# Rate of convergence towards Hartree dynamics

Save this PDF as:

Size: px
Start display at page:

Download "Rate of convergence towards Hartree dynamics"

## Transcription

1 Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski

2 1. Introduction boson system: quantum mechanical boson systems are described by a wave function ψ L 2 s (R3 ), symmetric w.r.t. permutations. ψ (x 1,..., x ) 2 = probability density ψ = 1 The dynamics is governed by the Schrödinger equation i t ψ,t = H ψ,t ψ,t = e ih t ψ. H is the Hamiltonian of the system, H = xj + i<j V (x i x j ) acts on L 2 s (R 3 ).

3 Mean field systems: boson system with Hamiltonian H = xj + 1 i<j V (x i x j ) on L 2 s (R 3 ) Evolution of a condensate: fix ϕ L 2 (R 3 ), consider the initial -particle state ψ (x) = ϕ(x j ) and its time-evolution ψ,t = e ih t ψ If factorization is approximately preserved ψ,t (x) ϕ t (x j ) 1 V (x i x j ) 1 V (x i y) ϕ t (y) 2 dy = (V ϕ t 2 )(x i ) j i Hartree equation: i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t

4 Marginal densities: let ψ,t = e ih t ψ and let γ,t = ψ,t ψ,t γ,t (x; x ) = ψ,t (x)ψ,t (x ) denote the density matrix associated with ψ,t. For k = 1,...,, define the k-particle marginal density γ (k),t (x k; x k ) = dx k γ,t (x k, x k ; x k, x k) with x k = (x 1,..., x k ), x k = (x 1,..., x k ), x k = (x k+1,..., x ) We use the convention Tr γ (k),t = 1 for all k,, t. For every k-particle observable J (k) on L 2 (R 3k ) ψ,t, ( J (k) 1 ( k)) ψ,t = Tr ( J (k) 1 ( k)) γ,t = Tr J (k) γ (k),t

5 Theorem (under appropriate assumptions on V ): Consider the factorized initial wave function ψ (x) = ϕ(x j ) for arbitrary ϕ H 1 (R 3 ) with ϕ = 1 and let γ (k),t be the k-particle marginal ass. with ψ,t = e ih t ψ. Then, for every fixed k 1 and t R, we have γ (k),t ϕ t ϕ t k as (in the trace class norm). Here ϕ t is the solution of the nonlinear Hartree equation with initial data ϕ t=0 = ϕ. i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t In particular, for every fixed k, and every bounded J (k), as. ψ,t, ( J (k) 1 ( k)) ψ,t ϕ k t, J (k) ϕ k t

6 Proof of the theorem: Spohn, 1980: bounded potential. Erdös-Yau, 2000: derivation of Hartree for Coulomb potential (partial results also by Bardos-Golse-Mauser). Elgart-S., 2005: derivation of relativistic Hartree i t ϕ t = ( ) 1 1 ϕ t + λ. ϕ t 2 ϕ t starting from H = 1 j + λ i<j 1 x i x j for all λ > 4/π (equation is critical) and ϕ H 1/2 (R 3 ).

7 Let H = j + 1 i<j 3β V ( β (x i x j )) Erdös-S.-Yau, 2006: for β < 1/2, derivation of nonlinear Schrödinger equation i t ϕ t = ϕ t + b 0 ϕ t 2 ϕ t with b 0 = dxv (x) Analogous results in 1-dim by Adami-Bardos-Golse-Teta. Erdös-S.-Yau, 2007: for β 1, derivation of i t ϕ t = ϕ t + σ ϕ t 2 ϕ t with σ = { b0 if β < 1 8πa 0 if β = 1

8 Spohn s strategy: study BBGKY hierarchy γ (k),t i t γ (k),t = k + [ j, γ (k),t ( 1 k ) k ] + 1 k i<j [ V (x i x j ), γ (k),t [ Tr k+1 V (x j x k+1 ), γ (k+1) is compact it has at least one limit point γ(k),t Limit points {γ (k),t } k 1 satisfy the infinite hierarchy i t γ (k),t = k Observe: γ (k) t [ j, γ (k),t ] + k ],t [ Tr k+1 V (x j x k+1 ), γ (k+1) = ϕ t ϕ t k, k 1 solves inf. hier. iff i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t Proving the uniqueness of the infinite hierarchy, it follows that γ (k),t ϕ t ϕ t k for all k, t ],t ]

9 2. Rate of convergence towards Hartree dynamics. Question: What can be said about the error γ (1),t ϕ t ϕ t? Expanding the solution of the BBGKY hierarchy, one gets Tr γ(1),t ϕ t ϕ t C 1 for t < T 0 Iterating only leads to very bad bounds Tr γ(1),t ϕ t ϕ t C 1 2 t

10 Theorem (Rodnianski-S., 2007): suppose that V 2 (x) const (1 ) Let ψ (x) = ϕ(x j ), for ϕ H 1 (R 3 ) and let ψ,t = e ih t ψ Moreover let ϕ t denote the solution of the Hartree equation i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t Then there exists constants C, K > 0 such that, for all t R, Tr γ(1),t ϕ t ϕ t C ekt 1/4 Remarks: we can prove analogous bounds for k-particle marginal density. V (x) = λ/ x is allowed for arbitrary λ R. dependence is not optimal (1/ is expected).

11 Fock space representation (Hepp (1973), Ginibre-Velo (1979)): Bosonic Fock space: F = n 0 L 2 s (R 3n, dx 1... dx n ). Vectors in F are sequences ψ = {ψ (n) } n 1 with ψ (n) L 2 s (R 3n ). Special vectors: vectors like {0, 0,..., 0, ψ n, 0,... } F with ψ n L 2 (R 3n ) have a fixed number of particles (example Ω = {1, 0,... }). umber of particle operator: defined by ( ψ) (m) = mψ (m). Hamiltonian: we define (H ψ) (m) = H (m) ψ(m) with ote that H (m) = m xj + 1 m i<j V (x i x j ) e ith {0, 0,..., ψ, 0,... } = {0, 0,..., e ith ψ, 0,... }

12 Creation and annihilation operators: for f L 2 (R 3 ), define ( a (f)ψ ) (n) (x1,..., x n ) = 1 n n (a(f)ψ) (n) (x 1,..., x n ) = n + 1 f(x j )ψ (n 1) (x 1,..., ˆx j,..., x n ) dx f(x)ψ (n+1) (x, x 1,..., x n ) CCR: [a(f), a (g)] = (f, g) L 2 [a(f), a(g)] = [a (f), a (g)] = 0 Introduce also the operator-valued distributions a x, a x s.t. a (f) = dx f(x)a x and a(f) = dx f(x) a x CCR: [a x, a y] = δ(x y) [a x, a y ] = [a x, a y] = 0

13 Then we have the representations = H = dx a xa x dx x a x x a x + 1 dxdyv (x y)a xa ya y a x Creation and annihilation operators are not bounded, but a(f)ψ f 1/2 ψ and a (f)ψ f ( +1) 1/2 ψ States in F can be obtained applying creation operators on the vacuum. For example, factorized states are given by ψ (x) = ϕ(x j ) {0,..., 0, ψ, 0,... } = a (ϕ)! Ω

14 Coherent states: for ϕ L 2 (R 3 ) define the Weyl operator W (ϕ) = exp(a (ϕ) a(ϕ)) The coherent state with wave function ϕ is then given by W (ϕ)ω = e ϕ 2 /2 j=0 a (ϕ) j Ω j! W (ϕ) = W (ϕ) 1 = W ( ϕ) W (ϕ)ω, W (ϕ)ω = ϕ 2 We have W (ϕ) a x W (ϕ) = a x + ϕ(x), W (ϕ) a x W (ϕ) = a x + ϕ(x) a x W (ϕ)ω = ϕ(x) W (ϕ)ω

15 Theorem (dynamics of coherent states): Suppose that V 2 (x) C(1 ) Choose ϕ H 1 (R 3 ), ϕ = 1. Consider the coherent initial state W ( { ϕ)ω = e /2 1, ϕ,..., j/2 } ϕ j,... j! and let Γ (1),t be the 1-part. density ass. with ψ,t = e ih t W ( ϕ) Ω. where ϕ t solves Tr Γ(1),t ϕ t ϕ t C ekt i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t with ϕ t=0 = ϕ.

16 One-particle density: in general, the kernel of the one-particle density associated with ψ F is given by γ (1) 1 ψ (x; y) = ψ, ψ ψ, a ya x ψ Trγ (1) ψ = 1 Therefore, we are interested in Γ (1),t (x, y) = 1 e ih t W ( ϕ)ω, a y a x e ih t W ( ϕ)ω We expand = 1 Ω, W ( ϕ)e ih t a y a x e ih t W ( ϕ)ω e ih t a ye ih t = ϕ t (y) + e ih t (a y ϕ t (x))e ih t e ih t a x e ih t = ϕ t (x) + e ih t (a x ϕ t (x))e ih t around the solution of i t ϕ t = ϕ t + (V ϕ t 2 )ϕ t

17 Fluctuation dynamics: it was observed by Hepp that W ( ϕ)e ih t ( a y ϕ t (y) ) e ih t W ( ϕ) = U (t) a y U (t) where i t U (t) = L (t)u (t) with U (0) = 1 with time-dependent generator L (t) = + dx ( x a x x a x + (V ϕ t 2 )(x)a xa x ) dxdy V (x y)ϕ t (x)ϕ t (y) a xa y + dxdy V (x y) ( ϕ t (x)ϕ t (y) a xa ) y + ϕ t (x)ϕ t (y)a x a y + 1 dxdy V (x y) a ( x ϕt (y)a y + ϕ t (y)a y) ax + 1 dxdy V (x y) a x a y a ya x

18 Therefore Γ (1),t (x, y) ϕ t(x)ϕ t (y) = 1 Ω, U (t)a y a x U (t)ω + ϕ t(x) Ω, U (t)a yu (t)ω + ϕ t(y) Ω, U (t)a xu (t)ω Main problem: show that Ω, U (t) U (t)ω Ce Kt

19 First attempt: d dt Ω,U (t) U (t)ω = Ω, U (t)[il (t), ]U (t)ω = 2 dxdy V (x y) ( ϕ t (x)ϕ t (y) Ω, U (t)a xa yu (t)ω + c.c. ) + 1/2 dxdyv (x y) ( ϕ t (y) Ω, U (t)a xa ya x U (t)ω + c.c. ) (since L (t) = + dx ( x a x x a x + (V ϕ t 2 )(x)a xa x ) dxdy V (x y)ϕ t (x)ϕ t (y) a xa y + dxdy V (x y) ( ϕ t (x)ϕ t (y) a x a y + ϕ ) t(x)ϕ t (y)a x a y + 1 dxdy V (x y) a ( x ϕt (y)a y + ϕ t (y)a y) ax + 1 dxdy V (x y) a xa ya y a x ) d dt Ω, U (t) U (t)ω C Ω, U (t) U (t)ω

20 Instead: define new Hamiltonian with cutoff in cubic term L (t) = + dx ( x a x xa x + (V ϕ t 2 )(x)a x a x dxdy V (x y)ϕ t (x)ϕ t (y) a xa y + dxdy V (x y) ( ϕ t (x)ϕ t (y) a xa ) y + ϕ t (x)ϕ t (y)a x a y + 1 dxdy V (x y) ( ϕ t (y) a xa y χ( ) a x + h.c. ) + 1 dxdy V (x y) a xa ya y a x and the unitary evolution i Ũ(t) = L (t)ũ(t) with Ũ(0) = 1 Conclude proof by showing that Ω, Ũ (t) Ũ(t)Ω Ce Kt ) Ω, U (t) U (t)ω C

21 Remark: Hepp (1973) and Ginibre-Velo (1979) proved that where s lim U (t) = U(t) with generator L(t) = + + i t U(t) = L(t)U(t) with U(0) = 1 dx ( x a x x a x + (V ϕ t 2 )(x)a xa x ) dxdy V (x y)ϕ t (x)ϕ t (y) a x a y dxdy V (x y) ( ϕ t (x)ϕ t (y) a xa ) y + ϕ t (x)ϕ t (y)a x a y They only need to control the growth of the number of particles w.r.t. the limiting dynamics L(t), in the sense Ω, U (t) U(t)Ω Ce Kt

22 From coherent states to factorized states: note that {0,..., 0, ϕ, 0,... } = c 2π 0 dθ 2π eiθ W ( e iθ ϕ)ω with c =! e /2 1/4 /2 In fact, using that we get 2π 0 W ( e iθ ϕ)ω = e /2 j 0 dθ 2π eiθ W ( e iθ ϕ)ω = e /2 j/2 j 0 j! = e /2 /2! e ijθ (a (ϕ)) j Ω j/2 j! a (ϕ) j Ω a (ϕ) Ω! 2π = 1 c {0, 0,..., 0, ϕ, 0,... } 0 dθ 2π ei( j)θ

23 Let γ (1),t be the 1-part. dens. of {0,.., 0, e ih t ϕ, 0,..}, then γ (1),t (x, y) = 1 {0,.., 0, ϕ, 0,..}, e ih t a ya x e ih t {0,.., 0, ϕ, 0,..} = c2 2π 2π dθ 1 dθ 2 0 2π 0 2π e iθ 1e iθ 2 Ω, W ( e iθ 1ϕ)e iht a y a x e iht W ( e iθ 2ϕ)Ω Expanding e ih t a y e ih t = e iθ 1 ϕ t (y) + e ih t ( a y e iθ 1ϕ t (y) ) e ih t e ih t a x e ih t = e iθ 2 ϕ t (x) + e ih t ( a x e iθ 2ϕ t (x) ) e ih t and using that c 2π 0 dθ 2π ei( 1)θ W ( e iθ ϕ)ω = ϕ ( 1)

24 we obtain γ (1),t (x, y) ϕ t(x)ϕ t (y) = c2 dθ1 dθ 2 (2π) 2 e iθ 1e iθ 2 Ω, W ( e iθ 1ϕ)e ih t ( a y e iθ 1ϕ t (y) ) + c ϕ t (y) ( a x e iθ 2ϕ t (x) ) W ( e iθ 2ϕ)Ω dθ2 (2π) eiθ 2 ϕ ( 1), e ih t (a x e iθ 2ϕ t (x))e ih t W ( e iθ 2ϕ)Ω + c ϕ t (x) dθ1 (2π) e iθ 1 Ω, W ( e iθ 1ϕ)e ih t (a y e iθ 1 ϕ t (y))e ih t ϕ ( 1) This implies the theorem for the evolution of factorized states.

25 Why V 2 (1 )? In the commutator [il (t), ] we have to deal with terms like because dxdy V (x y)ϕ t (x)ϕ t (y) Ω, U (t) a x a y U (t)ω = dxϕ t (x) a x U (t)ω, a (V (x.)ϕ t )U (t)ω dx ϕ t (x) a x U (t)ω a (V (x.)ϕ t )U (t)ω sup x V (x.)ϕ t L 2 ( + 1) 1/2 U (t)ω 2 a (f)ψ f L 2 ( + 1) 1/2 ψ With the assumption V 2 (1 ), we can bound V (x.)ϕ t 2 = uniformly in time. dyv 2 (x y) ϕ t (y) 2 ϕ t H 1 const

### DISTRIBUTIONS AND FOURIER TRANSFORM

DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### Time Ordered Perturbation Theory

Michael Dine Department of Physics University of California, Santa Cruz October 2013 Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field.

### Systems with Persistent Memory: the Observation Inequality Problems and Solutions

Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +

### Derivation of Hartree and Bogoliubov theories for generic mean-field Bose gases

Derivation of Hartree and Bogoliubov theories for generic mean-field Bose gases Mathieu LEWIN Mathieu.Lewin@math.cnrs.fr (CNRS & Université de Cergy-Pontoise) joint works with P.T. Nam (Cergy), N. Rougerie

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### 7 - Linear Transformations

7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

### Chapter 5. Banach Spaces

9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on

### The Essentials of Quantum Mechanics

The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum

### THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Small divisors for quasiperiodic linear equations

linear s Small divisors for linear s Institut Mathématique de Jussieu (Institut Mathématique de Jussieu) Small divisors for linear s 1 / 16 linear s 1 2 3 (Institut Mathématique de Jussieu) Small divisors

### The integrating factor method (Sect. 2.1).

The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable

### Harmonic Oscillator Physics

Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the time-independent Schrödinger equation: d ψx

### Derivation of Mean Field Equations for Classical Systems

Derivation of Mean Field Equations for Classical Systems Master Thesis iklas Boers Derivation of Mean Field Equations for Classical Systems Master Thesis iklas Boers Fakultät für Mathematik, Informatik

### Iterative Techniques in Matrix Algebra. Jacobi & Gauss-Seidel Iterative Techniques II

Iterative Techniques in Matrix Algebra Jacobi & Gauss-Seidel Iterative Techniques II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin

### Section 12.6: Directional Derivatives and the Gradient Vector

Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### Extremal equilibria for reaction diffusion equations in bounded domains and applications.

Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,

### 1 Variational calculation of a 1D bound state

TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,

### A Note on Di erential Calculus in R n by James Hebda August 2010.

A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the

### Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

### Mixed states and pure states

Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

### 1 Fixed Point Iteration and Contraction Mapping Theorem

1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Applications of Fourier series

Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=

### GAUSSIAN MEASURES OF DILATIONS OF CONVEX ROTATIONALLY SYMMETRIC SETS IN C n. (2π) n exp (x 2 k + y2 k ) dx 1 dy 1... dx n dy n,

GAUSSIAN MEASURES OF DILATIONS OF CONVEX ROTATIONALLY SYMMETRIC SETS IN C n TOMASZ TKOCZ Abstract. We consider the complex case of the S-inequality. It concerns the behaviour of Gaussian measures of dilations

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### Inner products on R n, and more

Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### , < x < Using separation of variables, u(x, t) = Φ(x)h(t) (4) we obtain the differential equations. d 2 Φ = λφ (6) Φ(± ) < (7)

Chapter1: Fourier Transform Solutions of PDEs In this chapter we show how the method of separation of variables may be extended to solve PDEs defined on an infinite or semi-infinite spatial domain. Several

### 1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

### University of Ostrava. Fuzzy Transforms

University of Ostrava Institute for Research and Applications of Fuzzy Modeling Fuzzy Transforms Irina Perfilieva Research report No. 58 2004 Submitted/to appear: Fuzzy Sets and Systems Supported by: Grant

### Class Meeting # 1: Introduction to PDEs

MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### 1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute

Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### Lecture 18 Time-dependent perturbation theory

Lecture 18 Time-dependent perturbation theory Time-dependent perturbation theory So far, we have focused on quantum mechanics of systems described by Hamiltonians that are time-independent. In such cases,

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Prof. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy.

Homework 13 Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y. 13.1. Prove or give a counterexample. f preserves convergent sequences if {x n } n=1 is a convergent sequence in X then

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### The Alternating Series Test

The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other

### Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

### Lecture 12 Basic Lyapunov theory

EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12

### Some ergodic theorems of linear systems of interacting diffusions

Some ergodic theorems of linear systems of interacting diffusions 4], Â_ ŒÆ êæ ÆÆ Nov, 2009, uà ŒÆ liuyong@math.pku.edu.cn yangfx@math.pku.edu.cn 1 1 30 1 The ergodic theory of interacting systems has

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Quasi-static evolution and congested transport

Quasi-static evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### 22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three

### Distributions, the Fourier Transform and Applications. Teodor Alfson, Martin Andersson, Lars Moberg

Distributions, the Fourier Transform and Applications Teodor Alfson, Martin Andersson, Lars Moberg 1 Contents 1 Distributions 2 1.1 Basic Definitions......................... 2 1.2 Derivatives of Distributions...................

### Cylinder Maps and the Schwarzian

Cylinder Maps and the Schwarzian John Milnor Stony Brook University (www.math.sunysb.edu) Bremen June 16, 2008 Cylinder Maps 1 work with Araceli Bonifant Let C denote the cylinder (R/Z) I. (.5, 1) fixed

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### Chapter 5: Application: Fourier Series

321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1

### Lecture 13: Martingales

Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

### (Notice also that this set is CLOSED, but does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)

Definition 3.1 Group Suppose the binary operation p is defined for elements of the set G. Then G is a group with respect to p provided the following four conditions hold: 1. G is closed under p. That is,

### arxiv:0706.1300v1 [q-fin.pr] 9 Jun 2007

THE QUANTUM BLACK-SCHOLES EQUATION LUIGI ACCARDI AND ANDREAS BOUKAS arxiv:0706.1300v1 [q-fin.pr] 9 Jun 2007 Abstract. Motivated by the work of Segal and Segal in [16] on the Black-Scholes pricing formula

### Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems

Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Wolfgang Wagner Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wias-berlin.de WIAS workshop,

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Universal Algorithm for Trading in Stock Market Based on the Method of Calibration

Universal Algorithm for Trading in Stock Market Based on the Method of Calibration Vladimir V yugin Institute for Information Transmission Problems, Russian Academy of Sciences, Bol shoi Karetnyi per.

### EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

### Quantum Computation as Geometry

Quantum Computation as Geometry arxiv:quant-ph/0603161v2 21 Mar 2006 Michael A. Nielsen, Mark R. Dowling, Mile Gu, and Andrew C. Doherty School of Physical Sciences, The University of Queensland, Queensland

### Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

### 5. Orthogonal matrices

L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.

2. Free Fields The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. Sidney Coleman 2.1 Canonical Quantization In quantum mechanics,

### Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

### Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Second postulate of Quantum mechanics: If a system is in a quantum state represented by a wavefunction ψ, then 2

. POSTULATES OF QUANTUM MECHANICS. Introducing the state function Quantum physicists are interested in all kinds of physical systems (photons, conduction electrons in metals and semiconductors, atoms,

### Advanced results on variational inequality formulation in oligopolistic market equilibrium problem

Filomat 26:5 (202), 935 947 DOI 0.2298/FIL205935B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Advanced results on variational

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Chapter 7 Nonlinear Systems

Chapter 7 Nonlinear Systems Nonlinear systems in R n : X = B x. x n X = F (t; X) F (t; x ; :::; x n ) B C A ; F (t; X) =. F n (t; x ; :::; x n ) When F (t; X) = F (X) is independent of t; it is an example

### From Fourier Series to Fourier Integral

From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this

### REPRESENTATION THEORY WEEK 12

REPRESENTATION THEORY WEEK 12 1. Reflection functors Let Q be a quiver. We say a vertex i Q 0 is +-admissible if all arrows containing i have i as a target. If all arrows containing i have i as a source,