On the Divergence of Perturbation Theory. Steps Towards a Convergent Series

The mechanism underlying the divergence of perturbation theory is exposed. This is done through a detailed study of the violation of the hypothesis of the Dominated Convergence Theorem of Lebesgue using familiar techniques of Quantum Field Theory. That theorem governs the validity (or lack of it) of the formal manipulations done to generate the perturbative series in the functional integral formalism. The aspects of the perturbative series that need to be modified to obtain a convergent series are presented. Useful tools for a practical implementation of these modifications are developed. Some resummation methods are analyzed in the light of the above mentioned mechanism.Comment: 42 pages, Latex, 4 figure

The power of perturbation theory

We study quantum mechanical systems with a discrete spectrum. We show that the asymptotic series associated to certain paths of steepest-descent (Lefschetz thimbles) are Borel resummable to the full result. Using a geometrical approach based on the PicardLefschetz theory we characterize the conditions under which perturbative expansions lead to exact results. Even when such conditions are not met, we explain how to define a different perturbative expansion that reproduces the full answer without the need of transseries, i.e. non-perturbative effects, such as real (or complex) instantons. Applications to several quantum mechanical systems are presented

Vertebroplasty: experimental characterization of polymethylmethacrylate bone cement spreading as a function of viscosity, bone porosity, and flow rate

STUDY DESIGN: This is an experimental study on an artificial vertebra model and human cadaveric spine. OBJECTIVE: Characterization of polymethylmethacrylate (PMMA) bone cement distribution in the vertebral body as a function of cement viscosity, bone porosity, and injection speed. Identification of relevant parameters for improved cement flow predictability and leak prevention in vertebroplasty. SUMMARY OF BACKGROUND DATA: Vertebroplasty is an efficient procedure to treat vertebral fractures and stabilize osteoporotic bone in the spine. Severe complications result from bone cement leakage into the spinal canal or the vascular system. Cement viscosity has been identified as an important parameter for leak prevention but the influence of bone structure and injection speed remain obscure. METHODS: An artificial vertebra model based on open porous aluminum foam was used to simulate bone of known porosity. Fifty-six vertebroplasties with 4 different starting viscosity levels and 2 different injection speeds were performed on artificial vertebrae of 3 different porosities. A validation on a human cadaveric spine was executed. The experiments were radiographically monitored and the shape of the cement clouds quantitatively described with the 2 indicators circularity and mean cement spreading distance. RESULTS: An increase in circularity and a decrease in mean cement spreading distance was observed with increasing viscosity, with the most striking change occurring between 50 and 100 Pas. Larger pores resulted in significantly reduced circularity and increased mean cement spreading distance whereas the effect of injection speed on the 2 indicators was not significant. CONCLUSION: Viscosity is the key factor for reducing the risk of PMMA cement leakage and it should be adapted to the degree of osteoporosis encountered in each patient. It may be advisable to opt for a higher starting viscosity but to inject the material at a faster rate

Partial differential matrix equations for the inverse problem of scattering theory

Sufficient conditions for the existence of a continuous translation operator are found in the case of a system of differential equations in which the matrix potential has the singularity of the centripetal term. The sufficient conditions are found in terms of moments of the nuclear potential. The method used employs the Riemann Green's function. Threshold energies introduce a threshold energy dependence into the translation kernel and lead to a requirement of an exponential decrease for terms of the matrix potential