Sharp bound for embedded eigenvalues of Dirac operators with decaying potentials

The Schrödinger operator given by

and the Dirac operator given by

are two basic models in mathematics and physics. We are interested in the embedded eigenvalue (eigenvalue embeds into the essential spectrum) problem of Schrödinger operators and Dirac operators. For Schrödinger operators, the problem is well understood. Kato’s classical results [9] show that if $\limsup_{x\to\infty}\left\lvert xV(x)\right\rvert=A$, then the Schrödinger operator has no eigenvalues larger than $A^{2}$. Wigner and von Neumann’s examples [25] imply that there exist potentials with $A=8$, such that $\lambda=1$ is an eigenvalue of the associated Schrödinger operator. Finally, (see the survey [23] for the history), Atkinson and Everitt [1] obtained the sharp bound $\frac{4A^{2}}{\pi^{2}}$. They proved that there are no eigenvalues larger than $\frac{4A^{2}}{\pi^{2}}$, and for any $0<\lambda<\frac{4A^{2}}{\pi^{2}}$, there are potentials with $\limsup_{x\to\infty}\left\lvert xV(x)\right\rvert=A$ so that $\lambda$ is an eigenvalue of the associated Schrödinger operator.

The connection between equations (1) and (2) is straightforward. For example, by letting $p_{11}=V,$ and $p_{12}=p_{21}=p_{22}=0$, one can directly obtain

by

In this article, we study embedded eigenvalue problems of a particular type of Dirac operators on $L^{2}[0,\infty)\bigoplus L^{2}[0,\infty)$, namely Dirac operators with canonical form,

where $p\in L^{2}[0,\infty)$ and $q\in L^{2}[0,\infty)$ are real functions (referred to as potentials). The canonical form of Dirac operators plays an important role in spectral theory [19, Theorem 5.1]. In the study of  asymptotics of eigenvalues and the inverse problems of Dirac operators, it is crucial to use the canonical form [10, pp. 185-187], [27, 28]. We refer readers to [5, 4, 3, 6, 20, 7] for more recent development about various types of Dirac operators.

For any $\phi_{0}\in[0,\pi)$, under the boundary condition

the Dirac operator $L_{p,q}$ defined by (3) is self-adjoint.

Denote by $\sigma_{ess}(L_{p,q})$ the essential spectrum of $L_{p,q}$. Recall that $\lambda\in\sigma_{ess}(L_{p,q})$ if and only if there is an orthonormal sequence $\{\varphi_{n}\}_{n=1}^{\infty}$ such that

It is well known that

and $L_{0,0}$ has no eigenvalues.

By [26, Theorem 6.4], if

as $x\to\infty,$ then

In the first part of our paper, under the assumption that $p$ and $q$ are Coulomb type potentials (we ask that potentials have no singularity at $x=0$), we study the question when $L_{p,q}$ has embedded eigenvalues.

We say that the potential is $C^{\infty}$ if $p,q$ are $C^{\infty}.$ In the second part of the paper, we will construct $C^{\infty}$ potentials with which $L_{p,q}$ has many embedded eigenvalues.

For Dirac operators with single embedded eigenvalue, Evans and Harris [2] obtained the sharp bound for the separated Dirac equation with the form

where their results are under the assumption that $q$ is locally absolutely continuous. For more results on embedded single eigenvalue, one can refer to [11, 15].

For many embedded eigenvalues of Schrödinger operators or Dirac operators, Naboko [18] constructed smooth potentials such that $L_{0,q}$ has dense (rationally independent) embedded eigenvalues. Naboko’s constructions work for Schrödinger operators as well. Simon [22] constructed potentials such that the associated Schrödinger operator has dense embedded eigenvalues. More recently, Jitomirskaya and Liu [8] introduced a novel idea to construct embedded eigenvalues for Laplacian on manifolds, which is referred to as piecewise constructions. This approach turns out to be quite robust. Liu and his collaborators developed the approach of piecewise constructions to construct embedded eigenvalues for various models [15, 16, 17, 13]. For more results on embedded eigenvalue problems, one can refer to [21, 14, 12].

In this paper, we adapt the approach of piecewise construction to study embedded eigenvalue problems of Dirac operators. The main strategy of proofs for our main theorems follow from that of [8, 17, 13]. In the current case of Dirac operators, new difficulties and challenges arise from the Dirac operator being vector valued and its potential consisting of a pair of functions $p$ and $q$ (unlike the models in [8, 15, 16, 17, 13]).

Let $(u(x),v(x))^{T}$ be a solution of

We define the Prüfer variables  $R(x)$ and $\theta(x)$ of  $\lambda$ by

and

Clearly, we have

By the equation

we obtain

and

Set $q(x)=V(x)\cos\varphi(x),p(x)=V(x)\sin\varphi(x)$. Note that $p$ and $q$ are completely determined by $V$ and $\varphi$. By (5) and (6), one has

and

It is obvious that equations (7) and (8) are equivalent to

By Proposition 2.1, we only need to study (7) and (8).

We assume that $\lambda$ and $\lambda_{j}$ are different values. Denote the Prüfer variables of $\lambda$ and $\lambda_{j}$ by $R(x),\theta(x)$ and $R_{j}(x),\theta_{j}(x)$, respectively.

Recall that $V(x)$ and $\varphi(x)$ uniquely determine $p$ and $q$. Define  $V(x)=V(x,b)$ and $\varphi(x)=\varphi(x,\lambda,a,\varphi_{0})$ on $[a,\infty)$ by

and

where $C$ is a constant will be defined later, $a>b$ and $\varphi_{0}=\theta(a).$

This work was completed as part of the 2022 High School and Undergraduate Research Program “STODO” (Spectral Theory Of Differential Operators) at Texas A&M University. We would like to thank Wencai Liu for managing the program, introducing this project and many inspiring discussions. The authors are also grateful to the anonymous referee, whose comments led to an improvement of our manuscript. This work was partially supported by NSF DMS-2015683 and DMS-2000345.