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џндекс.Њетрика

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.
Language: Russian Download the full text  
For citation: Gil A. V., Nogin V. A. Complex Powers of a Differential Operator Related to the Schrodinger Operator. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.3-11. DOI 10.23671/VNC.2017.1.5817
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