Spectrum of stochastic evolution operators: local matrix representation approach.

Area of Science:

• Nonlinear dynamics
• Stochastic processes
• Quantum mechanics
• Mathematical physics

Background:

• Analyzing the spectrum of nonlinear stochastic flows is computationally challenging.
• Existing methods like Feynman diagram perturbation theory can be complex to implement.
• Semiclassical approximations are crucial for understanding spectral properties in quantum and stochastic systems.

Purpose of the Study:

• To develop a more efficient matrix-based method for computing the spectrum of nonlinear stochastic flows.
• To formulate a perturbative expansion for the spectrum in the weak noise limit.
• To enable computation of perturbative corrections beyond current attainable orders.

Main Methods:

• Utilizing a matrix representation of the evolution operator for nonlinear stochastic flows with additive noise.
• Developing a perturbative expansion based on local matrix representations centered on classical periodic orbits.
• Comparing the computational ease with standard Feynman diagram perturbation theory.

Main Results:

• Successfully computed the spectrum of the nonlinear stochastic flow using the matrix representation.
• Formulated a perturbative expansion for the spectrum in the weak noise limit.
• Achieved computation of perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant to several higher orders.

Conclusions:

• The proposed matrix representation offers a computationally advantageous framework for spectral analysis of nonlinear stochastic systems.
• This method facilitates the calculation of higher-order perturbative corrections, advancing understanding in stochastic and quantum mechanics.
• The findings provide a new tool for exploring semiclassical spectral properties in complex dynamical systems.