Time-Dependent Schrödinger Operators and Simulated Annealing

Abstract

Let U : https://www.w3.org/1998/Math/MathML" display="inline"> U : ℝ n → [ 0 , ∞ ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419914/8d73d2ed-d0a7-4013-a8ba-50e0dc128626/content/ieq0522.tif "/> be a C 2 function. It will help intuition if U is interpreted as (potential) energy of some fictitious physical system. A point moving according to the equation d x t /dt = — ∇ U will climb down the “energy landscape” https://www.w3.org/1998/Math/MathML" display="inline"> ( x , z ) ∈ ℝ n + 1 : ≈ = U ( x ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419914/8d73d2ed-d0a7-4013-a8ba-50e0dc128626/content/ieq0523.tif "/> until it reaches a local minimum of U . If we wish to allow x t to escape from local minima and visit the global minima of U , we may add some small random contribution to d x t /dt (thermal fluctuations) and consider the stochastic differential equation 1.1 https://www.w3.org/1998/Math/MathML" display="block"> d x t = − ∇ U x t d t + [ 2 T ( t ) ] 1 2 d w t , x 0 = x , https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419914/8d73d2ed-d0a7-4013-a8ba-50e0dc128626/content/eqn1_1f.tif "/> where { ωt : t ∈ [0, ∞)} is n —dimensional standard Brownian motion and T(t) > 0 is to be interpreted as temperature at time t .