The complete set of symmetry operators of the Schrödinger equation (Q1185176)

The higher symmetries of equations in nonrelativistic quantum mechanics are studied enclosing the Schrödinger equation for a complex wave function \(\psi(x)\) \[ L\psi(x)=0,\qquad L=p_ 0+(p^ 2/2m), \tag{1} \] where \(x=(x_ 0,\dots,x_ 3)\), \(p_ 0=i\partial/\partial x_ 0\), \(p_ x=-i\partial/\partial x_ \alpha\), \(\alpha=1,2,3\). The linear differential operator \(Q^{(n)}\) of order \(n\) in the variables \(p_ 1,p_ 2,p_ 3\) with coefficients depending on \(x\) is called symmetry operator of the Schrödinger equation if \([Q^{(n)},L]\psi=0\). The following is proved: Equation (1) admits \((1/4!)(n+1)(n+2)^ 2(n+3)\) symmetry operators of order \(n\) and their explicit expressions are found in the algebra A Schr(1,3) of the 12-parametric Lie group of the Schrödinger equation. The Schrödinger equation is invariant relative to a 28-dimensional Lie algebra realized in the class of second order differential operators. The symmetry operators of the Levi-Leblond equation and of the equations for nonrelativistic particles of an arbitrary spin are also realized in the algebra A Schr(1,3).