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Operator extensions, Friedrichs model and some fitted zero-range solvable models

Only a slim part of linear partial differential equations have solutions represented by explicit formulas or by approximate explicit expressions, with a controllable errors. Strong demand for reliable data on spectral and transport characteristics of the corresponding quantum and classical dynamical systems is nowadays satisfied based on results of expensive and resource consuming scanning over the multi-dimensional space of essential geometrical and physical parameters of the systems. The area of the scanning may be essentially reduced based on use of the fitted solvable models. A large class of the “zero-range models” can be constructed based on John von Neumann operator extension technique. The corresponding theory, developed in previous century, is technically demanding and hard to fit into a compact text for the first reading. Fortunately there exist a representative Friedrichs model of a linear dynamical system where almost all typical difficulties of the corresponding theory can be reviewed as well as advanced techniques used to overcome them.