Entire functions, PT-symmetry and Voros’s quantization scheme

  • A.E. Eremenko Department of Mathematics, Purdue University West Lafayette, IN
Keywords: entire functions; P T -symmetry; exact quantization scheme; Stokes multipliers

Abstract

In this paper, A. Avila's theorem
on convergence of the exact quantization scheme of A.~Vo\-ros
is related to the reality proofs of eigenvalues of certain $PT$-symmetric
boundary value problems.
As a result, a special case
of a conjecture of C. Bender, S. Boettcher
and P. Meisinger on reality of eigenvalues is proved.
In particular the following Theorem~2 is proved:
{\sl Consider the eigenvalue problem
$$-w''+(-1)^\ell(iz)^mw=\lambda w,$$
where $m\geq 2$ is real, and $(iz)^m$ is the principal branch,
$(iz)^m>0$ when $z$ is on the negative imaginary ray,
with boundary conditions $w(te^{i\beta})\to 0,\ t\to\infty,$
where
$ \beta=\pi/2\pm\frac{\ell+1}{m+2}\pi.$
If $\ell=2$, and $m\geq 4$, then all eigenvalues are positive.
}\

References

A. Avila, Convergence of the exact quantization scheme, Comm. Math. Phys., 240 (2004), 305–318.
C. Bender, S. Boettcher, P. Meisinger, P T -symmetric quantum mechanics, J. Math. Phys., 40 (1999), 2201–2229.
C. Bender, S. Boettcher, P. Meisinger, Conjecture on reality of spectra of non-Hermitian Hamiltonians, in: W. Janke et all, eds, Fluctuating Paths and Fields, World Scientific, Singapore, 2001.
W. Bergweiler, A. Eremenko, A. Hinkkanen, Entire functions with two radially distributed values, preprint.
G. Gundersen, Questions on meromorphic functions and complex differential equations, preprint.
P. Dorey, C. Dunning, R. Tateo, On the relation of the Stokes multipliers and the T − Q systems of conformal field theory, Nuclear Physics B, 563 (1999), 573–602.
P. Dorey, C. Dunning, R. Tateo, Spectral equivalences, Bethe Ansatz equations, and reality properties in P T -symmetric quantum mechanics, J. Phys. A: Math. Gen., 34 (2001), 5679.
P. Dorey, C. Dunning, R. Tateo, The ODE/IM correspondences, J. Phys. A: Math. Theor., 40 (2007) R205–R283.
M. Fedoryuk, Asymptotic analysis. Linear ordinary differential equations, Springer-Verlag, Berlin, 1993.
K. Shin, On the reality of the eigenvalues for a class of P T -symmetric oscillators, Comm. math. phys., 229 (2002), 543–564.
K. Shin, Eigenvalues of P T -symmetric oscillators with polynomial potentials, J. Phys. A, 38 (2005), No27, 6147–6166.
K. Shin, The potential (iz)m generates real eigenvalues only, under symmetric rapid decay boundary conditions, J. Math. Phys., 46 (2005), 082110, 1–17.
K. Shin, Asymptotics of eigenvalues of non-self-adjoint Schrödinger operators on a half-line, Comput. Methods Funct. Theory, 10 (2010), No1, 111–133.
K. Shin, Anharmonic oscillators in the complex plane, P T -symmetry, and real eigenvalues, Potential Anal., 35 (2011), No2, 145–174.
Y. Sibuya, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North Holland, 1975.
Y. Sibuya, Non-trivial entire solutions of the functional equation f (λ) + f (ωλ)f (ω −1 λ) = 1, ω5 = 1, Analysis, 8 (1998), 271–295.
Y. Sibuya, R. Cameron, An entire solution of the functional equation f (λ) + f (ωλ)f (ω −1λ) = 1, Lecture Notes Math., Springer, Berlin, 312 (1973), 194–202.
T. Tabara, Asymptotic behavior of Stokes multipliers for y00 − (xσ + λ)y = 0, (σ ≥ 2) as λ → ∞, Dynamics of continuous, discrete and impulsive systems, 5 (1999), 93–105.
A. Voros, Exact quantization condition for anharmonic oscillators (in one dimension), J. Phys. A: Math. Gen., 27 (1994) 4653–4661.
A. Voros, Exact anharmonic quantization condition (in one dimension). In: Quasiclassical methods, (Minneapolis, MN, 1995), IMA Vol. Math. Appl., New York: Springer, 95 (1997), 189–224.
Published
2020-12-25
How to Cite
1.
Eremenko A. Entire functions, PT-symmetry and Voros’s quantization scheme. Mat. Stud. [Internet]. 2020Dec.25 [cited 2021Nov.28];54(2):203-10. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/161
Section
Articles