Subject: My main interests concern mathematical problems in natural sciences: quantum and classical mechanics, astrophysics, biology etc. My articles of pure mathematical origin are relatively rare. Several papers were devoted to the spectral theory of the Schrodinger operator H, to be more precise, to behavior of eigenfunctions of H at infinity. For a potential bounded below, the following assertion was proved. If there exists a bounded or slowly growing solution w(x) of the equation Hw=Ew, then the number E belongs to the spectrum of H. In one- dimensional case, for a Sturm-Liouville operator on a half- line, the following refined inverse assertion is valid: eigenfunctions w(x) of the operator H grow not faster than a power of x for almost all E (in the sense of the spectral measure). These and analogues results have stimulated the activity of many mathematicians. These are, apparently, the most known of my results. I was much engaged in problems of stability and studied some specific physical problems as well as issues of the general theory of stability . The series of papers (written with L.G.Khazin) concerns stability of equilibrium points of ordinary differential equations in the "critical" cases (i.e., when stability is not determined by the linearized equations). We considered all 20 critical cases that correspond to the degeneracy level 1, 2, 3 and studied cases close to the critical ones. For some important cases we have found criteria of stability. On the other hand, we have pointed out a case, where the algebraic stability criterion does not exist. The main results on this topic are summarized in the book: L.G.Khazin, E.E.Shnol. "Stability of Critical Equilibrium States." Manchester Univ. Press, 1991. During many years I worked in several research directions where computers served as the main tools of investigations. I mention here two of these directions. 1) Investigation of some physical and chemical phenomena by the computer simulation of molecular motion ("method of molecular dynamics"). Polymer chain motion and sorption onto a surface are examples of the phenomena studied. A.G.Grivtsov and I were among the pioneers of this direction in the USSR and, I think, some ideas of that time are still of interest. (See my lecture "Numerical experiments on molecules in motion" at a summer school in Moldavia, July, 1975. The lecture was recently reprinted as a chapter of the book "Method of molecular dynamics in physical chemistry". Moscow, Nauka,1996. ) 2) Investigation of non-linear waves in active media by a numerical solution of corresponding partial differential equations. We studied, in particular, the spiral waves in two-dimensional media and various phenomena, which arise when an autowave passes through apertures. Apparently, the following article has attracted a particular attention of specialists in this field: A.M.Pertsov, E.A.Ermakova, E.E.Shnol. "On the diffraction of autowaves". Physica, 1990, v. D44, p. 178-190. During last years I am occupied with the bifurcation theory for ordinary differential equations. I (in collaboration with E.V.Nikolaev) studied the equations having some groups of symmetries and have described the full bifurcation pictures for some simplest bifurcations. See, in particular, the article: E.E.Shnol and E.V.Nikolaev. "On the bifurcations of symmetric equilibria corresponding to double eigenvalues". Matem. sbornik, 1999, vol.190:9, p. 127-150 (English transl. Sbornik: Mathematics, v, 190:9, p. 1353- 1376.)
Main publications: - L. G. Khazin, E. E. Shnol. Stability of Critical Equilibrium States. Manchester Univ. Press, 1991, 208 p.
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