Symbolic-Manipulation Constructions of Hilbert-Space Metrics in Quantum Mechanics

The recently formulated quantum-mechanics problem of the determination of the Hilbert-space metric Θ which renders a given Hamiltonian H self-adjoint is addressed. Via an exactly solvable example of the so called Gegenbauerian quantum-lattice oscillator it is demonstrated that the construction (basically, the solution of the so called Dieudonne's operator equation) and analysis of suitable Θ = Θ(H) (i.e., the determination of their domain's "exceptional-point" boundary) may enormously be facilitated via symbolic algebraic manipulations and via the MAPLE-supported numerics and graphics.

[1]  R. Tateo,et al.  The ODE/IM correspondence , 2001, hep-th/0703066.

[2]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[3]  Complete set of inner products for a discrete PT-symmetric square-well Hamiltonian , 2009, 0911.0336.

[4]  M. Znojil.  Fundamental length in quantum theories with PT-symmetric Hamiltonians , 2009, 0907.2677.

[5]  M. Znojil.  Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators , 2009, 0908.4045.

[6]  F. Scholtz,et al.  Quasi-Hermitian operators in quantum mechanics and the variational principle , 1992 .

[7]  P. Siegl.  Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop , 2008 .

[8]  Scattering theory using smeared non-Hermitian potentials , 2009, 0903.1007.

[9]  Carolyn R. Bertozzi,et al.  Methods and Applications , 2009 .

[10]  A. Das.  Pseudo-Hermitian quantum mechanics , 2011 .

[11]  Miloslav Znojil,et al.  Three-Hilbert-Space Formulation of Quantum Mechanics , 2009, 0901.0700.

[12]  A. Messiah.  Quantum Mechanics , 1961 .

[13]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[14]  E. Davies,et al.  Linear Operators and their Spectra , 2007 .

[15]  Bruce W. Char,et al.  Maple V Language Reference Manual , 1993, Springer US.

[16]  Gegenbauer-solvable quantum chain model , 2010, 1011.4803.

[17]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[18]  Anomalous real spectra of non-Hermitian quantum graphs in strong-coupling regime , 2010, 1003.3738.

[19]  Ali Mostafazadeh,et al.  Pseudo-Hermitian Representation of Quantum Mechanics , 2008, 0810.5643.

[20]  John Stillwell,et al.  Symmetry , 2000, Am. Math. Mon..

[21]  Carl M. Bender,et al.  Making sense of non-Hermitian Hamiltonians , 2007, hep-th/0703096.