Systematic search for -symmetric potentials with real energy spectra

Changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions. Here we relax the standard Hermiticity requirement and find an innovative construction which leads to unusual, complex potentials. Their energy spectrum is shown to stay real after a weakening of the Hermiticity of the Schrodinger equation to its mere invariance under the combined (parity) and (time-reversal) symmetry. This ultimately results in richer bound-state spectra. Some of our new exactly solvable potentials generalize the current textbook models. Details are given for constructions based on the hypergeometric and confluent hypergeometric special functions.

[1]  M. Znojil,et al.  Supersymmetry without hermiticity within PT symmetric quantum mechanics , 2000, hep-th/0003277.

[2]  M. Znojil,et al.  The Coulomb – harmonic oscillator correspondence in symmetric quantum mechanics , 2000, quant-ph/0003081.

[3]  G. Mezincescu.  Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant , 2000, quant-ph/0002056.

[4]  M. Znojil.  Quasi-exactly solvable quartic potentials with centrifugal and Coulombic terms , 2000, math-ph/0002036.

[5]  R. Roychoudhury,et al.  LETTER TO THE EDITOR: A new PT-symmetric complex Hamiltonian with a real spectrum , 2000 .

[6]  Comment on “Conditionally exactly soluble class of quantum potentials” , 1998, quant-ph/9811088.

[7]  M. Znojil.  -symmetrically regularized Eckart, Pöschl-Teller and Hulthén potentials , 1999, quant-ph/9912027.

[8]  M. Znojil.  Exact solution for Morse oscillator in -symmetric quantum mechanics , 1999, quant-ph/9909003.

[9]  C. Bender,et al.  Variational ansatz for PT-symmetric quantum mechanics , 1999, quant-ph/9907008.

[10]  C. Bender,et al.  Complex square well - a new exactly solvable quantum mechanical model , 1999, quant-ph/9906057.

[11]  M. Znojil.  Non-Hermitian matrix description of the calPcalT-symmetric anharmonic oscillators , 1999, quant-ph/9906029.

[12]  M. Znojil.  Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable , 1999, math-ph/9904039.

[13]  R. E. Nettleton.  Shear-induced anisotropy in liquid thermal conduction , 1999 .

[14]  C. Bender,et al.  Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian , 1998, quant-ph/9812039.

[15]  F. Fernández,et al.  A FAMILY OF COMPLEX POTENTIALS WITH REAL SPECTRUM , 1998, quant-ph/9812026.

[16]  C. Bender,et al.  PT-symmetric quantum mechanics , 1998, 2312.17386.

[17]  Orsay,et al.  SUSY Quantum Mechanics with Complex Superpotentials and Real Energy Spectra , 1998, quant-ph/9806019.

[18]  Éric Delabaere,et al.  Eigenvalues of complex Hamiltonians with PT-symmetry. II , 1998 .

[19]  Miloslav Znojil,et al.  Strong-coupling expansions for the -symmetric oscillators , 1998 .

[20]  G. L'evai,et al.  Unified treatment of the Coulomb and harmonic oscillator potentials in D dimensions , 1998, quant-ph/9802012.

[21]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[22]  D. Baye,et al.  Phase-equivalent complex potentials , 1996 .

[23]  B. W. Williams,et al.  “Implicit” potentials associated with Jacobi polynomials: some novel aspects , 1995 .

[24]  A. Khare,et al.  Supersymmetry and quantum mechanics , 1994, hep-th/9405029.

[25]  G. Lévai.  Solvable potentials associated with su(1,1) algebras: a systematic study , 1994 .

[26]  V. Buslaev,et al.  Equivalence of unstable anharmonic oscillators and double wells , 1993 .

[27]  B. W. Williams,et al.  A simple method for generating exactly solvable quantum mechanical potentials , 1993 .

[28]  B. W. Williams,et al.  The generalized Coulomb problem: an exactly solvable model , 1993 .

[29]  C. Bender,et al.  Analytic continuation of eigenvalue problems , 1993 .

[30]  P. Cordero,et al.  Algebraic solution for the Natanzon confluent potentials , 1991 .

[31]  B. W. Williams.  A second class of solvable potentials related to the Jacobi polynomials , 1991 .

[32]  G. Lévai.  A class of exactly solvable potentials related to the Jacobi polynomials , 1991 .

[33]  C. A. Singh,et al.  Single-variable realisation of the SU(1,1) spectrum generating algebra and discrete eigenvalue spectra of a class of potentials , 1990 .

[34]  G. Lévai.  A search for shape-invariant solvable potentials , 1989 .

[35]  J. Ginocchio.  A class of exactly solvable potentials II. The three-dimensional Schrödinger equation , 1985 .

[36]  J. Ginocchio.  A class of exactly solvable potentials. I. One-dimensional Schrödinger equation☆ , 1984 .

[37]  E. Caliceti,et al.  Perturbation theory of odd anharmonic oscillators , 1980 .

[38]  G. Natanzon.  General properties of potentials for which the Schrödinger equation can be solved by means of hypergeometric functions , 1979 .

[39]  E. Sudarshan,et al.  A class of solvable potentials , 1962 .