Stability of spectral characteristics of boundary value problems for $2 \times 2$ Dirac type systems

2 Jul 2021, 17:05


Anton Lunyov (Facebook, Inc.)


Boundary value problems associated in $L^2([0,1]; \mathbb{C}^2)$ with the following $2 \times 2$ Dirac type equation

L_U(Q) y = -i B^{-1} y' + Q(x) y = \lambda y , \quad
B = \begin{pmatrix} b_1 & 0 \ 0 & b_2 \end{pmatrix}, \quad b_1 < 0 < b_2,\quad
y = {\rm col}(y_1, y_2),
with a potential matrix $Q \in L^p([0,1]; \mathbb{C}^{2 \times 2})$, $p \ge 1$, and subject to the regular boundary conditions $Uy :=\{U_1, U_2\}y=0$ has been investigated in numerous papers. If $b_2 = -b_1 =1$ this equation is equivalent to the one dimensional Dirac equation.

In this talk we present recent results concerning the stability property under the perturbation $Q \to \widetilde{Q}$ of different spectral characteristics of the corresponding operator $L_U(Q)$ obtained in our recent preprint [2]. Our approach to the spectral stability relies on the existence of triangular transformation operators for system (1) with $Q \in L^1$, which was established in our paper [1].

Assuming boundary conditions to be strictly regular, let $\Lambda_{Q} = \{\lambda_{Q,n}\}_{n \in \mathbb{Z}}$ be the spectrum of $L_U(Q)$. It happens that the mapping $Q \to \Lambda_Q - \Lambda_0$ sends $L^p([0,1]; \mathbb{C}^{2 \times 2})$ into the weighted space $\ell^p(\{(1+|n|)^{p-2}\})$ as well as into $\ell^{p'}$, $p'=p/(p-1)$. One of our main results is the Lipshitz property of this mapping on compact sets in $L^p([0,1]; \mathbb{C}^{2 \times 2})$, $p \in [1, 2]$. The proof of the inclusion into the weighted space $\ell^p(\{(1+|n|)^{p-2}\})$ involves as an important ingredient inequality that generalizes classical Hardy-Littlewood inequality for Fourier coefficients. Similar result is proved for the eigenfunctions of $L_U(Q)$ using the deep Carleson-Hunt theorem for ``maximal'' Fourier transform. Certain modifications of these spectral stability results are also proved for balls in $L^p([0,1]; \mathbb{C}^{2 \times 2})$, $p \in [1, 2]$.


[1] A.A. Lunyov and M.M. Malamud, On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators. J. Math. Anal. Appl. 441 (2016), pp. 57--103 (arXiv:1504.04954).

[2] A.A. Lunyov and M.M. Malamud, Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems. arXiv:2012.11170.

Primary author

Anton Lunyov (Facebook, Inc.)


Prof. Mark Malamud (Peoples Friendship University of Russia (RUDN University))

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