Gaps in the essential spectrum for second order systems

Abstract:

Consider the equation $({D^2} + A + \alpha E)f = 0$, where $\alpha$ is a positive real number, $E$ is the $n \times n$ identity matrix, $A$ is a continuously differentiable function from $[0,\infty )$ to the $n \times n$ Hermitian matrices, and $A$ and $A’$ are bounded. It is shown that, if $\alpha$ is large with respect to $||A|{|_\infty }$, there are small positive numbers $\lambda$ such that, for every solution $f$ to the equation $({D^2} + A + \alpha E)f = 0,{e^{ - \lambda t}}f(t)$ is square integrable, but ${e^{\lambda t}}f(t)$ is not. It is also shown that, if $\alpha$ is large with respect to $||A|{|_\infty }$, there is a real number $\lambda$ close to zero such that $\lambda$ is in the essential spectrum of any selfadjoint operator in ${L_2}$ associated with ${D^2} + A + \alpha E$. These results generalize the results of Hartman and Putnam, who proved these statements for the scalar case $n = 1$.