We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to "cure" the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary which include instanton configurations, characterized by nonanalytic factors exp(−a/g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a nonanalytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det(H − E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function. The same strategy could result in new conjectures for problems where our present understanding is more limited.

In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton constributions to the partition function, using the formalism introduced in the first part of the treatise [J. Zinn-Justin and U. D. Jentschura, e-print quantph/0501136]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker-Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is nonzero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton Afunction to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.

Usually, photons are described by plane waves with a definite 4-momentum. In addition to plane-wave photons, "twisted photons" have recently entered the field of modern laser optics; these are coherent superpositions of plane waves with a defined projection m of the orbital angular momentum onto the propagation axis, where m is integer. In this paper, we show that it is possible to produce high-energy twisted photons by Compton backscattering of twisted laser photons off ultra-relativistic electrons. Such photons may be of interest for experiments related to the excitation and disintegration of atoms and nuclei, and for studying the photo-effect and pair production off nuclei in previously unexplored experimental regimes.PACS numbers: 13.60. Fz, 42.65.Ky, 27.70.Jj, 12.20.Ds Introduction.-An interesting research direction in modern optics is related to experiments with so-called "twisted photons." These are states of the laser beam whose photons have a defined value m of the angular momentum projection on the beam propagation axis where m is a (large) integer [1]. An experimental realization [2] exists for states with projections as large as m = 200. Such photons can be created from usual laser beams by means of numerically computed holograms. The wavefront of such states rotates around the propagation axis, and their Poynting vector looks like a corkscrew (see Fig. 1 in Ref. [1]). It was demonstrated that micron-sized teflon and calcite "particles" start to rotate after absorbing twisted photons [3].In this Letter, we show that it is possible to convert twisted photons from an energy range of about 1 eV to a higher energies of up to a hundred GeV using Compton backscattering off ultra-relativistic electrons. In principle, Compton backscattering is an established method for the creation of high-energy photons and is used successfully in various application areas from the study of photonuclear reactions [4,5] to colliding photon beams of high energy [6]. However, the central question is how to treat Compton backscattering of twisted photons, whose field configuration is manifestly different from plane waves. Below, we use relativistic Gaussian units with c = 1, = 1, α ≈ 1/137. We denote the electron mass by m e and write the scalar product of 4-vectors k = (ω, k) andTwisted photon.-We wish to construct a twisted photon state with definite longitudinal momentum k z , absolute value of transverse momentum κ and projection m of the orbital angular momentum onto the z axis (propagation axis). We start from a plane-wave photon state with 4-momentum k = (ω, k) and helicity Λ = ±1,