Analytical treatment of SUSY Quasi-normal modes

J. ROMERO; P.SANCHEZ; M. BELLINI

EUROPEAN PHYSICAL JOURNAL C - PARTICLES AND FIELDS

SPRINGER

Lugar: Heidelberg; Año: 2011 vol. c p. 1256 - 1265

1434-6044

We use the Fock¨CIvanenko formalism to obtain the Dirac equation which describes the interaction of a massless 1/2-spin neutral fermion with a gravitational field around a Schwarzschild black hole (BH).We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the asymptotic states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature TH: E (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. around a Schwarzschild black hole (BH).We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the asymptotic states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature TH: E (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. around a Schwarzschild black hole (BH).We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the asymptotic states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature TH: E (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. /2-spin neutral fermion with a gravitational field around a Schwarzschild black hole (BH).We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the asymptotic states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature TH: E (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. TH: E (¡ü¡ý) n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. n = −2¦ÐinTH. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. the isolated bottom state has a real null energy eigenvalue. −2¦ÐinTH