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

Three types of discrete energy eigenvalues in complex $P T$ -symmetric scattering potentials

Zafar Ahmed, Sachin Kumar, and Dona Ghosh

Phys. Rev. A 98, 042101 – Published 1 October 2018

Abstract

For complex $PT$ -symmetric scattering potentials (CPTSSPs) $V (x) = V_{1} f_{even} (x) + i V_{2} f_{odd} (x), f_{even} (\pm \infty) = 0 = f_{odd} (\pm \infty), V_{1}, V_{2} \in Re$ , we show that complex $k$ poles of transmission amplitude $t (k)$ or zeros of $1 / t (k)$ of the type $\pm k_{1} + i k_{2}, k_{2} \geq 0$ 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: $E_{n} \pm i γ_{n}$ ) and real positive energy called spectral singularity (SS) at $E = E_{*}$ where the transmission and reflection coefficient 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 $V_{* m}, m = 0, 1, 2, \dots$ , 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 exists 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: $E_{l} ⪅ E_{*}$ , here $E_{l}$ corresponds to the last of CCPEs. If $V (x)$ has Kato's exceptional points (EPs: $V_{EP 1} < V_{EP 2} < V_{EP 3} < \dots < V_{EPl}$ ), 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 coexist.

Received 19 June 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042101

Physics Subject Headings (PhySH)

Quantum optics Scattering theory

Atomic, Molecular & Optical

Authors & Affiliations

Zafar Ahmed^1,2,*, Sachin Kumar^3,†, and Dona Ghosh^4,‡

¹Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India
²Homi Bhabha National Institute, Mumbai 400 094, India
³Theoretical Physics Section, Bhabha Atomic Research Centre, Mumbai 400 085, India
⁴Department of Mathematics, Jadavpur University, Kolkata 700032, India

^*zahmed@barc.gov.in
^†sachinv@barc.gov.in
^‡rimidonaghosh@gmail.com

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Contour plots of real and imaginary parts of $F (k) = M_{22} (k_{1} + i k_{2}) = 0$ for Scarf II [Eq. (4)] in the complex $k$ plane, when (a) $V_{1} = 5$ and $V_{2} = 5$ and (b) $V_{1} = - 5$ and $V_{2} = 19.75$ (see row 13 in Table 1). Solid red lines and dashed blue lines depict real and imaginary parts of $F (k)$ , respectively. In (a), notice that solid and dashed curves are intersecting only for $k_{2} < 0$ , so no physical zero is giving rise to discrete eigenvalues (SDSS or CCPE). In (b), these pairs of curves are intersecting three times for $k_{2} \geq 0$ . For the first point of intersection $k_{2} = 0$ (SDSS), two upper ones denote two CCPEs. Units of physical quantities are as in Table 1. Unit of $k$ is $Å^{- 1}$ .
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Figure 2
Same as Fig. 1, but for the Dirac $δ$ potential (9) $(a = 1 Å)$ . (a) When $V_{1} = 0, V_{2} = 5.653$ , there are three pairs of roots with $k_{2} > 0$ , these are $15.456 \pm i 0.133, 7.663 \pm i 1.949, 2.420 \pm i 0.280$ , The first pair is due to splitting of SDSS that occurs at $E_{*} = 15.421$ when $V_{2} = 5.553 = V_{*}$ (see row 3, in Table 2). (b) When $V_{1} = - 5$ and $V_{2} = 5.671 = V_{*} + 0.1$ , there are two CCPEs: $3.009 \pm i 0.004, 1.789 \pm i 14.177$ . The first one results from the splitting of SDSS (see row 13 in Table 2).
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Figure 3
Same as Fig. 1, but for the square well potential (13) ( $a = 2 Å$ ). (a) When $V_{1} = - 5$ and $V_{2} = 2.0$ , there are two negative real discrete eigenvalues as the dashed (blue) and solid (red) curves intersect on the upper $y$ axis ( $k_{1} = 0, k_{2} > 0$ ) and we have one CCPE (see row 11, in Table 3). (b) When $V_{1} = - 5$ and $V_{2} = 10.483$ , there are four pairs of roots with $k_{2} > 0$ , these pairs are $22.6139 \pm i 0.134, 9.762 + 5.309 i, 1.45 \pm 8.530 i, - 3.377 \pm i 10.041$ . The first pair results from the splitting of SDSS that occurs at $E_{*} = 22.578$ when $V_{2} = 10.383 = V_{*}$ (see row 14 in Table 3).
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Figure 4
Evolution of negative real discrete eigenvalues of the exponential potential (17) $E_{n} (V_{2})$ when $V_{1} = - 60$ eV and $a = 2 Å$ . Coordinates of the endpoints are $(12.82, - 0.22), (12.96, - 2.44), (13.17, - 7.73), (13.63, - 17.64), and (14.78, - 36.40)$ which are in units of eV. So EPs of the potential are $V_{EP n}$ (eV): 12.82, 12.96, 13.17, 13.36, and 14.78. This structure is temple-like (length of the bottom loop is largest); for a cone-like structure of evolution see [16]. Part (b) presents 10 real discrete negative eigenvalues which are negative energy poles of $T (E),$ Eq. (20), for the case of the potential (see row 19 of Table 4).
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Figure 5
Same as Figs. 1, 2, 3, but for the exponential potential (17) $(a = 1 Å$ ), see various rows in Table 4. (a) When $V_{1} = - 60$ and $V_{2} = 10$ , there are 10 physical zeros of $F (k)$ as 10 solid (red) curves are being intersected by the vertical dashed (blue) line. These roots on the $y$ axis represent bound states of the potential as $V_{2}$ is less than the fourth (smallest) EP which is $V_{EP 4} = 12.82$ [see Fig. 4]. There are no SDSSs or CCPEs in this case (row 18 in Table 4). In (b) we have $V_{1} = - 60$ and $V_{2} = 14$ , which lie between two EPs, $V_{EP 1} = 14.78$ and $V_{EP 2} = 13.63$ ; in this case there are two real negative energy bound states and the rest are CCPEs (row 19 in Table 4).
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