We use the reduction method, which allows one to reduce the study of perturbations of multiple eigenvalues to perturbations of simple eigenvalues, to analyze the general perturbation problem for Fredholm points of the discrete spectrum of linear operator functions analytically depending on the spectral parameter. The same method is used to study a perturbation of multiple Fredholm points of the discrete Schmidt spectrum (s-numbers) of a linear operator. We present an example of a problem on a perturbation of the domain of the Sturm–Liouville problem for a second-order differential operator.

Perturbation theory of eigenvalue problems

Perturbation of Fredholm eigenvalues of linear operators

Generalizing Courant's nodal domain theorem, the ``Extended Courant property'' is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in this case, the first eigenfunction is constant, and looking at the level sets of $u$ amounts to looking at the nodal sets $\{u-a=0\}$, where $a$ is a real constant. In the first part of the paper, we prove that the Extended Courant property is false for the subequilateral triangle and for regular $N$-gons ($N$ large), with the Neumann boundary condition. More precisely, we prove that there exists a Neumann eigenfunction $u_k$ of the $N$-gon, with labelling $k$, $4 \le k \le 6$, such that the set $\{u_k \not = 1\}$ has $(N+1)$ connected components.
In the second part, we prove that there exists a metric $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$), which can be chosen arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $u$ of the associated Laplace-Beltrami operator, such that the set $\{u \not = 1\}$ has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin. As for the positive direction, in Appendix~B, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in $\mathbb{R}^2$.

Non-boundedness of the number of super level domains of eigenfunctions

background

Infinite-dimensional Hamiltonian systems with continuous spectra : perturbation theory, normal forms, and Landau damping

The widespread idea that spin-density functional theory is based upon the extension of the Hohenberg–Kohn theorem to weak magnetic fields is contested. First, it is assumed that only the term linear in magnetic field can be kept in the Hamiltonian. Second, once this is done, two problems arise (1) not only the spin-dependent, but also the orbital-dependent term should be taken care of, and (2) the latter produces eigenvalues that are not bounded from below, thus invalidating the proof of the Hohenberg–Kohn theorem.

Absence of proof for the Hohenberg–Kohn theorem for a Hamiltonian linear in the magnetic field

We review basic results in the field of finding extremal metrics for spectral invariants of the Laplacian on both graphs and manifolds. Special attention is given to the special case of the Klein bottle. The nececery theory is developed to produce the result of Jakobson et all [J-N-P] regarding λ1 on the Klein bottle. Using similar techniques, a new result is established in proving that there is only one extremal metric of a certain kind for λ2 on the Klein bottle.

/pdf/extremal-metrics-on-graphs-and-manifolds-1r5fdj3yqe.pdf

Perturbation theory of eigenvalue problems

Citations

Perturbation of Fredholm eigenvalues of linear operators

Non-boundedness of the number of super level domains of eigenfunctions

Infinite-dimensional Hamiltonian systems with continuous spectra : perturbation theory, normal forms, and Landau damping

Absence of proof for the Hohenberg–Kohn theorem for a Hamiltonian linear in the magnetic field

Extremal metrics on graphs and manifolds

Related Papers (5)

Perturbation theory for linear operators

Methods of Modern Mathematical Physics

Matrix Analysis

The algebraic eigenvalue problem

Matrix computations