Abstract. This work is concerned with asymptotic approximations of the semi-classical Schrödinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Blochband based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-functions, we prove that the first-order Gaussian beam superposition converges to the original wave field at a rate of ǫ 1/2 , with ǫ the semiclassically scaled constant, as long as the initial data for Gaussian beam components in each band are prepared with same order of error or smaller. For a natural choice of initial approximation, a rate of ǫ 1/2 of initial error is verified. IntroductionWe consider the semiclassically scaled Schrödinger equation with a periodic potential:subject to the two-scale initial condition:where Ψ(t, x) is a complex wave function, ε is the re-scaled Planck constant, V e (x)-smooth external potential, S 0 (x)-real-valued smooth function, g(x, y) = g(x, y + 2π)-smooth function, compactly supported in x, i.e., g(x, y) = 0, x ∈ K 0 , K 0 -is a bounded set. V (y) is periodic with respect to the crystal lattice Γ = (2πZ) d , it models the electronic potential generated by the lattice of atoms in the crystal [14].A typical application arises in solid state physics where (1.1) describes the quantum dynamics of Bloch electrons moving in a crystalline lattice (generated by the ionic cores) [39]. The asymptotics of (1.1) as ε → 0+ is a well-studied two-scale problem in the physics and mathematics literature [8,16,20,41,23,13,33,1,14]. On the other hand, the computational challenge because of the small parameter ε has prompted a search for asymptotic model based numerical methods, see e.g., [29,37].1991 Mathematics Subject Classification. 35A21, 35A35, 35Q40.
I would like to take this opportunity to express my gratitude to those who helped me with conducting my research and writing this thesis. First of all, my advisor Prof. Hailiang Liu for his guidance, patience, support and keeping me motivated to work hard. I learned a lot from him during my graduate study and this knoweledge will help me in the future work. I appreciate his time and energy he devoted for my training during these years. Also I want to thank my coleagues and classmates, in particular, our members of the Focused Research Interaction seminar organized by Prof. Liu: Dr. Hui Yu, Nattapol Ploymaklam and Yongki Lee; presenting our research and communicating with each other helped me to understand my research topic better. I appreciate all suggestions and comments from my committee members, in particular Prof. Paul Sacks for his careful review and also for his deep questions which would be interesting to address in the future. I would like to thank all my ommittee members Prof. G. Lieberman, Prof. A. Roiterstein and Prof. M. Smiley. I feel very grateful to the Department of Mathematics of Iowa State University for providing me this life changing opportunity for graduate study and work. Finally, I would like to thank the members of "SQuaRE" research group, which is working on Gaussian beam methods. In particular,
This work is concerned with asymptotic approximations of the semi-classical Schrödinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Blochband based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-functions, we prove that the first-order Gaussian beam superposition converges to the original wave field at a rate of ǫ 1/2 , with ǫ the semiclassically scaled constant, as long as the initial data for Gaussian beam components in each band are prepared with same order of error or smaller. For a natural choice of initial approximation, a rate of ǫ 1/2 of initial error is verified.
In recent years multiple papers on borehole torsion, well curvature, and well profile energy have been published, highlighting the potential value in describing the well path with these new indices. (Samuel et al. 2021; Samuel and Zhang 2019; Samuel and Liu 2009; Samuel et al. 2005) In practice, these well path calculations have proven more challenging to calculate in comparison to more common measures such as dogleg severity. To enable the use of these new metrics, a web application was developed to make the outputs accessible to drilling engineers globally. Once available, the use of torsion and curvature aids in interpreting and anticipating downhole trajectory related issues and assists in differentiating between borehole cleaning and trajectory problems. This paper presents an overview of the web application, the method used for calculation, and two case studies demonstrating the real time application of torsion and torsion/curvature ratio representing how they provide additional insight beyond what is available from evaluating dogleg severity alone. A real-time methodology has been implemented by combining operational experience with multiple trajectory metrics. Proactive wellpath evaluation proved to be key in identifying measured depth intervals with higher likelihood for operational challenges during casing running and tripping operations in Permian Delaware Basin laterals.
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