arXiv subjects

Dona Ghosh

Publications and source records attributed to Dona Ghosh.

11 recordsLinked to original sources

arXiv2019

Solvable model of bound states in the continuum (BIC) in one dimension

Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann's Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier $V(x)=-V_0[\exp(2|x|/a)-1]$ which for $E V_0$, there is again a continuum of complex scattering solutions $\psi(x)$ whose real and imaginary parts though solutions of Schr{\"o}dinger equation yet their parities cannot be ascertained as $C\psi(x)$ is also a solution where $C$ is an arbitrary complex non-real number.

quant-ph↗

arXiv2018

Three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials

For complex PT-symmetric scattering potentials (CPTSSPs) $V(x)= V_1 f_{even}(x) + iV_2 f_{odd}(x), f_{even}(\pm \infty) = 0 = f_{odd}(\pm \infty), V_1,V_2 \in \Re $, we show that complex $k$-poles of transmission amplitude $t(k)$ or zeros of $1/t(k)$ of the type $\pm k_1+ik_2, k_2\ge 0$ are physical which yield three types of discrete energy eigenvalues of the potential. These discrete energies are real negative, complex conjugate pair(s) of eigenvalues (CCPEs: ${\cal E}_n \pm i γ_n$) and real positive energy called spectral singularity (SS) at $E=E_*$ where the transmission and reflection co-efficient of $V(x)$ become infinite for a special critical value of $V_2=V_*$. Based on four analytically solvable and other numerically solved models, we conjecture that a parametrically fixed CPTSSP has at most one SS. When $V_1$ is fixed and $V_2$ is varied there may exist Kato's exceptional point(s) $(V_{EP})$ and critical values $V_{*m}, m=0,1,2,..$, so when $V_2$ crosses one of these special values a new CCPE is created. When $V_2$ equals a critical value $V_{*m}$ there exist one SS at $E=E_*$ along with $m$ or more number of CCPEs. Hence, this single positive energy $E_*$ is the upper (or rough upper) bound to the CCPEs: ${\cal E}_l \lessapprox E_*$, here ${\cal E}_l$ corresponds to the last of CCPEs. If $V(x)$ has Kato's exceptional points (EPs: $V_{EP1}<V_{EP2}<V_{EP3}<...<V_{EPl}$), the smallest of critical values $V_{*m}$ is always larger than $V_{EPl}$. Hence, in a CPTSSP, real discrete eigenvalue(s) and the SS are mutually exclusive whereas CCPEs and the SS can co-exist .

quant-ph↗

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 $V(x)=-U(x) \Theta(x)$, where $\Theta(x)$ 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

Coherent scattering from semi-infinite non-Hermitian potentials

When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection $(r_L,r_R)$ and transmission $(t_L,t_R)$ amplitudes are non-reciprocal. In a parametric domain, there exists Spectral Singularity (SS) at a real energy $E=E_*$ and the determinant of the time-reversed two port S-matrix i.e., $|\det(S)|=|t_L t_R-r_L r_R|$ vanishes sharply at $E=E_*$ displaying the phenomenon of Coherent Perfect Absorption (CPA). In the complimentary parametric domain, the potential becomes either left or right reflectionless at $E=E_z$. But we rule out the existence of Invisibility despite $r_R(E_i)=0$ and $t_R(E_i)=1$ in these new models. We present two simple exactly solvable models where the expressions for $E_*$, $E_z$, $E_i$ and the parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the novel phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and having $t_L=t_R$.

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↗

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{\"o}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↗

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↗

arXiv2014

A solution to Ahmed's Integral(II)

In the year 2000, Ahmed proposed a family of integrals in the American Mathematical Monthly which invoked a considerable response then. Here I would like to present another solution to this family of integrals. I propose to call this as Ahmed's Integral (II) in the light of the well known Ahmed's Integral.

math.HO↗