On Nonintersection of Spectra of some Functionals on Spaces $\mathop{W}\limits^\circ{}^2_n$, $\mathop{W}\limits^\circ{}^2_{n+1}$, $\mathop{W}\limits^\circ{}^2_{n+2}$

Abstract

Spectra of functionals $$\Phi(u)=\frac{\left\langle u^{(n)}u^{(n)}\right\rangle}{\left\langle u^{(n-p)}u^{(n-p)}\right\rangle}$$ in spaces ${\mathop{W}\limits^\circ}^2_n$ are considered for different $n$. One has shown that for even functions in $\mathop{W}\limits^\circ{}^2_n$ and $\mathop{W}\limits^\circ{}^2_m$ spectra of functionals do not intersect for $m=n+1, n+2$. The neccesary conditions for two spectra to intersect are written for $\Delta=m-n>2$.