Exact semiclassical expansions for one-dimensional quantum oscillators

A set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrödinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansion

Résurgence de Voros et périodes des courbes hyperelliptiques

The aim of this article is to formulate in a geometrical way the master idea of Voros [ in Ann. Inst. Henri Poincaré, Sect. A 39, 211-238 (1983) ] : the solutions of the one dimensional stationary Schrödinger equation with a polynomial potential are exactly encoded in the complex domain by their WKB expansions (formal divergent expansions in powers of Planck’s constant) in a way which can be read in the geometry of periods of the differential form p d q ( q = position variable, ( p =classicial momentum)

Universal behavior of quantum Green's functions

We consider a general one-particle Hamiltonian H = - \Delta_r + u(r) defined in a d-dimensional domain. The object of interest is the time-independent Green function G_z(r,r') = . Recently, in one dimension (1D), the Green's function problem was solved explicitly in inverse form, with diagonal elements of Green's function as prescribed variables. The first aim of this paper is to extract from the 1D inverse solution such information about Green's function which cannot be deduced directly from its definition. Among others, this information involves universal, i.e. u(r)-independent, behavior of Green's function close to the domain boundary. The second aim is to extend the inverse formalism to higher dimensions, especially to 3D, and to derive the universal form of Green's function for various shapes of the confining domain boundary.Comment: 46 pages, the shortened version submitted to J. Math. Phy

Convergence Radii for Eigenvalues of Tri--diagonal Matrices

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z= 0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq \alpha <11/6.$

Non-perturbative calculations for the effective potential of the PTPT symmetric and non-Hermitian (−gϕ4)(-g\phi^{4}) field theoretic model

We investigate the effective potential of the $PT$ symmetric $(-g\phi^{4})$ field theory, perturbatively as well as non-perturbatively. For the perturbative calculations, we first use normal ordering to obtain the first order effective potential from which the predicted vacuum condensate vanishes exponentially as $G\to G^+$ in agreement with previous calculations. For the higher orders, we employed the invariance of the bare parameters under the change of the mass scale $t$ to fix the transformed form totally equivalent to the original theory. The form so obtained up to $G^3$ is new and shows that all the 1PI amplitudes are perurbative for both $G\ll 1$ and $G\gg 1$ regions. For the intermediate region, we modified the fractal self-similar resummation method to have a unique resummation formula for all $G$ values. This unique formula is necessary because the effective potential is the generating functional for all the 1PI amplitudes which can be obtained via $\partial^n E/\partial b^n$ and thus we can obtain an analytic calculation for the 1PI amplitudes. Again, the resummed from of the effective potential is new and interpolates the effective potential between the perturbative regions. Moreover, the resummed effective potential agrees in spirit of previous calculation concerning bound states.Comment: 20 page

Two-parametric PT-symmetric quartic family

We describe a parametrization of the real spectral locus of the two-parametric family of PT-symmetric quartic oscillators. For this family, we find a parameter region where all eigenvalues are real, extending the results of Dorey, Dunning, Tateo and Shin.Comment: 23 pages, 15 figure

Higher-Order Corrections to Instantons

The energy levels of the double-well potential receive, beyond perturbation theory, contributions which are non-analytic in the coupling strength; these are related to instanton effects. For example, the separation between the energies of odd- and even-parity states is given at leading order by the one-instanton contribution. However to determine the energies more accurately multi-instanton configurations have also to be taken into account. We investigate here the two-instanton contributions. First we calculate analytically higher-order corrections to multi-instanton effects. We then verify that the difference betweeen numerically determined energy eigenvalues, and the generalized Borel sum of the perturbation series can be described to very high accuracy by two-instanton contributions. We also calculate higher-order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with analytic results for the two-instanton effect and with exact data for the first 200 perturbative coefficients.Comment: 7 pages, LaTe

Exponential Type Complex and non-Hermitian Potentials in PT-Symmetric Quantum Mechanics

Using the NU method [A.F.Nikiforov, V.B.Uvarov, Special Functions of Mathematical Physics, Birkhauser,Basel,1988], we investigated the real eigenvalues of the complex and/or $PT$- symmetric, non-Hermitian and the exponential type systems, such as Poschl-Teller and Morse potentials.Comment: 14 pages, Late

Physical Aspects of Pseudo-Hermitian and PTPT-Symmetric Quantum Mechanics

For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT-symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h, the physical observables O, the localized states, and the conserved probability density for the non-Hermitian PT-symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT-symmetric quantum mechanics and clarify its relationship with both the conventional quantum mechanics and the classical mechanics.Comment: 45 pages, 13 figures, 2 table

Topological Expansion and Exponential Asymptotics in 1D Quantum Mechanics

Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological,is constructed for the corresponding Borel functions. Its main property is to order the singularity structure of the Borel plane in a hierarchical way by an increasing complexity of this structure starting from the analytic one. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic and anharmonic potentials. Together with the best approximation provided by the semiclassical series the exponentially small contribution completing the approximation are considered. A natural method of constructing such an exponential asymptotics relied on the Borel plane singularity structures provided by the topological expansion is developed. The method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator.Comment: 46 pages, 22 EPS figure