Quadratic conservation laws for equations of mathematical physics

Abstract

Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the focus of interest in this paper: the existence of other quadratic integrals, the Hamiltonian property of a linear system, and the complete integrability of such a system. For non-degenerate linear systems in a finite-dimensional space essentially exhaustive answers to all these questions are known. Results of a general nature are applied to linear evolution equations of mathematical physics: the wave equation, the Liouville equation, and the Maxwell and Schrödinger equations. Bibliography: 60 titles.

Linear System of Differential Equations with a Quadratic Invariant as the Schrödinger Equation

“…Let A be a linear operator with a dense domain ; A is not assumed to be bounded. The operator A is associated with the linear differential equation (1) Assume that this linear system has the quadratic invariant (2) This means that the derivative of this function vanishes by virtue of system (1), i.e., for all . However, this means that the operator A is skew-symmetric: for all .…”

“…Below, we describe the properties of eigenvectors of A that are important in what follows (for details and proofs, see [1]). Let…”

“…Define these are points of π n . The restriction of the original linear system (1) to π n has the form (5) This linear system describes the dynamics of a onedimensional harmonic oscillator with frequency . Obviously, it has the first integral .…”

“…It was shown in [1] that, under these assumptions, linear system (1) is a completely integrable Hamiltonian system. The Hamiltonian property also holds under weaker assumptions [2].…”

“…In addition to the Schrödinger equations, they include wave equations, the Liouville equation from statistical mechanics, and the Maxwell equations. All of them were examined in detail in [1] from the point of view of their complete integrability as Hamiltonian equations. Apparently, a reduction to the Schrödinger equation is also possible under more general conditions (see Section 4).…”