• Zafar Ahmed

      Articles written in Pramana – Journal of Physics

    • Random matrix model for disordered conductors

      Zafar Ahmed Sudhir R Jain

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      We present a random matrix ensemble where real, positive semi-definite matrix elements, x, are log-normal distributed, exp[−log2(x)]. We show that the level density varies with energy, E, as 2/(1+E) for large E, in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here). An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed and its connection with level dynamics is brought out. It is further proved that Dyson’s Coulomb gas analogy breaks down whenever the confining potential is given by a transcendental function for which there exist orthogonal polynomials.

    • Scarcity of real discrete eigenvalues in non-analytic complex $\mathcal{PT}$-symmetric potentials

      Zafar Ahmed

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      We find that a non-differentiability occurring whether in real or imaginary part of a complex $\mathcal{PT}$-symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials $x^{2}$ and $|x|$ by imaginary $\mathcal{PT}$ -symmetric potentials $ix|x|$ and $ix$, respectively.

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