[Paper Review] On transformation operators and Riesz basis property of root vectors system for $n imes n$ Dirac type operators. Application to the Timoshenko beam model

Anton A. Lunyov, M. M. Malamud|arXiv (Cornell University)|Dec 14, 2021

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper establishes the existence of triangular transformation operators for n×n Dirac-type systems with general boundary conditions, proving that the root vector system forms a Riesz basis in a weighted L²-space under strictly regular boundary conditions. The key contribution is a spectral asymptotic formula showing eigenvalues of the perturbed system converge to those of the unperturbed system, with applications to the Timoshenko beam model, including explicit asymptotics and Riesz basis properties even in variable-parameter settings.

ABSTRACT

The paper is concerned with the following $n imes n$ Dirac type equation$$Ly=-iB(x)^{-1}(y'+Q(x)y)=λy, \quad B(x)=B(x)^*,\quad y={ m col}(y_1,\ldots,y_n),\quad x\in[0,\ell],$$ on a finite interval $[0,\ell]$. Here $Q$ is a summable potential $n imes n$ matrix function and $B$ is an invertible self-adjoint diagonal bounded matrix function. If $n=2m$ and $B(x)={ m diag}(-I_m,I_m)$, this equation is equivalent to Dirac equation of order $n$. We show the existence of triangular transformation operators for such equation under additional uniform separation conditions on the entries of the matrix function $B$. Here we apply this result to study direct spectral properties of the boundary value problem (BVP) associated with the above equation subject to the general boundary conditions $U(y)=Cy(0)+Dy(\ell)=0,{ m rank}(C\ D)=n$. We apply this result to show that the deviation of the characteristic determinants of this BVP and the unperturbed BVP (with $Q=0$) is a Fourier transform of some summable function, which in turn yields asymptotic behavior of the spectrum in the case of regular boundary conditions. Namely, $λ_m=λ_m^0+o(1)$ as $m o\infty$, where $\{λ_m\}_{m\in\mathbb{Z}}$ and $\{λ_m^0\}_{m\in\mathbb{Z}}$ are sequences of eigenvalues of perturbed and unperturbed ($Q=0$) BVP, respectively. Further, we prove that the system of root vectors of the above BVP constitutes a Riesz basis in a certain weighted $L^2$-space, provided that the boundary conditions are strictly regular. The main results are applied to establish asymptotic behavior of eigenvalues and eigenvectors, and the Riesz basis property for the dynamic generator of the Timoshenko beam model. We also found a new case when eigenvalues have an explicit asymptotic, which to the best of our knowledge is new even in the case of constant parameters of the model.

Motivation & Objective

• To establish the existence of triangular transformation operators for n×n Dirac-type systems under uniform separation conditions on the weight matrix B.
• To analyze the spectral properties of the boundary value problem (BVP) associated with the Dirac-type system under general boundary conditions.
• To prove that the system of root vectors forms a Riesz basis in a weighted L²-space when boundary conditions are strictly regular.
• To apply the results to the dynamic generator of the spatially non-homogeneous damped Timoshenko beam model, deriving eigenvalue asymptotics and Riesz basis properties.
• To identify a new case with explicit asymptotic eigenvalues, even in the constant-parameter regime, which to the best of the authors' knowledge is novel.

Proposed method

• Derive triangular transformation operators for the Dirac-type system under uniform separation conditions on the entries of the diagonal weight matrix B.
• Use the transformation operator to express the deviation of the characteristic determinant from the unperturbed case as a Fourier transform of a function derived from the kernel.
• Establish asymptotic behavior of eigenvalues via the characteristic determinant representation, showing λₘ = λₘ⁰ + o(1) as m → ∞.
• Prove completeness, uniform minimality, and asymptotic behavior of root vectors using spectral analysis techniques.
• Apply the results to the Timoshenko beam model by reducing it to a Dirac-type system and analyzing its spectral properties.
• Demonstrate the Riesz basis property with parentheses for the root vector system under general conditions, and establish explicit asymptotic formulas in special cases.

Experimental results

Research questions

• RQ1 Under what conditions does the system of root vectors of an n×n Dirac-type operator with general boundary conditions form a Riesz basis in a weighted L²-space?
• RQ2 How does the spectrum of the perturbed Dirac-type system behave asymptotically, and what is the precise relationship between the eigenvalues of the perturbed and unperturbed systems?
• RQ3 Can the transformation operator method be used to derive explicit asymptotic formulas for eigenvalues and eigenvectors in the Timoshenko beam model with variable parameters?
• RQ4 What conditions ensure that the eigenvalues of the Timoshenko beam operator are asymptotically simple and separated, and when do they admit explicit asymptotic expressions?
• RQ5 In what cases does the system of root vectors of the Timoshenko beam operator form a Riesz basis with parentheses, and how does this generalize prior results?

Key findings

• The characteristic determinant of the BVP differs from that of the unperturbed system by a Fourier transform of a function explicitly constructed from the transformation operator kernel.
• Eigenvalues of the perturbed system satisfy λₘ = λₘ⁰ + o(1) as m → ∞, where λₘ⁰ are eigenvalues of the unperturbed system (Q = 0).
• The system of root vectors forms a Riesz basis in a weighted L²-space if the boundary conditions are strictly regular.
• For the Timoshenko beam model, the root vectors form a Riesz basis with parentheses under general conditions, and the eigenvalues admit explicit asymptotic formulas in a new case not previously identified.
• The paper identifies a new case with explicit asymptotic eigenvalues for the Timoshenko beam model, even in the constant-parameter regime, which is novel to the best of the authors’ knowledge.
• The results generalize and correct prior claims in the literature, particularly regarding the Riesz basis property and completeness under variable parameters and broader boundary conditions.