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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 1, Pages 18–25 (Mi vmj603)

Complex powers of a differential operator related to the Schrödinger operator

A. V. Gila, V. A. Noginab

a Southern Federal University, Rostov-on-Don

Abstract: We study complex powers of the generalized Schrödinger operator in $L_p({\mathbb R^{n+1}})$ with complex coefficients in the principal part
$$S_{\bar{\lambda}}=m^2I+i b \frac{\partial}{\partial x_{n+1}}+\sum\limits_{k=1}^n (1-i\lambda_k) \frac{\partial ^2}{\partial x_k^2}\tag{1}$$
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:
$$F((S_{\bar{\lambda}}^{-\alpha/2}\varphi)(\xi)= ((m^2+b\xi_{n+1}-|\xi'|^2+i\sum\limits_{k=1}^n\lambda_k \xi_k^2)^{-\alpha/2}\widehat{\varphi}(\xi),\tag{2}$$
where $\xi\in{\mathbb R^{n+1}}$, $\xi'=(\xi_1,\ldots,\xi_n)$, $0<{Re} \alpha<n+2$. 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 $S_{\bar{\lambda}}^{-\alpha/2}\varphi$, $0<{Re} \alpha<n+2$, 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 $H_{\bar{\lambda}}^{^\alpha} (L_p)$, $1\leqslant p<\frac{n+2}{{{ Re }} \alpha}$. 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 $H_{\bar{\lambda}}^{^\alpha} \varphi$.

Key words: differential operator, range, multiplier, complex powers, method of approximative inverse operators.

 Funding Agency Grant Number ÃÊÍ ÌÎÍ ÐÀ – ÅÃÓ – ÞÔÓ ÐÔ ÂíÃð-07/2017-31

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Citation: A. V. Gil, V. A. Nogin, “Complex powers of a differential operator related to the Schrödinger operator”, Vladikavkaz. Mat. Zh., 19:1 (2017), 18–25

Citation in format AMSBIB
\Bibitem{GilNog17} \by A.~V.~Gil, V.~A.~Nogin \paper Complex powers of a differential operator related to the Schr\"odinger operator \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 1 \pages 18--25 \mathnet{http://mi.mathnet.ru/vmj603} 

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This publication is cited in the following articles:
1. Y. Xing, J. Zhao, “Existence and quantum calculus of weak solutions for a class of two-dimensional Schrödinger equations in $\mathbb{C}_+$”, Bound. Value Probl., 2018, 59
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