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A weak value of an observable is studied for a quantum system which
is placed under the influence of an environment, where a quantum system
irreversibly evolves from a pre-selected state to a post-selected state. A
general expression for a weak value influenced by an environment is provided.
For a Markovian environment, the weak value is calculated in terms of the
predictive and retrodictive density matrices, or by means of the quantum
regression theorem. For a non-Markovian environment, a weak value is examined
by making use of exactly solvable models. It is found that although the
anomalous property is significantly suppressed by a Markovian
environment, it can survive a non-Markovian environment.
As concerns about energy and environment emerge the concept of sustainable development, this paper develops a five-sector endogenous technological change economic growth model considering sustainable use of energy and pollution control. We introduce energy and environment into production function and utility function, solve the conditions for sustainable economic growth with the optimal control method, further reveal the dynamic relationship which should be met by energy consumption rate, pollution control and sustainable economic growth, prove the possibility of sustainable economic growth and reveal the path and conditions of sustainable economic growth under the dual constraints of energy and environment.