Martin-Luther-Universität Halle-Wittenberg


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11. Juli 2023

Nenad Teofanov (Universität Novi Sad)

Localization Operators and Density Matrices

This lecture can be considered as a brief introduction to time-frequency localization operators.

As a warm up and motivation we first discuss Fourier multipliers. Such operators are used for design of frequency filters in signal analysis. However, for different purposes it is of interest to treat time-frequency plane as one geometric whole rather than as two separate spaces.

Operators which perform simultaneous localization in time and in frequency arise as a natural extension of Fourier multipliers. Historically, such operators were first observed by Felix Berezin in the context of quantization problem in quantum mechanics in early 1970’s. 35 years ago Ingrid Daubechies published an influential paper on localization operators and their applications in optics and signal analysis. A convenient framework for the study of localization operators as short-time Fourier transform multipliers is given by Elena Cordero and Karlheinz Gröchenig in 2003, that is 20 years ago.

In the first part of the talk we will survey basic results of localization operators and highlight their connection to pseudodifferential operators. We will also discuss bilinear localization operators.

In the second part of the talk we will reconsider localization operators as continuous frame multipliers defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. Finally, we consider the tensor product setting for continu- ous frame multipliers. A specific feature in such context is the notion of partial trace. By using the partial trace theorem we will offer an interpretation of tensor product continuous frame multipliers as density operators for bipartite quantum systems in quantum mechanics.

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