Complex powers of a differential operator related to the Schr\"odinger operator

A. V. Gil ; V. A. Nogin

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 1, pp. 18-25

Résumé

We study complex powers of the generalized Schrödinger operator in $L_p({\mathbb R^{n+1}})$ with complex coefficients in the principal part \begin{equation} 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}\tag{1} \end{equation} where $m>0$, $b>0$ $\bar{\lambda}=(\lambda_1,\ldots,\lambda_n)$, $\lambda_k>0$, $1\leqslant k\leqslant n$. 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: \begin{equation} 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),\tag{2} \end{equation} where $\xi\in{\mathbb R^{n+1}}$, $\xi'=(\xi_1,\ldots,\xi_n)$, $0{\rm{Re}}\,\alpha$. We obtain integral representations for complex powers (2) as potential-type operators with non-standard metric. The corresponding fractional potentials have the form $H_{\bar{\lambda}}^{^\alpha} \varphi$. Complex powers $S_{\bar{\lambda}}^{-\alpha/2}\varphi$, $0{\rm{Re}}\,\alpha$, are interpreted as distributions: $$\langle S_{\bar{\lambda}}^{-\alpha/2}\varphi,\omega\rangle= \langle\varphi, \overline{S_{\bar{\lambda}}^{-\alpha/2}}\omega\rangle,\quad \varphi\in \Phi,$$ 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 $H_{\bar{\lambda}}^{^\alpha} (L_p)$, $1\leqslant p\frac{n+2}{{{\rm Re\,}}\,\alpha}$. 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 $H_{\bar{\lambda}}^{^\alpha} \varphi$.