Exact phonon quasibound states around an optical black hole

Abstract

In this work, we present an exact solution of phonon quasibound states in a vortex photon flow that creates an analogous black hole background. We worked out and successfully solve the governing differential equation of the systen that is similar with the relativistic Klein–Gordon equation. The exact radial solutions are discovered in terms of the confluent Heun functions that behaves as ingoing waves near the black hole’s horizon and decaying far away from the black hole’s horizon. The polynomial condition of the confluent Heun function leads to the discovery of the the complex valued quantized energy levels expression of the quasibound state of which depends on the phonon mass Ω0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _0$$\\end{document}, the photon vortex horizon’s spin ΩH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _H$$\\end{document} and the azimuthal and the main quantum number (m, n). In the last section, by using the Damour–Ruffini method, the Hawking radiation of the analog black hole’s horizon is investigated to derive the phononic radiation distribution function from where, the Hawking temberature is obtained.