Yurii S. Osipov

К пятидесятилетию научной и педагогической деятельности Виктора Павловича Маслова@@@Fiftieth anniversary of research and teaching by Viktor Pavlovich Maslov

Viktor Pavlovich Maslov began teaching in 1957 after defending his candidate dissertation under the supervision of Professor S. V. Fomin. This moment can also be considered the start of his independent research because from then on, his strongest side began appearing, the ability to uncover the mathematics underlying the most important modern problems in the natural sciences and to pose relevant problems whose solutions advance not only mathematics itself but also science as a whole. More than 50 years ago, physicists discovered and constructed microwave masers. To build nanowave (light) lasers, in addition to an active substance that when illuminated would not absorb but amplify the beam passing through it, required developing a reflecting system that would retain only one mode while allowing the rest of the beam to leave the device. The young mathematician Maslov suggested a possible type of such a construction in 1957. It turned out that if a sufficiently narrow planar waveguide is bent slightly, then it becomes possible to ensure that only one mode is retained while the others “jump out” of the waveguide. Maslov showed that bending the axis of a very narrow waveguide results in a certain effective potential (refractive index) in the longitudinal direction. This potential contains one stable term −k/4, where k is the curvature of the waveguide axis. If the axis is straight in some waveguide segment, then the curvature and potential of this segment are zero. Bending the waveguide then results in increasing the curvature (and decreasing the potential −k/4). Next, the waveguide axis is straight again, and the curvature and potential are again equal to zero. A negative potential well is thus formed. The Helmholtz equation for a monochromatic electromagnetic wave in a bent narrow waveguide leads to the one-dimensional Schrödinger equation with the potential describing this well. A discrete spectrum exists in this case. If the bend is small, then the depth of the well is sufficiently small, and the well contains a single eigenvalue and therefore exactly one eigenmode. For a somewhat greater bend, there can be two, three, and so on eigenmodes. A wave trap is thus formed. (We note that the standard quasistationary modes, which can be present in such a system because of reflection from the open ends, are exponentially small compared to the waveguide width-to-length ratio.) Maslov was the speaker at three consecutive seminars headed by A. M. Prokhorov, who was determined to comprehend this theory. Toward the end of the discussion, one of the participants bluntly concluded that it was technologically impossible, for example, to construct a single-mode planar laser based on this theory because techniques for creating thin films and ultranarrow channels in them did not exist. At the same time, Maslov’s patent application was rejected (by the same opponent) on the same grounds. But already in 1959, the world saw the birth of the first planar transistor, and R. Feynman published a paper on the prospects of advancing into the nanoscale domain and uttered his now legendary phrase while speaking at Caltech: “There’s plenty of room at the bottom.” It can be said that Maslov’s work published in 1958 as a short, purely mathematical paper in Doklady Akademiki Nauk SSSR was the first example of a nanostructure suggested by a mathematician. The development of this idea has been very fruitful within mathematics itself. Maslov gave a formal mathematical description of this type of mixed system with deterministic (classical) and simultaneously wave (quantum) behavior in his book Perturbation Theory and Asymptotic Methods (MSU Publ., Moscow, 1965; English translation: Mir, Moscow, 1965). The principal mathematical objects introduced in that book are Lagrangian submanifolds in phase space (so named by Maslov), the “canonical operator” assigning wave functions to such submanifolds, and an integer index and