We complexify a 1-d potential V (x) = V 0 cosh 2 µ{tanh[(x − µd)/d] + tanh(µ)} 2 which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters (µ, d) becomes imaginary. For the case of µ → iµ, we have entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from left (or right) at certain discrete energies. The penetrating states in the other case (d → id) are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. Presence of MSS and reflectionlessness are also discussed for the free states in the later case.
It appears that there is a genuine shortage of radio pulsars with surface magnetic fields significantly smaller than ∼108 G. We propose that the pulsars with very low magnetic fields are actually strange stars locked in a state of minimum free energy and therefore at a limiting value of the magnetic field which cannot be lowered by the system spontaneously.
The single glitch observed in PSR B1821−24, a millisecond pulsar in M28, is unusual on two counts. First, the magnitude of this glitch is at least an order of magnitude smaller (Δν/ν∼ 10−11) than the smallest glitch observed to date. Secondly, all other glitching pulsars have strong magnetic fields with B≳ 1011 G and are young, whereas PSR B1821−24 is an old recycled pulsar with a field strength of 2.25 × 109 G. We have earlier suggested that some of the recycled pulsars could actually be strange quark stars. In this work, we argue that the crustal properties of such a strange pulsar are just right to give rise to a glitch of this magnitude, explaining the scarcity of larger glitches in millisecond pulsars.
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