On the condition number for a system of eigenfunctions of a nonlinear spectral problem (Q1594256)

The stability and asymptotic behavior of solutions to evolution equations is related to the spectral properties of families of functions to the corresponding spectral problems, in particular to the degree of nonorthogonality of the corresponding eigenfunctions. If they form a Riesz basis then the quantity \(\Upsilon\) proportional to the norm of the inverse operator and to the norm of the orthogonalizer operator measures the degree of the nonorthogonality. The physical problem generating this study is the oscillation of plasma. The plasma is oscillating in a uniform magnetic field directed along the \(x_3\)-axis, while the plasma density depends only on the variable \(x_1\). Separating variables results in the following equation describing the variation of the amplitude \(y\): \[ i\varepsilon\lambda(y''- k^2 y)+ (q(x)- \lambda^2) y= 0\tag{1} \] satisfying the boundary conditions: \[ y'(a)= y'(b)= 0.\tag{2} \] Here, \(\varepsilon> 0\) is a parameter inversely proportional to the conductivity, and \(q(x)\) is a function describing the plasma density. As \(\varepsilon\to 0\) the magnetohydrodynamic model approaches the one of classical hydrodynamics. Assuming that \(q(x)> 0\), and \(q'(x)> 0\), \(x\in [a,b]\), the author denotes the spectral parameter \(\mu= i\lambda\), and the operators \(A_0= -d^2/dx^2+ k^2\), acting on \({\mathcal H}= L_2(a, b)\), with boundary conditions (2) and \(G= (A_0)^{-1}\), \(H= A_0^{-1/2} qA^{-1/2}_0\). Then a vector \(\varphi\) satisfies \[ \varepsilon\varphi- \mu G\varphi \mu- \mu^{-1} H\varphi= 0 \] if and only if \(y= A_0^{-1/2}\varphi\) solves problem (1), (2). Thus, the original problem (1), (2) reduces to the spectral analysis of the pencil \(T= T_\varepsilon(\mu)= \mu^2 G-\varepsilon\mu I+ H\) in \({\mathcal H}\). A theorem of Shkalikov and Piev asserts that under compact perturbation of strongly damped operator pencils the normalized eigenvectors of a pencil split into two subsystems \(E^\pm\) of a Riesz basis on \({\mathcal H}\). Each consists of respectively positive and negative eigenvectors \(\varphi_j\in \text{Ker }T(\mu)\) corresponding to eigenvalues whose imaginary part is respectively positive or negative. Thus, each element \(h\in{\mathcal H}\) can be written as a sum \(h= \sum^\infty\alpha_j \varphi_j\), with \(\alpha_j= (h,\psi_j)\), where \(\{\psi_j\}\) is a biorthogonal system. This allows the author to make some estimates, to fix the domains and to introduce norms to the ``\(1/2\) power operators''. He shows that the lower semicontinuity of the spectrum is violated as \(\varepsilon\to 0\). Specifically, there is a jump in the spectrum at the point \(\varepsilon= 0\). Finally, he proves that the Riesz constants \(\Upsilon^\pm\) of the bases \(A^{-1/2}_0 E^\pm\) become unbounded as \(\varepsilon\to 0\).