The aim of this paper is to find the exact solutions of the Schrödinger equation. As is known, the Schrödinger equation can be reduced to the continuum equation. In this paper, using the non-linear Legendre transform the equation of continuity is linearized. Particular solutions of such a linear equation are found in the paper and an inverse Legendre transform is considered for them with subsequent construction of solutions of the Schrödinger equation. Examples of the classical and quantum systems are considered.

Russian FederationModern accelerator systems and detectors contain magnetic systems of complex geometrical configuration. Design and optimization of the magnetic systems demands solving a nonlinear boundary-value problem of magnetostatic.The region in which the boundary-value problem is solved, consists of two sub-domains: a domain of vacuum and a domain of ferromagnetic. In view of the complex geometrical configuration of magnetic systems, the ferromagnetic/vacuum boundary can be nonsmooth, i.e. it contains a corner point near of which the boundary is formed by two smooth curves crossed in a corner point at some angle.Thereby, the solution of such a problem has to be found by numerical methods, a question arises about the behavior of the boundary value problem solution around the angular point of the ferromagnetic. This work shows that if the magnetic permeability function meets certain requirements, the corresponding solution of the boundary value problem will have a limited gradient. In this paper, an upper estimate of maximum possible growth of the magnetic field in the corner domain is given.In terms of this estimate, a method of condensing the differential mesh near the corner domain is proposed. This work represents an algorithm of constructing an adaptive mesh in the domain with a boundary corner point of ferromagnetic taking into account the character of behavior of the solution of the boundary value problem. An example of calculating a model problem in the domain containing a corner point is given.Key words and phrases: magnet systems, mathematical modeling, boundary value problem, elliptic equations, the behavior of solutions in the corner domain

Abstract. Homogeneous magnetic fields are important requisites in modern physics research. In this paper we discuss the problem of magnetic field homogeneity area maximization for solenoid magnets. We discuss A-model and B-model, which are basic types of solenoid magnets used to provide a homogeneous field, and methods for their optimization. We propose C-model which can be used for the NICA project. We have also carried out a cross-check of the C-model with the parameters stated for the CLEO II detector.

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