Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 015, 9 pages      arXiv:1002.0798
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

Kwang C. Shin
Department of Mathematics, University of West Georgia, Carrollton, GA, 30118, USA

Received October 11, 2009, in final form January 28, 2010; Published online February 03, 2010

We study the eigenvalue problem −u''+V(z)uu in the complex plane with the boundary condition that u(z) decays to zero as z tends to infinity along the two rays arg z=−π/2± 2π(m+2), where V(z)=−(iz)mP(iz) for complex-valued polynomials P of degree at most m−1≥2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.

Key words: anharmonic oscillators; asymptotic formula; infinitely many real eigenvalues; PT-symmetry.

pdf (232 kb)   ps (158 kb)   tex (12 kb)


  1. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT-symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  2. Bender C.M., Hook D.W., Mead L.R., Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra, J. Phys. A: Math. Theor. 41 (2008), 392005, 9 pages, arXiv:0807.0424.
  3. Caliceti E., Graffi S., Sjöstrand J., PT symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum, J. Phys. A: Math. Theor. 40 (2007), 10155-10170, arXiv:0705.4218.
  4. Conway J.B., Functions of one complex variable. I, 2nd ed., Springer-Verlag, New York, 1995.
  5. Dorey P., Dunning C., Tateo R., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 34 (2001), 5679-5704, hep-th/0103051.
  6. Hille E., Lectures on ordinary differential equations, Addison-Wesley Publ. Co., Reading, Massachusetts, 1969.
  7. Lévai G., Siegl P., Znojil M., Scattering in the PT-symmetric Coulomb potential, J. Phys. A: Math. Theor. 42 (2009), 295201, 9 pages, arXiv:0906.2092.
  8. Mostafazadeh A., Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205-214, math-ph/0107001.
  9. Shin K.C., On the reality of the eigenvalues for a class of PT-symmetric oscillators, Comm. Math. Phys. 229 (2002), 543-564, math-ph/0201013.
  10. Shin K.C., Eigenvalues of PT-symmetric oscillators with polynomial potentials, J. Phys. A: Math. Gen. 38 (2005), 6147-6166, math.SP/0407018.
  11. Shin K.C., Asymptotics of eigenvalues of non-self adjoint Schrödinger operators on a half-line, Comput. Methods Funct. Theory, to appear, arXiv:1001.5120.
  12. Sibuya Y., Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, Vol. 18, North-Holland Publishing Co., Amsterdam - Oxford, 1975.
  13. Znojil M., Siegl P., Lévai G., Asymptotically vanishing PT-symmetric potentials and negative-mass Schrödinger equations, Phys. Lett. A 373 (2009), 1921-1924, arXiv:0903.5468.

Previous article   Next article   Contents of Volume 6 (2010)