Designing lasing and perfectly absorbing potentials

Existence of a spectral singularity (SS) in the spectrum of a Schrödinger operator with a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, $\mathcal{PT}$-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.

Figure 1

Even non-Hermitian potential (15) having a SS at $k_0=\pi$ (a) and potential (17) having a SS at $k_0=\pi /2$ (b). The real and imaginary parts of the potentials are shown by blue solid and red dashed lines, respectively. (c) and (d) The absolute values of the transmission coefficient $\left|t\right|$ (solid blue lines) and of the reflection coefficients $|r_{\mathrm{L}}|=|r_{\mathrm{R}}|$ (dashed red lines) for the potentials shown in (a) and (b), respectively.

Figure 2

Examples of the $\mathcal{PT}$-symmetric potentials, generated by the function (24), with SSs $k_0=1.1$ (a) and $k_0=3$ (b). The real and imaginary parts of the potentials are shown by blue solid and red dashed lines, respectively.

Figure 3

Examples of non-Hermitian potentials generated by the function (31) having SS of the second order at $k_0=\pi$, which correspond to two different values of $a$: $a\approx -.850249$ (a) and $a\approx -.381869$ (b). The real and imaginary parts of the potentials are shown by blue solid and red dashed lines, respectively.

Figure 4

Imaginary (solid lines 1 and 3) and real (dashed lines 2 and 4) parts of (a) ${\zeta }_1$ (red lines 1 and 2) and ${\zeta }_2$ (blue lines 3 and 4) for which there exists a positive SS of the second order with $k_0>0$, and (b) coefficients ${\zeta }^{\prime \prime }/2$ in Eq. (38) describing splitting of second-order SSs.