Generalized Squeeze Equation and Its Symmetry

I study the solutions, symmetry, and the map under time inversion (t ↦ 1/t) of the following generalized squeeze equation (GSQE)♡n,mu≡[∂∂t−k(∑k=1n∂2∂xk2−γt2∑r=1m∂2∂pr2)]u(t,xk,pr)=0,which is a formal generalization of the squeeze equation SQE,♡Q≡[∂t−14∂x2+14t2∂p2]Q(t,x,p)=0.I determine the Lie symmetry algebra g _n,m of the GSQE, which yields a deeper understanding of the Lie symmetry algebra g _n,0 of the n-dimensional heat equation. I introduced the parameter γ to obtain an ‘internal contraction’ of so(n,m) to iso(n,m), similar to that of the hydrogen atom.