The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Ei…

Bose-Einstein condensation

In this paper, we mainly review recent results on mathematical theory and
numerical methods for Bose-Einstein condensation (BEC),
based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then
carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for
spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

http://osada.csp.escience.cn/dct/attach/Y2xiOmNsYjpwZGY6NjI5NzY=

Mathematical theory and numerical methods for Bose-Einstein condensation

We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional integer lattice $\mathbb {Z}^d$. We are interested in the spatial decay of the quenched elliptic Green function $G(a;x,y)$. Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $\langle \cdot \rangle $. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $\langle |\nabla _x G(x,y)|^p\rangle $ and $\langle |\nabla _x\nabla _y G(x,y)|^p\rangle $, have the same decay rates in $|x-y|\gg 1$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358–390, 2005), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $G$, that is, $\langle |\nabla _x G(x,y)|^2\rangle $ and $\langle |\nabla _x\nabla _y G(x,y)|\rangle $. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

/pdf/annealed-estimates-on-the-green-function-29omxk2oxa.pdf

Annealed estimates on the Green function

We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional integer lattice $\mathbb{Z}^d$. We are interested in the spatial decay of the quenched elliptic Green function $G(a;x,y)$. Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $\langle\cdot\rangle$. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $\langle|\nabla_x G(x,y)|^p\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|^p\rangle$, have the same decay rates in $|x-y|\gg 1$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel \cite{DeuschelDelmotte}, which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $G$, that is, $\langle|\nabla_x G(x,y)|^2\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|\rangle$. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrodinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for
superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without
any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.

On the Cauchy problem for nonlinear Schrodinger equations with rotation

A mathematical framework for optimal bilinear control of nonlinear Schr\"odinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used $L^2$- or $H^1$-norms for the cost of the control action. Well-posedness of the problem and existence of an optimal control is proven. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

Optimal bilinear control of Gross-Pitaevskii equations

A mathematical framework for optimal bilinear control of nonlinear Schrodinger equations of Gross--Pitaevskii type arising in the description of Bose--Einstein condensates is presented. The obtaine...

/pdf/optimal-bilinear-control-of-gross-pitaevskii-equations-4xmgyxzy6i.pdf

Daniel Marahrens

Papers

Annealed estimates on the Green function

Annealed estimates on the Green function

On the Cauchy problem for nonlinear Schrodinger equations with rotation

Optimal bilinear control of Gross-Pitaevskii equations

Optimal Bilinear Control of Gross--Pitaevskii Equations