Complex powers of a differential operator related to the Schrodinger operator

We study complex powers of the generalized Schr$\rm \ddot{o}$dinger operator in $L_p({\mathbb R^{n+1}})$ with complex coefficients in the principal part $$ S_{\bar{\lambda}}=m^2I+i b \frac{\partial}{\partial x_{n+1}}+\sum\limits_{k=1}^n (1-i\lambda_k) \frac{\partial ^2}{\partial x_k^2}\eqno(1) $$ where $m>0$, $b>0$ $\bar{\lambda}=(\lambda_1,\ldots,\lambda_n)$, $\lambda_k>0$, $1\leq k\leq n$. Complex powers of the operator $S_{\bar{\lambda}}$ with negative real parts on > functions $\varphi(x)$ are defined as multiplier operators, whose action in the Fourier pre-images is reduced to multiplication by the corresponding power of the symbol of the operator under consideration: $$ F\left((S_{\bar{\lambda}}^{-\alpha/2}\varphi\right)(\xi)= \left((m^2+b\xi_{n+1}-|\xi'|^2+i\sum\limits_{k=1}^n\lambda_k \xi_k^2\right)^{-\alpha/2}\widehat{\varphi}(\xi),\eqno(2) $$ where $\xi\in{\mathbb R^{n+1}}$, $\xi'=(\xi_1,\ldots,\xi_n)$, $0