We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrödinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable. The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space. / eSSENCE

Li Kin-kuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 52-54). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Variational Formulation --- p.10 / Chapter 3 --- The Existence Of A Mountain Pass Solution --- p.12 / Chapter 4 --- Ground States --- p.21 / Chapter 5 --- The Projections Of v And w --- p.35 / Chapter 6 --- Computation Of The Energy: An Upper Bound --- p.37 / Chapter 7 --- Convergence: The First Approximation --- p.40 / Chapter 8 --- Convergence: The Second Approximation --- p.44 / Chapter 9 --- Computation Of The Energy: A Lower Bound --- p.48 / Chapter 10 --- Comparing The Energy: Completion Of The Proof Of Theorem 1.2 --- p.51 / Bibliography --- p.52

Será explorada a versão hidrodinâmica da equação de Schrödinger não-linear e não-local, descrevendo condensados de Bose-Einstein com auto-interações de longo alcance. Tais sistemas têm despertado interesse tendo em vista a busca da realização da condensação de Bose-Einstein sem necessidade de um potencial externo confinante e nos quais as interações atômicas locais não são suficientes. Para obter a descrição hidrodinâmica, a transformação de Madelung para a função de onda será utilizada, reduzindo o problema a uma equação da continuidade e a uma equação de transporte de momentum. Esta última é similar à equação de Euler em fluidos ideais, porém contendo um potencial quântico efetivo e um termo não local, o qual advém da interação atômica. Tais equações de fluido traduzem, respectivamente, a conservação da probabilidade e do momentum total. O método hidrodinâmico permitirá o estudo de excitações elementares, entre os quais os modos de Bogoliubov, segundo uma abordagem macroscópica. / The hydrodynamic version of the Schrödinger equation nonlinear and nonlocal will be explored, describing Bose-Einstein condensates with long-range self-interactions. Such systems have aroused interest with a view to pursuing the realization of Bose-Einstein condensation without an external confining potential and in which local atomic interactions are not enough. For the hydrodynamic description, the eikonal decomposition of the wave function is used, reducing the problem to one equation of continuity and to a transport of momentum equation. The latter is similar to the Euler equation in ideal fluid but containing an effective quantum potential and a nonlocal term, which comes from the atomic interaction. Such fluid equations translate, respectively, conservation of probability and total momentum. The hydrodynamic method will allow the study of elementary excitations, including Bogoliubov modes according to a macroscopic approach.

The DMPO type spin trap 5,5-Dimethyl-1-pyrroline N-oxide (DMPO) and the exceptionally similar spin trap 2-ethoxycarbonyl-2-methyl-3,4-dihydro-2H-pyrrole-l-oxide (EMPO) are widely studied in computational and theoretical works. This particular study examines the addition reactions that both these molecules undergo with the carcinogenic hydroxyl radical. This work used a relatively new approximation method, called the correlation consistent composite approach or ccCA, for carrying out quantum mechanical calculations to give the free energies of the products and reactants of the reactions. The free energies are to be used to extrapolate the rate constants of the reactions from the Arrhenius equation. Though both the spin traps studied have been widely examined and assessed in both theoretical and experimental work, accurately calculated rate constants have not been previously obtained using computational methods. The results obtained here will help to assess the efficiency and the accuracy of the ccCA method, as well as lead to the design of better, more novel spin traps.

Computational chemists are concerned about two aspects when choosing between the myriad of theoretical methodologies: the accuracy (the "truth") and the computational cost (the tractability). Among the least expensive methods are the Hartree-Fock (HF), density functional theory (DFT), and second-order Moller-Plesset perturbation theory (MP2) methods. While each of these methods yield excellent results in many cases, the inadequate inclusion of certain types of electron correlation (either high-orders or nondynamical) can produce erroneous results. The compromise for the computation of noncovalent interactions often comes from empirically scaling DFT and/or MP2 methods to fit benchmark data sets. The DFT method with an empirically fit dispersion term (DFT-D) often yields semi-quantitative results. The spin-component scaled MP2 (SCS-MP2) method parameterizes the same- and opposite-spin correlation energies and often yields less than 20% error for prototype noncovalent systems compared to chemically accurate CCSD(T) results. There is no simple fix for cases with a large degree of nondynamical correlation (such as transition metal-salen complexes). While testing standard and new DFT functionals on the spin-state energy gaps of transition metal-salen complexes, no DFT method produced reliable results compared to very robust CASPT3 results. Therefore each metal-salen complex must be evaluated on a case-by-case basis to determine which methods are the most reliable. Utilizing a combination of DFT-D and SCS-MP2 methods, the reaction mechanism for the addition of cyanide to unsaturated imides catalyzed by the Al(Cl)-salen complex was performed. Various experimental observations are rationalized through this mechanism.

abstract: In the traditional setting of quantum mechanics, the Hamiltonian operator does not depend on time. While some Schrödinger equations with time-dependent Hamiltonians have been solved, explicitly solvable cases are typically scarce. This thesis is a collection of papers in which this first author along with Suslov, Suazo, and Lopez, has worked on solving a series of Schrödinger equations with a time-dependent quadratic Hamiltonian that has applications in problems of quantum electrodynamics, lasers, quantum devices such as quantum dots, and external varying fields. In particular the author discusses a new completely integrable case of the time-dependent Schrödinger equation in R^n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. The author also considers several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. Finally, the author constructs integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2011

Será explorada a versão hidrodinâmica da equação de Schrödinger não-linear e não-local, descrevendo condensados de Bose-Einstein com auto-interações de longo alcance. Tais sistemas têm despertado interesse tendo em vista a busca da realização da condensação de Bose-Einstein sem necessidade de um potencial externo confinante e nos quais as interações atômicas locais não são suficientes. Para obter a descrição hidrodinâmica, a transformação de Madelung para a função de onda será utilizada, reduzindo o problema a uma equação da continuidade e a uma equação de transporte de momentum. Esta última é similar à equação de Euler em fluidos ideais, porém contendo um potencial quântico efetivo e um termo não local, o qual advém da interação atômica. Tais equações de fluido traduzem, respectivamente, a conservação da probabilidade e do momentum total. O método hidrodinâmico permitirá o estudo de excitações elementares, entre os quais os modos de Bogoliubov, segundo uma abordagem macroscópica. / The hydrodynamic version of the Schrödinger equation nonlinear and nonlocal will be explored, describing Bose-Einstein condensates with long-range self-interactions. Such systems have aroused interest with a view to pursuing the realization of Bose-Einstein condensation without an external confining potential and in which local atomic interactions are not enough. For the hydrodynamic description, the eikonal decomposition of the wave function is used, reducing the problem to one equation of continuity and to a transport of momentum equation. The latter is similar to the Euler equation in ideal fluid but containing an effective quantum potential and a nonlocal term, which comes from the atomic interaction. Such fluid equations translate, respectively, conservation of probability and total momentum. The hydrodynamic method will allow the study of elementary excitations, including Bogoliubov modes according to a macroscopic approach.

Made available in DSpace on 2015-05-15T11:46:18Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 899692 bytes, checksum: d08b5615c89c2ad179f1f6dda0a8c410 (MD5) Previous issue date: 2014-06-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study issues related the existence, nonexistence and regularity of solutions to semilinear Schrödinger equations of type u + a(x)u = jujp2u; u 2 H1(RN); where N 2, p > 2 if N = 2 and 2 < p < 2N=(N 2) if N 3 and the potential a(x) is a positive function that belongs to L1(RN). To obtain the results, we use a Linking Theorem and the Principle of Symmetric Criticality. / Neste trabalho, estudamos questões relacionadas a existência, não-existência e regularidade de soluções para equações de Schrödinger semilineares do tipo u + a(x)u = jujp2u; u 2 H1(RN); onde N 2, p > 2 se N = 2 e 2 < p < 2N=(N 2) se N 3 e o potencial a(x) é uma função positiva que pertence a L1(RN). Para obtenção dos resultados, usamos um Teorema de Linking e o Princípio da Criticalidade Simétrica.

Made available in DSpace on 2015-05-15T11:46:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1404779 bytes, checksum: fa23dbf1324f104548ef91fcbbf20fba (MD5) Previous issue date: 2014-09-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study questions related to the existence of positive solutions for some classes of quasilinear Schrödinger equations, with hypotheses on the potential that permit this potential to vanish at infinity. In order to use variational methods to obtain our results, we make some changes of variables to obtain some semilinear equations, whose associated functionals are well defined in a classical Sobolev spaces. We also work with these equations on an Orlicz type space whose energy functional satisfy the geometric properties of the Mountain Pass Theorem. We still use the penalty technique due to Del Pino and Felmer and the Moser iteration method to obtain estimates in L1 norm, which are fundamental to our study. / Neste trabalho, estudamos questões relacionadas à existência de soluções positivas para algumas classes de equações de Schrödinger quasilineares, com hipóteses sobre o potencial que o possibilita se anular no infinito. Afim de usarmos métodos variacionais na obtenção de nossos resultados, aplicamos mudança de variáveis para reduzirmos as equações quasilineares a equações semilineares. Os funcionais associados a essas novas equações estão bem definidos em espaços de Sobolev clássicos e em espaços tipo Orlicz e satisfazem as propriedades geométricas do Teorema do Passo da Montanha. Ainda utilizamos a técnica de penalização de Del Pino e Felmer e o método de iteração de Moser para obtenção de estimativas, fundamentais para o nosso estudo, na norma L1.

Será explorada a versão hidrodinâmica da equação de Schrödinger não-linear e não-local, descrevendo condensados de Bose-Einstein com auto-interações de longo alcance. Tais sistemas têm despertado interesse tendo em vista a busca da realização da condensação de Bose-Einstein sem necessidade de um potencial externo confinante e nos quais as interações atômicas locais não são suficientes. Para obter a descrição hidrodinâmica, a transformação de Madelung para a função de onda será utilizada, reduzindo o problema a uma equação da continuidade e a uma equação de transporte de momentum. Esta última é similar à equação de Euler em fluidos ideais, porém contendo um potencial quântico efetivo e um termo não local, o qual advém da interação atômica. Tais equações de fluido traduzem, respectivamente, a conservação da probabilidade e do momentum total. O método hidrodinâmico permitirá o estudo de excitações elementares, entre os quais os modos de Bogoliubov, segundo uma abordagem macroscópica. / The hydrodynamic version of the Schrödinger equation nonlinear and nonlocal will be explored, describing Bose-Einstein condensates with long-range self-interactions. Such systems have aroused interest with a view to pursuing the realization of Bose-Einstein condensation without an external confining potential and in which local atomic interactions are not enough. For the hydrodynamic description, the eikonal decomposition of the wave function is used, reducing the problem to one equation of continuity and to a transport of momentum equation. The latter is similar to the Euler equation in ideal fluid but containing an effective quantum potential and a nonlocal term, which comes from the atomic interaction. Such fluid equations translate, respectively, conservation of probability and total momentum. The hydrodynamic method will allow the study of elementary excitations, including Bogoliubov modes according to a macroscopic approach.