The sech^2 x potential is a celebrated object in soliton theory. In this talk We consider the Schroedinger operator on the real line, H=-d^2/dx^2+V, where V(x) is a negative hyperbolic sechant potential. Using biorthogonal dyadic system, we introduce Besov spaces and Triebel-Lizorkin spaces associated with H. Our approach is based on eigenfunction expansion method, where the eigenfunctions of H are chosen to be the solutions of the Lippman-Schwinger equation in scattering theory. We also prove a Mikhlin-Hormander type spectral operator theorem on these spaces, including the L^p boundedness result. As a brief overview of current developments in this area we shall compare the Hermite and Laguerre wavepacket case, as well as recent general spectral multiplier results based on heat kernel approach.
In the second part of my talk, I am going to give a short introduction of the conductivity model for our TAIP program (Thin Film Analog Imaging Processor). The TAIP device is a hybrid analog/digital device concept which promises to perform approximate image filtering operations very rapidly and with very low power consumption compared to standard digital processors. Our proposal is to provide efficient wavelet computational method of the solution of the anisotropic heat equation, which arises in the context of a static electromagnetic field for our model.