AM  Vol.3 No.10 , October 2012
Real Eigenvalue of a Non-Hermitian Hamiltonian System
Abstract: With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x1+ip2, p=p1+ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT-symmetric condition is satisfied. Thus PT- symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.
Cite this paper: R. Singh, "Real Eigenvalue of a Non-Hermitian Hamiltonian System," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1117-1123. doi: 10.4236/am.2012.310164.

[1]   H. Feshbach, C. E. Porter and V. F. Weisskopf, “Model for Nuclear Reactions with Neutrons,” Physical Review, Vol. 96, No. 2, 1954, pp. 448-464. doi:10.1103/PhysRev.96.448

[2]   R. S. Kaushal, “Classical and Quantum Mechanics of Noncentral Potentials,” Narosa Publishing House, New Delhi, 1998.

[3]   F. Verheest, “Nonlinear Wave Interaction in a Complex Hamiltonian Formalism,” Journal of Physics A: Mathematical and General, Vol. 20, No. 1, 1987, pp. 103-110. doi:10.1088/0305-4470/20/1/019

[4]   N. N. Rao, B. Buti and S. B. Khadkikar, “Hamiltonian Systems with Indefinite Kinetic Energy,” Pramana: Journal of Physics, Vol. 27, No. 4, 1986, pp. 497-505.

[5]   R. S. Kaushal and H. J. Korsch, “Some Remarks on Complex Hamiltonian Systems,” Physics Letters A, Vol. 276, No. 1-4, 2000, pp. 47-51. doi:10.1016/S0375-9601(00)00647-2

[6]   R. S. Kaushal and S. Singh, “Construction of Complex Invariants for Classical Dynamical Systems,” Annals of Physics, Vol. 288, No. 2, 2001, pp. 253-276. doi:10.1006/aphy.2000.6108

[7]   C. M. Bender and A. Turbiner, “Analytic Continuation of Eigenvalue Problems,” Physics Letters A, Vol. 173, No. 6, 1993, pp. 442-446. doi:10.1016/0375-9601(93)90153-Q

[8]   C. M. Bender and S. Boettcher, “Real Spectra in NonHermitian Hamiltonians Having -Symmetry,” Physical Review Letters, Vol. 80, No. 24, 1998, pp. 5243-5246. doi:10.1103/PhysRevLett.80.5243

[9]   C. M. Bender, S. Boettcher and P. N. Meisinger, “ Symmetric Quantum Mechanics,” Journal of Mathematical Physics, Vol. 40, No. 5, 1999, pp. 2201-2229. doi:10.1063/1.532860

[10]   F. M. Fernandez, R. Gujardiola, J. Ross and M. Zonjil, “Strong Coupling Expansion for the -Symmetric Oscillator V(x)=a(ix)+b(ix)2+c(ix)3,” Journal of Physics A: Mathematical and General, Vol. 31, No. 50, 1998, pp. 10105-10112. doi:10.1088/0305-4470/31/50/008

[11]   R. S. Kaushal, “On the Quantum Mechanics of Complex Hamiltonian Systems in One Dimension,” Journal of Physics A: Mathematical and General, Vol. 34, No. 49, 2001, pp. L709-L714. doi:10.1088/0305-4470/34/49/104

[12]   A. L. Xavier Jr. and M. A. M. de Aguiar, “Complex Trajectories in the Quartic Oscillator and Its Semiclassical Coherent-State,” Annals of Physics, Vol. 252, No. 2, 1996, pp. 458-476. doi:10.1006/aphy.1996.0141

[13]   A. L. Xavier Jr. and M. A. M. de Aguiar, “Phase Space Approach to the Tunnel Effect: A New Semiclassical Traversal Time,” Physical Review Letters, Vol. 79, No. 18, 1997, pp. 3323-3326. doi:10.1103/PhysRevLett.79.3323

[14]   T. J. Hollowood, “Solitons in Affine Toda Theories,” Nuclear Physics B, Vol. 384, No. 3, 1992, pp. 523-540. doi:10.1016/0550-3213(92)90579-Z

[15]   D. R. Nelson and N. M. Shnerb, “Non-Hermitian Localization and Population Biology,” Physical Review E, Vol. 58, No. 2, 1998, pp. 1383-1403. doi:10.1103/PhysRevE.58.1383

[16]   N. Hatano and D. R. Nelson, “Localization Transitions in Non-Hermitian Quantum Mechanics,” Physical Review Letters, Vol. 77, No. 3, 1996, pp. 570-573. doi:10.1103/PhysRevLett.77.570

[17]   N. Hatano and D. R. Nelson, “Vortex Pinning and NonHermitian Quantum Mechanics,” Physical Review B, Vol. 56, No. 14, 1997, pp. 8651-8673. doi:10.1103/PhysRevB.56.8651

[18]   R. S. Kaushal and Parthasarthi, “Quantum Mechanics of Complex Hamiltonian Systems in One Dimension,” Journal of Physics A: Mathematical and General, Vol. 35, No. 41, 2002, pp. 8743-8761. doi:10.1088/0305-4470/35/41/308

[19]   Parthasarthi and R. S. Kaushal, “Quantum Mechanics of Complex Sextic Potential in One Dimension,” Physica Scripta, Vol. 68, No. 2, 2003, pp. 115-127. doi:10.1238/Physica.Regular.068a00115

[20]   L. I. Schiff, “Quantum Mechanics,” Tata McGraw-Hill Publishing Company Limited, New York, 1968.