Generalized Squeeze Equation and Its Symmetry

Abstract

I study the solutions, symmetry, and the map under time inversion (t ↦ 1/t) of the following generalized squeeze equation (GSQE) $$_{n,m} u \equiv \left[ {\frac{\partial }{{\partial t}} - k\left( {\sum\limits_{k = 1}^n {\frac{{\partial ^2 }}{{\partial x_k^2 }} - \frac{\gamma }{{t^2 }}\sum\limits_{r = 1}^m {\frac{{\partial ^2 }}{{\partial p_r^2 }}} } }\right)} \right]u\left( {t,x_k ,p_r } \right) = 0,$$ which is a formal generalization of the squeeze equation SQE, $$Q \equiv \left[ {\partial t - \frac{1}{4}\partial x^2 + \frac{1}{{4t^2 }}\partial p^2 } \right]Q\left( {t,x,p} \right) = 0.$$ I determine the Lie symmetry algebra gn,m of the GSQE, which yields a deeper understanding of the Lie symmetry algebra gn,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.