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

## TL;DR

This paper investigates the spectra of specific functionals in certain Sobolev-like spaces, proving non-intersection for adjacent spaces and establishing conditions for intersection when the space indices differ by more than two.

## Contribution

It demonstrates non-intersection of spectra between spaces with consecutive indices and derives necessary conditions for spectral intersection at larger index differences.

## Key findings

- Spectra of functionals do not intersect for adjacent spaces.
- Necessary conditions for spectral intersection are identified for larger index differences.
- Results apply to even functions in the considered function spaces.

## 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 $\Delta=m-n>2$.