Vladimir Mandelshtam

Chemistry Department, University of California at Irvine, Irvine, CA 92697, USA

The Fourier Transform (FT) is commonly accepted as the central (and often the only) method of spectral analysis. It is also a common belief that the "uncertainty principle", which is only a property of the Fourier Transform, is the very general one that defines the spectral resolution in terms of the length of the time signal. Under certain assumptions about the time signal which are often
justified in quantum dynamics problems the spectral information can be extracted much more efficiently using the Filter Diagonalization Method (FDM), which is a linear algebraic technique using local spectral analysis. I will present several recent developments of FDM, in particular, the multi-scale FDM based on using a non-uniform Fourier basis that can perform a local spectral analysis in high resolution while describing the rest of the spectrum in low resolution. The problem of high resolution (non-linear) spectral analysis is often very ill-defined in that the result is highly sensitive to the input data, usually making the application of the corresponding technique (e.g., the FDM) a state-of-the-art. This instability problem will be addressed. In particular, I will discuss the noise averaging idea to stabilize FDM. Another way to circumvent the said instability problem corresponds to the recently proposed Regularized Resolvent Transform (RRT) which is a direct linear algebraic transformation of the signal from the time to frequency domain, that has the high resolution property, is stable, fast, robust, etc., and can be used to replace FT. I will discuss some applications of these general techniques to several diverse problems, such as calculating the quantum bound and resonance spectra of small molecules, spectral analysis of semiclassically obtained time correlation functions, extracting the spectral information from experimentally measured signals, etc.