing as the modulus of the determinant of the $S$-matrix, $|\det S|$, becomes 1, for all positive energies. Next we show that in the parametric regimes where the PT-symmetry is unbroken, the eigenvalues, $s_\pm$ of $S$, can become unitary (uni-modular) for all energies. Then the potential becomes coherent perfect emitter on both sides for any energy of coherent injection. We call this property Transparency.

quant-ph↗