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DOI: 10.23671/VNC.2017.1.5817

# Complex Powers of a Differential Operator Related to the Schrodinger Operator

Gil A. V. , Nogin V. A.
Vladikakazian Mathematical Journal 2017. Vol. 19. Issue 1.
Abstract:
We study complex powers of the generalized Schrodinger operator in
$$L_p({\mathbb R^{n+1}})$$ with complex coefficients in the
principal part

$$S_{\bar{\lambda}}=m^2I+i b \frac{\partial}{\partial x_{n+1}}+\sum\limits_{k=1}^n (1-i\lambda_k) \frac{\partial ^2}{\partial x_k^2},$$

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 "sufficiently  nice" 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:

$$F\left((S_{\bar{\lambda}}^{-\alpha/2}\varphi\right)(\xi)= \left((m^2+b\xi_{n+1}-|\xi'|^2+i\sum\limits_{k=1}^n\lambda_k \xi_k^2\right)^{-\alpha/2}\widehat{\varphi}(\xi),$$
where $$\xi\in{\mathbb R^{n+1}}$$, $$\xi'=(\xi_1,\ldots,\xi_n)$$, $$0<\operatorname{Re} \,\alpha$$ $$\langle S_{\bar{\lambda}}^{-\alpha/2}\varphi,\omega\rangle= \langle\varphi, \overline{S_{\bar{\lambda}}^{-\alpha/2}}\omega\rangle,\quad \varphi\in \Phi,$$

where $$\Phi$$ is the Lizorkin space of functions in $$S$$,  whose Fourier transforms vanish on coordinate hyperplanes.  Within the framework of the method of approximative inverse operators we  describe the range $$H_{\bar{\lambda}}^{^\alpha} (L_p)$$, $$1\leq p<\frac{n+2}{{{\rm Re\,}}\,\alpha}$$. Recently a number of papers related to complex powers of
second order degenerating differential operator was published (see survey papers [1-3], and also  [6-11]). The case considered in our work is the most difficult,  because of non-standard expressions for the potentials $$H_{\bar{\lambda}}^{^\alpha} \varphi$$.
Keywords: differential operator, range, multiplier, complex powers, method of approximative inverse operators.