Estimates for eigenvalues of Aharonov-Bohm operators with varying poles and non-half-interger circulation

TL;DR

This paper analyzes how the eigenvalues of Aharonov-Bohm operators change as the pole moves within a domain, providing estimates and detailed blow-up analysis for non-half-integer circulations.

Contribution

It introduces new estimates for eigenvalue variation and a precise blow-up analysis for eigenfunctions with non-half-integer circulation.

Findings

Eigenvalue variation rate depends on eigenfunction vanishing order.

Sharp convergence estimates for scaled eigenfunctions.

Detailed blow-up analysis for eigenfunctions near the pole.

Abstract

We study the behavior of eigenvalues of a magnetic Aharonov-Bohm operator with non-half-integer circulation and Dirichlet boundary conditions in a planar domain. As the pole is moving in the interior of the domain, we estimate the rate of the eigenvalue variation in terms of the vanishing order of the limit eigenfunction at the limit pole. We also provide an accurate blow-up analysis for scaled eigenfunctions and prove a sharp estimate for their rate of convergence.

Equations

$$A_{a}(x)=\alpha\bigg{(}-\frac{x_{2}-a_{2}}{(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}},\frac{x_{1}-a_{1}}{(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}}\bigg{)},\quad x=(x_{1},x_{2})\in\mathbb{R}^{2}\setminus\{a\},$$

$$(i\nabla +A_a{)}^2:=-\Delta +2i{A}_a\cdot \nabla +|A_a{|}^2.$$

$$\alpha\in(0,1)\setminus\Big{\{}\frac{1}{2}\Big{\}}.$$

$$\{u\in H^1(\Omega ,\mathbb{C})\cap C^{\infty }(\Omega ,\mathbb{C}):u\text{ vanishes in a neighborhood of }a\}$$

$$\|u\|_{H^{1,a}(\Omega,\mathbb{C})}=\left(\left\|\nabla u\right\|^{2}_{L^{2}(\Omega,\mathbb{C}^{2})}+\|u\|^{2}_{L^{2}(\Omega,\mathbb{C})}+\Big{\|}\frac{u}{|x-a|}\Big{\|}^{2}_{L^{2}(\Omega,\mathbb{C})}\right)^{\!\!1/2}.$$

$${\left({\Vert (i\nabla +A_a)u\Vert }_{L^2(\Omega ,{\mathbb{C}}^2)}^2+\Vert u{\Vert }_{L^2(\Omega ,\mathbb{C})}^2\right)}^{1/2},$$

$$\int_{D_{r}(a)}|(i\nabla+A_{a})u|^{2}\,dx\geq\Big{(}\min_{j\in\mathbb{Z}}|j-\alpha|\Big{)}^{2}\int_{D_{r}(a)}\frac{|u(x)|^{2}}{|x-a|^{2}}\,dx,$$

$$\int_{D_{r_{1}}(a)\setminus D_{r_{2}}(a)}|(i\nabla+A_{a})u|^{2}\,dx\geq\Big{(}\min_{j\in\mathbb{Z}}|j-\alpha|\Big{)}^{2}\int_{D_{r_{1}}(a)\setminus D_{r_{2}}(a)}\frac{|u(x)|^{2}}{|x-a|^{2}}\,dx.$$

$$H_0^{1,a}(\Omega ,\mathbb{C})=\left\{u\in H_0^1(\Omega ,\mathbb{C}):\frac{u}{|x-a|}\in L^2(\Omega ,\mathbb{C})\right\}.$$

$$E_a$$

$${\lambda }_1^a\le {\lambda }_2^a\le \cdots \le {\lambda }_j^a\le \dots$$

$${\lambda }_{n_0}^0\text{ is simple},$$

$${\lambda }_0={\lambda }_{n_0}^0\text{and}{\lambda }_a={\lambda }_{n_0}^a$$

$$\text{if λj0 is simple, the function a↦λja is analytic in a neighborhood of 0}.$$

$$\{\begin{array}{ll}(i\nabla +A_0{)}^2{\phi }_0={\lambda }_0{\phi }_0,& \text{in }\Omega ,\\ {\phi }_0=0,& \text{on }\partial \Omega ,\\ {\int }_{\Omega }|{\phi }_0(x){|}^{2}dx=1.& \end{array}$$

$${\phi }_0\text{ vanishes at }0\text{ with a vanishing order equal to }|\alpha -k|\text{ for some }k\in \mathbb{Z},$$

$$r^{-|\alpha -k|}{\phi }_0(r(\cos t,\sin t))\to \beta \frac{e^{ikt}}{\sqrt{2\pi }}\text{in }{C}^{1,\tau }([0,2\pi ],\mathbb{C})$$

$$|\lambda_{a}-\lambda_{0}|=O\left(|a|^{1+\big{\lfloor}2|\alpha-k|\big{\rfloor}}\right)\quad\text{ as }|a|\to 0,$$

$$\frac{{\lambda }_0-{\lambda }_a}{|a{|}^{2|\alpha -k|}}$$

$$p\in {\mathbb{S}}^1:=\{x\in {\mathbb{R}}^2:|x|=1\}.$$

$$\underset{a=|a|p\to 0}{\lim }\frac{{\lambda }_0-{\lambda }_a}{|a{|}^{2|\alpha -k|}}=\underset{a=|a|p\to 0}{\lim }\frac{{\lambda }_0-{\lambda }_a}{|a{|}^{|1-2k|}}$$

$${\lambda }_0-{\lambda }_a=P(a)+o(|a{|}^{|1-2k|}),\text{ as }|a|\to 0^{+},$$

$${\theta }_b(b+r(\cos t,\sin t))=t\text{for all }r>0\text{ and }t\in [\vartheta ,\vartheta +2\pi ),$$

$${\theta }_0^b(r(\cos t,\sin t))=t\text{ for all }r>0\text{ and }t\in [\vartheta ,\vartheta +2\pi ),$$

$${\Gamma }_b:=\{tb:t\in [0,1]\}.$$

$$\{\begin{array}{ll}(i\nabla +A_a{)}^2{\phi }_a={\lambda }_a{\phi }_a,& \text{in }\Omega ,\\ {\phi }_a=0,& \text{on }\partial \Omega ,\end{array}$$

$${\int }_{\Omega }|{\phi }_a(x){|}^{2}dx=1\text{ and }{\int }_{\Omega }{e}^{i\alpha ({\theta }_0^a-{\theta }_a)(x)}{\phi }_a(x)\overline{{\phi }_0(x)}dx\text{ is a positive real number}.$$

$${\phi }_a\to {\phi }_0\text{in }{H}^1(\Omega ,\mathbb{C})\text{ and in }{C}_{\mathrm{loc}}^2(\Omega \setminus \{0\},\mathbb{C}),$$

$$(i\nabla +A_a){\phi }_a\to (i\nabla +A_0){\phi }_0\text{ in }{L}^2(\Omega ,\mathbb{C}).$$

$$(i\nabla +A_p{)}^2{\Psi }_p=0\text{ in }{\mathbb{R}}^2\text{ in a weak }{H}^{1,p}\text{-sense},$$

Full text

1. Introduction and statement of the main results

An infinitely long thin solenoid perpendicular to the plane $(x_{1},x_{2})$ at the point $a=(a_{1},a_{2})\in\mathbb{R}^{2}$ produces a point-like magnetic field as the radius of the solenoid goes to zero and the magnetic flux remains constantly equal to $\alpha\in\mathbb{R}\setminus\mathbb{Z}$. This magnetic field is a $2\pi\alpha$-multiple of the Dirac delta at $a$ orthogonal to the plane $(x_{1},x_{2})$ and is generated by the Aharonov-Bohm vector potential

[TABLE]

see e.g. [5, 6, 17]. We are interested in the spectral properties of Schrödinger operators with Aharonov-Bohm vector potentials, i.e. of operators

[TABLE]

Since $\mathop{\rm curl}A_{a}\equiv 0$ in $\mathbb{R}^{2}\setminus\{a\}$, the magnetic field is concentrated at the pole $a$. If the circulation $\alpha$ is an integer number, then the potential $A_{a}$ can be gauged away by a phase transformation so that the operator $(i\nabla+A_{a})^{2}$ becomes spectrally equivalent to the standard Laplacian. On the other hand, if $\alpha\not\in\mathbb{Z}$, the vector potential $A_{a}$ cannot be eliminated by gauge transformations and the spectrum of the operator is modified by the presence of the magnetic field: this produces the so-called Aharonov-Bohm effect, i.e. the magnetic potential affects charged quantum particles moving in the region $\Omega\setminus\{a\}$, even if the magnetic field $B_{a}=\mathop{\rm curl}A_{a}$ is zero there.

The dependence on the pole $a$ of the spectrum of the Schrödinger operator $(i\nabla+A_{a})^{2}$ in a bounded domain $\Omega$ was investigated in [1, 2, 4, 10, 18, 19] under homogeneous Dirichlet boundary conditions. In particular, in [1, 2] sharp asymptotic estimates for eigenvalues were given in the case of half-integer circulation $\alpha\in\mathbb{Z}+\frac{1}{2}$ as the pole $a$ moves towards a fixed point $\bar{a}\in\Omega$; analogous sharp estimates were derived in [4] in the case $\bar{a}\in\partial\Omega$.

The case $\alpha\in\mathbb{Z}+\frac{1}{2}$ studied in the aforementioned papers presents several peculiarities which allow approaching the problem with a perspective and a technique which are not completely adaptable to a general circulation $\alpha\in\mathbb{R}\setminus\mathbb{Z}$. Indeed, if $\alpha\in\mathbb{Z}+\frac{1}{2}$ the problem can be reduced by gauge transformation to the case $\alpha=\frac{1}{2}$ and, in this case, the eigenfunctions of $(i\nabla+A_{a})^{2}$ can be identified, up a complex phase, with the antisymmetric eigenfunctions of the Laplace Beltrami operator on the twofold covering manifold of $\Omega$, see [13, 19]. As a consequence, if $\alpha=\frac{1}{2}$, the magnetic eigenfunctions have an odd number of nodal lines ending at the pole $a$. It has been proved in [14] that the corresponding nodal domains are related to optimal partition problems. We refer to [9] and references therein for related numerical simulations.

The special features characterizing Aharonov-Bohm operators with circulation $\frac{1}{2}$ played a crucial role in [1, 2, 4, 10, 18, 19]. In particular, in [18] local energy estimates for eigenfunctions near the limit pole are performed by studying an Almgren type quotient (see [7]), which is estimated using a representation formula by Green’s functions for solutions to the corresponding Laplace problem on the twofold covering. Moreover, in [1, 2, 4] a limit profile vanishing on the special directions determined by the nodal lines of limit eigenfunctions is constructed: this allows establishing a sharp relation between the asymptotics of the eigenvalue function and the number of nodal lines, which is strongly related to the order of vanishing of the limit eigenfunction.

In this paper we will focus on the case of non-integer and non-half-integer circulation, i.e. we will assume $\alpha\in\mathbb{R}\setminus\frac{\mathbb{Z}}{2}$. A reduction to the Laplacian on the twofold covering manifold is no more available in this case; moreover, magnetic eigenfunctions vanish at the pole $a$ but they do not have nodal lines ending at $a$ (see Proposition 2.1). The lack of the special features of Aharonov-Bohm operators with half-integer circulation described above requires alternative methods and produces a less precise estimate. In particular, in order to estimate the Almgren frequency function, we will give a detailed description of the behaviour of eigenfunctions at the pole and we will study the dependence of the coefficients of their asymptotic expansion with respect to the moving pole $a$, see Lemma 2.2.

By gauge invariance, if $\alpha\in\mathbb{R}\setminus\frac{\mathbb{Z}}{2}$ it is not restrictive to assume that

[TABLE]

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, open and simply connected domain. For every $a\in\Omega$, we introduce the functional space $H^{1,a}(\Omega,\mathbb{C})$ as the completion of

[TABLE]

with respect to the norm

[TABLE]

The norm (1.2) is equivalent, under condition (1.1), to the norm

[TABLE]

in view of the Hardy type inequality proved in [15] (see also [8] and [11, Lemma 3.1 and Remark 3.2])

[TABLE]

which holds for all $r>0$, $a\in\mathbb{R}^{2}$ and $u\in H^{1,a}(D_{r}(a),\mathbb{C})$. Here we denote as $D_{r}(a)$ the disk of center $a$ and radius $r$; we will denote as $D_{r}:=D_{r}(0)$ the disk with radius $r$ centered at the origin.

It is also worth mentioning the following formulation of the magnetic Hardy inequality proved in [8, Lemma 4.1]: for all $r_{1}>r_{2}>0$, $a\in\mathbb{R}^{2}$, and $u\in H^{1,a}(D_{r_{1}}(a)\setminus D_{r_{2}}(a),\mathbb{C})$,

[TABLE]

We also consider the space $H^{1,a}_{0}(\Omega,\mathbb{C})$ as the completion of $C^{\infty}_{\rm c}(\Omega\setminus\{a\},\mathbb{C})$ with respect to the norm $\|\cdot\|_{H^{1}_{a}(\Omega,\mathbb{C})}$, so that

[TABLE]

From classical spectral theory, for every $a\in\Omega$, the eigenvalue problem

[TABLE]

admits a diverging sequence of real eigenvalues $\{\lambda_{k}^{a}\}_{k\geq 1}$ with finite multiplicity; in the enumeration

[TABLE]

we repeat each eigenvalue as many times as its multiplicity. We are interested in the behavior of the function $a\mapsto\lambda_{j}^{a}$ in a neighborhood of a fixed point $\bar{a}\in\Omega$. Up to a translation and a dilation, it is not restrictive to assume that $\bar{a}=0\in\Omega$ and $\overline{D_{2}}\subset\Omega$.

Let us assume that there exists $n_{0}\geq 1$ such that

[TABLE]

and denote

[TABLE]

for any $a\in\Omega$. In [16, Theorem 1.3] it is proved that

[TABLE]

In particular the function $a\mapsto\lambda_{a}$ is continuous and, if $a\to 0$, then $\lambda_{a}\to\lambda_{0}$ (see also [10]). Let $\varphi_{0}\in H^{1,0}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ be a $L^{2}(\Omega,\mathbb{C})$-normalized eigenfunction of problem $(E_{0})$ associated to the eigenvalue $\lambda_{0}=\lambda_{n_{0}}^{0}$, i.e. satisfying

[TABLE]

From [11, Theorem 1.3] (see also Proposition 2.1) it is known that

[TABLE]

in the sense that there exist $k\in\mathbb{Z}$ and $\beta\in\mathbb{C}\setminus\{0\}$ such that

[TABLE]

as $r\to 0^{+}$ for any $\tau\in(0,1)$.

Our first result provides an estimate of the rate of convergence of $\lambda_{0}-\lambda_{a}$ in terms of the order of vanishing of $\varphi_{0}$ at [math]; in particular we have that higher vanishing orders imply faster convergence of eigenvalues.

Theorem 1.1**.**

Let \alpha\in(0,1)\setminus\big{\{}\frac{1}{2}\big{\}} and $\Omega\subset\mathbb{R}^{2}$ be a bounded, open and simply connected domain such that $0\in\Omega$. Let $n_{0}\in\mathbb{N}$ be such that the $n_{0}$-th eigenvalue $\lambda_{n_{0}}^{0}=\lambda_{0}$ of problem $(E_{0})$ is simple and let $\varphi_{0}\in H^{1,0}_{0}(\Omega,\mathbb{C})$ be an associated eigenfunction satisfying (1.7). Let $k\in\mathbb{Z}$ be such that $|\alpha-k|$ is the order of vanishing of $\varphi_{0}$ at [math] as in (1.9). For $a\in\Omega$, let $\lambda_{n_{0}}^{a}=\lambda_{a}$ be the $n_{0}$-th eigenvalue of problem ($E_{a}$). Then

[TABLE]

where $\lfloor\cdot\rfloor$ denotes the floor function $\lfloor t\rfloor:=\max\{k\in\mathbb{Z}:k\leq t\}$.

To prove Theorem 1.1, we will study the quotient

[TABLE]

as $a$ approaches the origin along a straight line $\{tp\,:\,t>0\}$ for any direction

[TABLE]

We will prove that, for every $p\in\mathbb{S}^{1}$, the quotient (1.10) is bounded as $a=|a|p\to 0$. Then (1.6) and the fact that $2|\alpha-k|$ is non-integer imply that the Taylor polynomials of the function $\lambda_{0}-\lambda_{a}$ with center [math] and degree less or equal than $\lfloor 2|\alpha-k|\rfloor$ vanish, thus yielding the conclusion of Theorem 1.1.

In the case of half-integer circulation $\alpha=\frac{1}{2}$ the special nodal structure of the limit problem allows proving instead that the limit

[TABLE]

is different from [math] along some special directions $p$ corresponding to tangents to the nodal lines of the limit eigenfunction. As a consequence, the leading term of the Taylor expansion of the eigenvalue variation $\lambda_{0}-\lambda_{a}$ has order exactly $|1-2k|$, i.e.

[TABLE]

for some homogeneous polynomial $P\not\equiv 0$ of degree $|1-2k|$, see [1, Theorem 1.2]. In [2, Theorem 2] the exact value of all coefficients of the polynomial $P$ is determined proving that $P(|a|(\cos t,\sin t))=C_{0}|a|^{|1-2k|}\cos(|1-2k|(t-t_{0}))$ for some $t_{0}$ and $C_{0}>0$. In particular the leading polynomial $P$ is harmonic.

In this paper we will also describe the behaviour of the eigenfunctions as $a\to 0$, proving a blow-up result for scaled eigenfunctions and giving a sharp rate of the convergence to the limit eigenfunction $\varphi_{0}$. In order to state these results more precisely, we need to introduce some notations.

For every $b=(b_{1},b_{2})=|b|(\cos\vartheta,\sin\vartheta)\in\mathbb{R}^{2}\setminus\{0\}$ with $\vartheta\in[0,2\pi)$, we define the polar angle centered at $b$, $\theta_{b}:\mathbb{R}^{2}\setminus\{b\}\to[\vartheta,\vartheta+2\pi)$ as

[TABLE]

and the function $\theta_{0}^{b}:\mathbb{R}^{2}\setminus\{0\}\to[\vartheta,\vartheta+2\pi)$ as

[TABLE]

in such a way that the difference function $\theta_{0}^{b}-\theta_{b}$ is regular except for the segment

[TABLE]

For all $a\in\Omega$, let $\varphi_{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ be an eigenfunction of problem ($E_{a}$) associated to the eigenvalue $\lambda_{a}$, i.e. solving

[TABLE]

such that the following normalization conditions hold

[TABLE]

Using (1.5), (1.7), (1.14), (1.15), and standard elliptic estimates (see e.g. [12, Theorem 8.10]), it is easy to prove that

[TABLE]

and

[TABLE]

To give a precise description of the behavior of the eigenfunction $\varphi_{a}$ for $a$ close to [math], we consider a homogeneous scaling of order $|a|^{|\alpha-k|}$ of $\varphi_{a}$ along a fixed direction $p\in{\mathbb{S}}^{1}$. Theorem 1.2 below gives the convergence of scaled eigenfunctions to a nontrivial limit profile $\Psi_{p}\in H^{1,{p}}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$, which can be characterized as the unique solution to the problem

[TABLE]

satisfying

[TABLE]

where $\psi_{k}:\mathbb{R}^{2}\to\mathbb{C}$ is defined as

[TABLE]

The existence and uniqueness of a limit profile satisfying (1.18) and (1.19) will be proved in Lemma 5.3. We notice that the function $\psi_{k}$ in (1.20) is the unique (up to a multiplicative constant) $H^{1,{0}}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$-solution to $(i\nabla+A_{0})^{2}\psi_{k}=0$ in $\mathbb{R}^{2}$ which is homogeneous of degree $|\alpha-k|$.

Theorem 1.2**.**

Under the same assumptions as in Theorem 1.1, for all $a\in\Omega$ let $\varphi_{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})$ be an eigenfunction of problem ($E_{a}$) associated to the eigenvalue $\lambda_{a}$ and satisfying (1.15). Let $p\in{\mathbb{S}}^{1}$. Then

[TABLE]

in $H^{1,p}(D_{R},\mathbb{C})$ for every $R>1$, almost everywhere in $\mathbb{R}^{2}$ and in $C^{2}_{\rm loc}(\mathbb{R}^{2}\setminus\{p\},\mathbb{C})$, with $\beta\neq 0$ and $k\in\mathbb{Z}$ being as in (1.9) and $\Psi_{p}$ being as in (1.18)–(1.19).

Finally, we describe the sharp rate of convergence (1.17), which also turns out to depend strongly on the order of vanishing of $\varphi_{0}$ at [math], as stated in the following theorem.

Theorem 1.3**.**

Under the same assumptions as in Theorems 1.1 and 1.2, for every $p\in{\mathbb{S}}^{1}$ there exists ${\mathfrak{L}}_{p}>0$ such that

[TABLE]

We observe that Theorem 1.3 extends to the case of non-half-integer circulation an analogous result obtained in [3] for half-integer circulation.

The paper is organized as follows. In Section 2 we perform a detailed description of the behavior of the eigenfunction $\varphi_{a}$ near the pole $a$, which is crucial in Section 3 to prove an Almgren type monotonicity formula and to derive local energy estimates for eigenfunctions uniformly with respect to the moving pole. In Section  4 we obtain some upper and lower bounds for the difference $\lambda_{0}-\lambda_{a}$ by exploiting the Courant-Fisher minimax characterization of eigenvalues and testing the Rayleigh quotient with suitable competitor functions. Section 5 contains a blow-up analysis for scaled eigenfunctions which allows proving Theorems 1.1 and 1.2. Finally, in Section  6 we prove Theorem 1.3.

**Notation. ** We list below some notation used throughout the paper.

For all $r>0$ and $a\in\mathbb{R}^{2}$, $D_{r}(a)=\{x\in\mathbb{R}^{2}\,:\,|x-a|<r\}$ denotes the disk of center $a$ and radius $r$.

• -

For all $r>0$, $D_{r}=D_{r}(0)$ and $\mathbb{S}^{1}=\partial D_{1}$.

• -

$ds$ denotes the arc length on $\partial D_{r}(a)$.

• -

For every complex number $z\in\mathbb{C}$, $\overline{z}$ denotes its complex conjugate.

• -

For $z\in\mathbb{C}$, $\mathop{\mathfrak{Re}}z$ denotes its real part and $\mathop{\mathfrak{Im}}z$ its imaginary part.

2. Local asymptotics of eigenfunctions

We recall from [11] the description of the asymptotics at the singularity of solutions to elliptic equations with Aharonov-Bohm potentials. In the case of Aharonov-Bohm potentials with circulation $\alpha\in(0,1)\setminus\{\frac{1}{2}\}$, such asymptotics is described in terms of eigenvalues and eigenfunctions of the following operator $\mathcal{H}$ acting on $2\pi$-periodic functions

[TABLE]

It is easy to verify that the eigenvalues of $\mathcal{H}$ are \big{\{}(\alpha-j)^{2}:\,j\in\mathbb{Z}\big{\}}; each eigenvalue $(\alpha-j)^{2}$ has multiplicity $1$ and the eigenspace associated is generated by the function $\frac{e^{ijt}}{\sqrt{2\pi}}$. Let us enumerate the eigenvalues $(\alpha-j)^{2}$ as \big{\{}(\alpha-j)^{2}\,:\,j\in\mathbb{Z}\big{\}}=\{\mu_{j}\,:\,j=1,2,\dots\} with $\mu_{1}<\mu_{2}<\mu_{3}<\ldots$, so that

[TABLE]

and $\mu_{2}=\max\{\alpha^{2},(1-\alpha)^{2}\}$.

Proposition 2.1** ([11], Theorem 1.3).**

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set containing $b$, $\lambda\in\mathbb{R}$, and $u\in H^{1,b}_{0}(\Omega,\mathbb{C})$ be a nontrivial weak solution to the problem

[TABLE]

i.e.

[TABLE]

Then there exists $j\in\mathbb{Z}$ such that

[TABLE]

Furthermore, there exists $\beta(b,u,\lambda)\neq 0$ such that

[TABLE]

as $r\to 0^{+}$ for any $\tau\in(0,1)$.

Let us fix $n\in\mathbb{N}$, $n\geq 1$. For all $a\in\Omega$, let $\varphi_{n}^{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ be an eigenfunction of problem ($E_{a}$) associated to the eigenvalue $\lambda_{n}^{a}$, i.e. solving

[TABLE]

such that

[TABLE]

Since $a\in\Omega\mapsto\lambda_{n}^{a}$ admits a continuous extension on $\overline{\Omega}$ as proved in [10, Theorem 1.1], we have that

[TABLE]

Moreover, from (2.3), (2.4), and (1.3) it follows that

[TABLE]

which, by (2.3) and classical elliptic regularity theory, implies that, for each $\omega\subset\subset\Omega\setminus\{0\}$, there exists $\rho_{\omega}>0$ such that

[TABLE]

The following lemma provides a detailed description of the behaviour of the Fourier coefficients of the function $t\mapsto\varphi_{n}^{a}(a+r(\cos t,\sin t))$ as $a$ is close to [math].

Lemma 2.2**.**

For $n\geq 1$ fixed and $a$ varying in $\Omega$, let $\varphi_{n}^{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ satisfy (2.3) and (2.4). For all $j\in\mathbb{Z}$ and $a\in\Omega$, let

[TABLE]

Then there exists $\rho_{0}>0$ such that, for all $a$ with $|a|\leq\rho_{0}$, the following properties hold.

1. (i)

For all $j\in\mathbb{Z}$, $v_{j}^{a}(r)=O(r^{|\alpha-j|})$ as $r\to 0^{+}$. In particular, for all $j\in\mathbb{Z}$ and for all $R>0$ such that $\{x\in\mathbb{R}^{2}\,:\,|x-a|\leq R\}\subset\Omega$, the value

[TABLE]

is well defined and independent of $R$. 2. (ii)

For all $j\in\mathbb{Z}$, $|\beta_{j}^{a}|\leq B$ for some $B>0$ independent of $j$ and $a$. 3. (iii)

For all $j\in\mathbb{Z}$,

[TABLE]

where $|R_{j,a}(r)|+|\widetilde{R}_{j,a}(r)|\leq{\rm const\,}r^{2}$ for some ${\rm const}>0$ independent of $j$ and $a$. 4. (iv)

$\varphi_{n}^{a}$* can be expanded as*

[TABLE]

being $R_{j,a}(r)$ as in (iii), where the convergence of the above series is uniform on disks $D_{R}(a)$ for each $R\in(0,1)$. 5. (v)

Letting ${\boldsymbol{\nu}}(t)=(\cos t,\sin t)$ and ${\boldsymbol{\tau}}(t)=(-\sin t,\cos t)$, $(i\nabla+A_{a})\varphi_{n}^{a}$ can be expanded as

[TABLE]

being $R_{j,a}(r),\widetilde{R}_{j,a}(r)$ as in (iii), where the above series converges absolutely in $L^{2}(D_{R}(a),\mathbb{C})$ and point-wise in $D_{R}(a)$ for each $R\in(0,1)$.

Proof.

The functions \big{\{}\frac{e^{ijt}}{\sqrt{2\pi}}\big{\}}_{j\in\mathbb{Z}} form an orthonormal basis of $L^{2}((0,2\pi),\mathbb{C})$. Hence, recalling that we are assuming that $\overline{D_{2}}\subset\Omega$, if $|a|$ sufficiently small $\varphi_{n}^{a}$ can be expanded as

[TABLE]

where $v_{j}^{a}$ is defined in (2.7). Equation (1.14) implies that, for every $j\in\mathbb{Z}$,

[TABLE]

or equivalently

[TABLE]

Integrating twice between $r$ and $1$, we obtain, for some $c_{1,j}^{a},c_{2,j}^{a}\in\mathbb{C}$,

[TABLE]

for all $r\in(0,1]$.

The convergence (2.2) in Proposition 2.1 implies that, for all $a$, $|\varphi_{n}^{a}(a+r(\cos t,\sin t))|=O(r^{\sqrt{\mu_{1}}})$ as $r\to 0^{+}$, with $\mu_{1}$ as in (2.1) (not necessarily uniformly with respect to $a$). Hence, for every $a$ in a sufficiently small neighborhood of [math], there exists a constant $C(a)>0$ such that, for all $j\in\mathbb{Z}$,

[TABLE]

We deduce that each function $v_{j}^{a}$ is bounded near [math], hence (2.11) necessarily yields

[TABLE]

We can therefore rewrite

[TABLE]

If $\sqrt{\mu_{1}}+2\geq|\alpha-j|$, using (2.12) to estimate the right hand side of (2.14) we obtain the improved estimate $|v_{j}^{a}(r)|\leq C(j,a)r^{|\alpha-j|}$. Otherwise, if $\sqrt{\mu_{1}}+2<|\alpha-j|$, we can use (2.12) to estimate the right hand side of (2.14) to obtain the improved estimate $|v_{j}^{a}(r)|\leq C(j,a)r^{\sqrt{\mu_{1}}+2}$, for some constant $C(j,a)>0$ depending on $a$ and $j$. By iterating the process $m+1$ times, with $m$ the largest natural number such that $\sqrt{\mu_{1}}+2m<|\alpha-j|$, we obtain that $|v_{j}^{a}(r)|\leq C(j,a)r^{|\alpha-j|}$, possibly for a different constant $C(j,a)$. We deduce that the quantity $\beta_{j}^{a}$ introduced in (2.8) is well defined. The fact that $\beta_{j}^{a}$ is independent of $R$ is a direct consequence of (2.10) and (2.14). This proves statement (i).

Using the independence of $\beta_{j}^{a}$ with respect to $R$, we choose $R=1$ in (2.8) and $r=1$ in (2.14) and obtain that

[TABLE]

so that (2.14) can be rewritten as

[TABLE]

From (2.16) it follows that, for all $r\in(0,1]$,

[TABLE]

Hence the Gronwall Lemma applied to the function $r\mapsto r^{-|\alpha-j|}|v_{j}^{a}(r)|$ yields that

[TABLE]

for some constant $C>0$ independent of $j$, $a$, and $r$.

From (2.11), (2.7), and (2.6) it follows that

[TABLE]

for some $\textrm{const\,}>0$ independent of $j$ and $a$; moreover from (2.13) and (2.4) we deduce that

[TABLE]

for some $\textrm{const\,}>0$ independent of $j$ and $a$. Hence

[TABLE]

for some $\widetilde{C}>0$ independent of $j$ and $a$.

Let $K>0$ be such that

[TABLE]

with $C$ being as in (2.17) and $\Lambda_{n}$ being as in (2.5). Hence, from (2.5), (2.15), (2.17) and (2.18) it follows that, if $|\alpha-j|>K$, then

[TABLE]

Let us choose $R_{0}\in(0,1)$ such that

[TABLE]

From (2.8) and (2.17) it follows that, if $|\alpha-j|\leq K$,

[TABLE]

Since, in view of (2.6), $v_{j}^{a}(R_{0})$ is bounded uniformly with respect to $a$ and $j$, we conclude that, for all $j$ such that $|\alpha-j|\leq K$, $|\beta_{j}^{a}|$ is bounded uniformly with respect to $a$ and $j$. This, together with (2.19), yields (ii).

From (2.16) and (2.17) it follows that

[TABLE]

where $|R_{j,a}(r)|\leq\textrm{const\,}r^{2}$ for some $\textrm{const\,}>0$ independent of $j$ and $a$, thus proving the first estimate in (iii). Differentiating (2.16) and using the above estimate (2.20), we easily obtain that

[TABLE]

where $|\widetilde{R}_{j,a}(r)|\leq\textrm{const\,}r^{2}$ for some $\textrm{const\,}>0$ independent of $j$ and $a$. Hence the proof of (iii) is complete.

From (2.9) and (iii) we have that the series $\frac{1}{\sqrt{2\pi}}\sum_{j\in\mathbb{Z}}r^{|\alpha-j|}\beta_{j}^{a}(1+R_{j,a}(r))e^{ijt}$ converges in $L^{2}((0,2\pi),\mathbb{C})$ to $\varphi_{n}^{a}(a+r(\cos t,\sin t))$ for all $r\in(0,1]$. In view of the estimates obtained in (ii)-(iii), Weierstrass M-Test ensures that the series is uniformly convergent in $D_{R}(a)$ for every $R\in(0,1)$, thus proving (iv).

Let $f_{j}^{a}(a+r(\cos t,\sin t))=v_{j}^{a}(r)\frac{e^{ijt}}{\sqrt{2\pi}}$. Since

[TABLE]

the above estimates also imply that, for every $R\in(0,1)$, the series of functions $\sum_{j}(i\nabla+A_{a})f_{j}^{a}$ is convergent absolutely in $L^{2}(D_{R}(a),\mathbb{C})$ and point-wise in $D_{R}(a)$ to $(i\nabla+A_{a})\varphi_{n}^{a}$ for every $R\in(0,1)$. Hence (v) follows from (iii). ∎

Corollary 2.3**.**

Under the same assumptions and with the same notation as in Lemma 2.2, let $R\in(0,1)$. Then, for all $r\in(0,R)$ and $t\in[0,2\pi]$,

[TABLE]

where $|\mathcal{R}_{a}(r,t)|\leq\text{\rm const\,}r^{1+\sqrt{\mu_{1}}}$ and $|\widetilde{\mathcal{R}}_{a}(r,t)|\leq\text{\rm const\,}r^{\sqrt{\mu_{1}}}$ for some $\text{\rm const\,}>0$ independent of $a,r,t$.

Proof.

From part (iv) of Lemma 2.2 we have that

[TABLE]

where

[TABLE]

Let us fix $R\in(0,1)$. Estimates (ii)–(iii) of Lemma 2.2 imply that, for some $\textrm{const\,}>0$ independent of $a,r,t$ (possibly varying from line to line),

[TABLE]

for all $r\in(0,R)$, thus proving (2.21).

From part (v) of Lemma 2.2 we have that

[TABLE]

where

[TABLE]

From Lemma 2.2 (ii)–(iii) we have that, for all $r\in(0,R)$,

[TABLE]

for some $\text{\rm const\,}>0$ independent of $a,r,t$ (possibly varying from line to line), thus proving (2.22). ∎

We now describe some consequences of Lemma 2.2 and Corollary 2.3, which will be needed in section 3 to prove a monotonicity type formula.

Lemma 2.4**.**

Under the same assumptions and with the same notation as in Lemma 2.2, we have that

[TABLE]

Proof.

Let $R\in(0,1)$ fixed. From (2.22) we have that, for all $r\in(0,R)$,

[TABLE]

where $|\widehat{\mathcal{R}}_{a}(r,t)|\leq\textrm{const\,}r^{2\sqrt{\mu_{1}}-1}$ for some $\textrm{const\,}>0$ independent of $a,r,t$. It follows that

[TABLE]

Moreover, from (2.22) we have that

[TABLE]

as $\varepsilon\to 0^{+}$, and hence, taking into account that $\int_{0}^{2\pi}a\cdot{\boldsymbol{\nu}}(t)\,dt=\int_{0}^{2\pi}a\cdot{\boldsymbol{\tau}}(t)\,dt=0$, we obtain

[TABLE]

from which the conclusion follows. ∎

Lemma 2.5**.**

For $n\geq 1$ fixed and $a$ varying in $\Omega$, let $\varphi_{n}^{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ satisfy (2.3) and (2.4). Let us assume that $\varphi_{n}^{a}\to\varphi_{n}^{0}$ in $L^{2}(\Omega,\mathbb{C})$ as $a\to 0$ (or respectively along a sequence $a_{\ell}\to 0$). Let $k\in\mathbb{Z}$ be such that $|\alpha-k|$ is the order of vanishing of $\varphi_{n}^{0}$ at [math]. For all $j\in\mathbb{Z}$ and $a\in\Omega$, let $v_{j}^{a}$ be as in (2.7) and $\beta_{j}^{a}$ be as in (2.8). Then there exists $\rho_{0}>0$ such that, for all $a$ with $|a|\leq\rho_{0}$ (respectively for $a=a_{\ell}$ with $\ell$ sufficiently large), the following properties hold.

1. (i)

For all $j\in\mathbb{Z}$, $\beta_{j}^{a}\to\beta_{j}^{0}$ as $a\to 0$ (respectively along the sequence $a_{\ell}\to 0$). 2. (ii)

There holds

[TABLE]

where $|\widehat{R}_{a}(r)|\leq h(r)$ for some function $h(r)$ independent of $a$ such that $h(r)\to 0$ as $r\to 0^{+}$, and

[TABLE]

where $|R_{j,a}(r)|\leq{\rm const\,}r^{2}$ for some ${\rm const}>0$ independent of $j$ and $a$, and $|R_{a}(r,t)|\leq f(r)$ for some function $f(r)$ independent of $a$ and $t$ such that $f(r)\to 0$ as $r\to 0^{+}$. 3. (iii)

Let ${\boldsymbol{\nu}}(t)=(\cos t,\sin t)$ and ${\boldsymbol{\tau}}(t)=(-\sin t,\cos t)$. There holds

[TABLE]

where $|\widetilde{R}_{a}(r)|\leq p(r)$ for some function $p(r)$ independent of $a$ such that $p(r)\to 0$ as $r\to 0^{+}$, and

[TABLE]

where $|{\mathbf{R}}_{j,a}(r,t)|\leq\textrm{const }r^{2}$ for some positive constant $\textrm{const }>0$ independent of $j$ and $a$ and $|{\widetilde{\mathbf{R}}}_{a}(r,t)|\leq g(r)$ for some function $g(r)$ independent of $a$ and $t$ such that $g(r)\to 0$ as $r\to 0^{+}$.

Proof.

In order to prove statement (i), we notice that (2.8) evaluated at $R=1$ provides

[TABLE]

From Lemma 2.2 (statements (ii) and (iii)) it follows that, for $|a|\leq\rho_{0}$ with $\rho_{0}>0$ sufficiently small,

[TABLE]

for some constant $C^{\prime}>0$ independent of $j$, $a$, and $r$. Moreover (2.3), (2.4), the convergence $\varphi_{n}^{a}\to\varphi_{n}^{0}$ in $L^{2}(\Omega,\mathbb{C})$, and standard elliptic estimates (see e.g. [12, Theorem 8.10]), imply that

[TABLE]

From (2.23), (2.24), (2.25), and the Dominated Convergence Theorem we obtain that, for all $j\in\mathbb{Z}$,

[TABLE]

thus proving (i).

If $k\in\mathbb{Z}$ is such that $|\alpha-k|$ is the order of vanishing of $\varphi_{n}^{0}$ at [math], from Lemma 2.2 (iii) it follows that $\beta_{k}^{0}\neq 0$ and $\beta_{j}^{0}=0$ for all $j\in\mathbb{Z}$ such that $|\alpha-j|<|\alpha-k|$; in particular, in view of (i), we have that $\lim_{a\to 0}\beta_{k}^{a}\neq 0$ and hence $\inf_{|a|\leq\rho_{0}}|\beta_{k}^{a}|>0$ for $\rho_{0}$ sufficiently small. Then, from Lemma 2.2 (iv) and the Parseval identity we deduce that

[TABLE]

with

[TABLE]

so that the first estimate in (ii) follows from Lemma 2.2 (ii) and (iii). From Lemma 2.2 (iii) we also deduce that

[TABLE]

where $|R_{a}(r,t)|\leq f(r)$ for some function $f(r)$ independent of $a$ and $t$ such that $f(r)\to 0$ as $r\to 0$. Then the second estimate in (ii) follows from Lemma 2.2 (iv).

From Lemma 2.2 (v) and the Parseval identity we deduce that

[TABLE]

so that the first estimate in (iii) follows from Lemma 2.2 (ii) and (iii) arguing as above. In a similar way, the second estimate in (iii) follows from statements (iii) and (v) of Lemma 2.2. ∎

Remark 2.6**.**

In the particular case $n=n_{0}$ with $n_{0}$ such that (1.5) holds, the above lemma applies to the family of eigenfunctions $\varphi_{a}=\varphi_{n_{0}}^{a}$ satisfying (1.14) and (1.15). Indeed, in this case (1.16) holds, i.e. the eigenfunctions $\varphi_{a}$ converge as $a\to 0^{+}$, so that the assumptions of Lemma 2.5 are fulfilled. In particular we deduce that, if $\varphi_{0}$ satisfies (1.7), (1.8), and (1.9) and if $\varphi_{a}$ is as in (1.14)–(1.15), then, for $a$ sufficiently close to [math], the vanishing order of $\varphi_{a}$ is less or equal than the vanishing order of $\varphi_{0}$.

Lemma 2.7**.**

For $n\geq 1$ fixed and $a$ varying in $\Omega\setminus\{0\}$, let $\varphi_{n}^{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ satisfy (2.3) and (2.4). Then there exist $\sigma>0$ and $C>0$ such that, for all $R>1$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma}{R}$,

[TABLE]

and

[TABLE]

Proof.

Let us prove the first estimate arguing by contradiction: assume that there exist sequences $R_{\ell}>1$ and $a_{\ell}\in\Omega$ such that $R_{\ell}|a_{\ell}|<\frac{1}{\ell}$ and

[TABLE]

It is easy to verify that, up to extracting a subsequence, $\varphi_{n}^{a_{\ell}}\to\varphi_{n}^{0}$ in $L^{2}(\Omega,\mathbb{C})$ as $\ell\to\infty$ for some $\varphi_{n}^{0}\in H^{1,0}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ satisfying

[TABLE]

Let $k\in\mathbb{Z}$ be such that $|\alpha-k|$ is the order of vanishing of $\varphi_{n}^{0}$ at [math]. Then, from Lemma 2.5 (first estimate in (ii)) it follows that, for $\ell$ sufficiently large,

[TABLE]

for some positive constant $\textrm{const }>0$ independent of $\ell$, while

[TABLE]

thus contradicting (2.26) as $\ell\to\infty$.

To prove the second estimate, let us assume by contradiction that there exist sequences $R_{\ell}>1$ and $a_{\ell}\in\Omega$ such that $R_{\ell}|a_{\ell}|<\frac{1}{\ell}$ and

[TABLE]

As above we have that, up to extracting a subsequence, $\varphi_{n}^{a_{\ell}}\to\varphi_{n}^{0}$ in $L^{2}(\Omega,\mathbb{C})$ as $\ell\to\infty$ for some $\varphi_{n}^{0}\in H^{1,0}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ satisfying (2.27). Then, from Lemma 2.5 (first estimate in (iii)) it follows that, for $\ell$ sufficiently large and for some positive constant $\textrm{const }>0$ independent of $\ell$,

[TABLE]

which, in view of (2.28), contradicts (2.29) as $\ell\to\infty$. ∎

Remark 2.8**.**

Arguing as in Lemma 2.7, we can also prove the following similar estimate (possibly taking a smaller $\sigma$ and a larger $C$ if necessary): for all $R>1$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma}{R}$

[TABLE]

and

[TABLE]

Lemma 2.9**.**

For $n\geq 1$ fixed, let $\varphi_{n}^{a}$ be a solution to (2.3)–(2.4). Let $\sigma>0$ and $C>0$ be as in Lemma 2.7 and Remark 2.8. Then, for all $R>2$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma}{R}$,

[TABLE]

Proof.

We note that

[TABLE]

where

[TABLE]

and

[TABLE]

We note that $\hat{\nu}$ is the outer unit normal vector on $\partial\mathcal{L}_{1,R}^{a}$ and $-\hat{\nu}$ is the outer unit normal vector on $\partial\mathcal{L}_{2,R}^{a}$. By denoting $\nu_{1}(x)=x/|x|$, we can rewrite the right hand side of (2.30) as

[TABLE]

We observe that

[TABLE]

Moreover, since $\nu_{1}$ is smooth in $\overline{\mathcal{L}_{1,R}^{a}}\cup\overline{\mathcal{L}_{2,R}^{a}}$, we can apply the Divergence Theorem to the first two terms in the right hand side of (2.31), thus rewriting the right hand side of (2.30) as

[TABLE]

Estimate of the first term in (2.32). Parametrizing $\partial D_{R|a|}(a)$ as $x=a+R|a|(\cos t,\sin t)$ and writing $a=|a|(\cos\theta_{a},\sin\theta_{a})$ for some angle $\theta_{a}\in[0,2\pi)$, we get

[TABLE]

on $\partial D_{R|a|}(a)$. Therefore,

[TABLE]

Estimate of the second term in (2.32). The second term in (2.32) splits into two parts

[TABLE]

Since $D_{R|a|}(0)\subset D_{(R+1)|a|}(a)$, we have that $\mathcal{L}_{1,R}^{a}\subseteq D_{(R+1)|a|}(a)\setminus D_{R|a|}(a)$. Let $\sigma>0$ and $C>0$ be as in Lemma 2.7 and Remark 2.8. Hence by Lemma 2.7 we have that, for all $R>1$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma}{R}$,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

for all $R>1$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma}{R}$.

Estimate of the third term in (2.32). The estimate of the third term can be derived in a similar way, observing that, since $D_{R|a|}(0)\supset D_{(R-1)|a|}(a)$, $\mathcal{L}_{2,R}^{a}\subseteq D_{R|a|}(a)\setminus D_{(R-1)|a|}(a)$ and using Remark 2.8 to obtain

[TABLE]

for all $R>2$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma}{R}$ (by possibly changing $C$ and $\sigma$).

Therefore combining (2.33), (2.34) and (2.35) we complete the proof. ∎

3. Monotonicity formula

3.1. Almgren type frequency function

Arguing as in [1, Lemma 3.1], one can easily prove the following Poincaré type inequality

[TABLE]

which holds for every $r>0$, $a\in D_{r}$, and $u\in H^{1,a}(D_{r},\mathbb{C})$. Furthermore, defining, for every $b\in D_{1}$,

[TABLE]

we have that the infimum $m_{b}$ is attained and $m_{b}>0$. Arguing as in [1], we can prove that $b\mapsto m_{b}$ is continuous in $D_{1}$ and that $m_{0}=\sqrt{\mu_{1}}$ (with $\mu_{1}$ as in (2.1)). Therefore a standard dilation argument yields that, for any $\delta\in(0,\sqrt{\mu_{1}})$, there exists some sufficiently large $\Upsilon_{\delta}>1$ such that, for every $r>0$ and $a\in D_{r}$ such that $\frac{|a|}{r}<\frac{1}{\Upsilon_{\delta}}$,

[TABLE]

For $\lambda\in\mathbb{R}$, $b\in\mathbb{R}^{2}$, $u\in H^{1,b}(D_{r},\mathbb{C})$ and $r>|b|$, we define the Almgren type frequency function as

[TABLE]

where

[TABLE]

For all $1\leq n\leq n_{0}$ and $a\in\Omega$, let $\varphi_{n}^{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})\setminus\{0\}$ be an eigenfunction of problem ($E_{a}$) associated to the eigenvalue $\lambda_{n}^{a}$, i.e. solving (2.3), such that

[TABLE]

For $n=n_{0}$, we choose

[TABLE]

with $\varphi_{a}$ as in (1.14)–(1.15). Let

[TABLE]

We recall that $\Lambda$ is finite in view of the continuity result of the eigenvalue function $a\mapsto\lambda_{n}^{a}$ in $\overline{\Omega}$ proved in [10, Theorem 1.1].

Arguing as in [1, Lemma 5.2], we can prove that there exists $0<R_{0}<\left(\Lambda\left(1+\tfrac{2}{\sqrt{\mu_{1}}}\right)\right)^{-1/2}$ such that $D_{R_{0}}\subset\Omega$ and, if $|a|<R_{0}$,

[TABLE]

Furthermore, for every $r\in(0,R_{0}]$ there exist $C_{r}>0$ and $\alpha_{r}\in(0,r)$ such that

[TABLE]

Thanks to (3.4), the function $r\mapsto N(\varphi_{n}^{a},r,\lambda_{n}^{a},A_{a})$ is well defined in $(|a|,R_{0})$. By direct calculations (see [18] for details), we can prove that

[TABLE]

where

[TABLE]

Lemma 2.9 together with Lemmas 2.2 and 2.4 allow us to give an estimate of the quantity $M_{n}^{a}$ defined in (3.8). We notice that the techniques used in [1, 18] to estimate the term $M_{n}^{a}$ for $\alpha=1/2$ were based on the possibility of rewriting the problem as a Laplace equation on the twofold covering; hence it is not possibile here to extend such proofs to the case $\alpha\not\in\frac{\mathbb{Z}}{2}$ and a new strategy of proof is needed.

Lemma 3.1**.**

There exist $\sigma_{0}>0$ and $c_{0}>2$ such that, for every $1\leq n\leq n_{0}$, $R>c_{0}$ and $a\in\Omega$ such that $|a|<\frac{\sigma_{0}}{R}$,

[TABLE]

Proof.

Let us fix $n\in\{1,2,\dots,n_{0}\}$ and define, for $|a|$ small and $r\in(0,1]$,

[TABLE]

From the Parseval identity and Lemma 2.2 (iv) it follows that there exists $\sigma_{n}>0$ such that, for every $R>2$ and $a\in\Omega$ such that $|a|<\frac{\sigma_{n}}{R}$,

[TABLE]

where the $\beta_{j}^{a}$’s are the coefficients defined in (2.8) for the eigenfunction $\varphi_{n}^{a}$ (with $n$ fixed). From the elementary inequality $ab\leq\frac{1}{2}(a^{2}+b^{2})$, it follows that

[TABLE]

Combining (3.9) and (3.10) we obtain that

[TABLE]

Moreover, Lemma 2.4 implies that

[TABLE]

Lemma 2.9 provides some constant $c_{n}$ (independent of $a$ and $R$) such that, for a possibly smaller $\sigma_{n}$ and for all $R>2$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma_{n}}{R}$,

[TABLE]

Therefore, by combining (3.11), (3.12), and (3.13), we obtain

[TABLE]

for all $R>c_{n}+2$ and $a\in\Omega$ such that $0<|a|<\frac{\sigma_{n}}{R}$.

The conclusion then follows by repeating the argument for all $n\in\{1,2,\dots,n_{0}\}$ and choosing $\sigma_{0}=\min\{\sigma_{n}:1\leq n\leq n_{0}\}$ and $c_{0}=\max\{2+c_{n}:1\leq n\leq n_{0}\}$. ∎

Lemma 3.2**.**

For $\delta\in(0,\frac{1}{2}\sqrt{\mu_{1}})$, let $\Upsilon_{\delta}$ be such that (3.2) holds. Let $R_{0}$ be as above, $r_{0}\leq R_{0}$ and $n\in\{1,\dots,n_{0}\}$. If $\Upsilon_{\delta}|a|\leq r_{1}<r_{2}\leq r_{0}$ and $\varphi_{n}^{a}$ is a solution to (2.3) satisfying (3.3), then

[TABLE]

Proof.

Combining (3.1) with (3.2) we obtain that, for every $\Upsilon_{\delta}|a|<r<r_{0}$,

[TABLE]

From above, (3.6) and (3.2), we have that for every $\Upsilon_{\delta}|a|<r<r_{0}$

[TABLE]

so that, in view of (3.4),

[TABLE]

Integrating between $r_{1}$ and $r_{2}$ we obtain the desired inequality. ∎

Lemma 3.3**.**

For $n\in\{1,\dots,n_{0}\}$ and $a\in\Omega$, let $\varphi_{n}^{a}$ be a solution of (2.3) satisfying (3.3). Let $R_{0}$ be as above, $\sigma_{0}$ and $c_{0}>0$ be as in Lemma 3.1 and let $r_{0}\leq\min\{R_{0},\sigma_{0}\}$. For $\delta\in(0,\frac{\sqrt{\mu_{1}}}{2})$, let $\Upsilon_{\delta}>1$ be such that (3.2) holds. Then, there exists $c_{r_{0},\delta}>0$ such that for all $R>\max\{\Upsilon_{\delta},c_{0}\}$, $|a|<r_{0}/R$, $R|a|\leq r<r_{0}$ and $n\in\{1,\dots,n_{0}\}$,

[TABLE]

Proof.

By direct calculations, using the expressions for the derivatives of the functions $H(\varphi_{n}^{a},r)$ and $E(\varphi_{n}^{a},r,\lambda_{n}^{a},A_{a})$ written in (3.6) and (3.7) and the Cauchy-Schwarz inequality, we obtain that

[TABLE]

By Lemmas 3.2 and 3.1 the first term can be estimated as

[TABLE]

for all $R>\max\{\Upsilon_{\delta},c_{0}\}$, $|a|<r_{0}/R$, $R|a|\leq r<r_{0}$ and $n\in\{1,\dots,n_{0}\}$.

For the second term, the Poincaré inequality (3.1) leads to

[TABLE]

for $r<r_{0}$, which implies

[TABLE]

for $r<r_{0}$. Using (3.15) and (3.16) we can estimate the right hand side of (3.14) thus obtaining

[TABLE]

for all $R|a|\leq r<r_{0}$ with $R>\max\{\Upsilon_{\delta},c_{0}\}$. Integrating between $r$ and $r_{0}$ and using the fact that $R|a|\leq r<r_{0}$, we obtain the statement with

[TABLE]

Lemma 3.4**.**

Let $\varphi_{a}$ be a solution of (1.14)–(1.15) and let $k$ be as in (1.8). For every $\delta\in(0,\frac{\sqrt{\mu_{1}}}{2})$, there exist $r_{\delta}\in(0,R_{0})$ and $K_{\delta}>\Upsilon_{\delta}$ such that, if $R>K_{\delta}$, $|a|<r_{\delta}/R$ and $R|a|\leq r<r_{\delta}$, then

[TABLE]

Proof.

From (1.16)–(1.17) it follows that, for every $r<R_{0}$,

[TABLE]

Moreover, from [11, Theorem 1.3] we know that, under assumption (1.9),

[TABLE]

Then, the proof is a direct consequence of Lemma 3.3, see [18, Lemma 7.2], [1, Lemma 5.7], [4, Lemma 5.7] for details. ∎

3.2. Local energy estimates

Corollary 3.5**.**

For $\delta\in(0,\frac{1}{2}\sqrt{\mu_{1}})$ let $r_{\delta},K_{\delta}$ be as in Lemma 3.4 and $\alpha_{r_{\delta}}$ be as in (3.5). Then there exists $C_{\delta}>0$ such that

[TABLE]

Proof.

From (3.6), the definition of $\mathcal{N}$, and Lemma 3.4 we have that

[TABLE]

so that estimate (3.17) follows by integration over $[K_{\delta}|a|,R|a|]$ and estimate (3.18) from integration over $[K_{\delta}|a|,r_{\delta}]$ and (3.5). Finally (3.19) is a direct consequence of Lemma 3.2. ∎

Lemma 3.6**.**

For $n\in\{1,\dots,n_{0}\}$ and $a\in\Omega$, let $\varphi_{n}^{a}$ be a solution to (2.3) satisfying (3.3). Let $R_{0}>0$ be as in (3.4). For every $\delta\in(0,\sqrt{\mu_{1}}/2)$, there exist $\tilde{K}_{\delta}>1$ and $\tilde{C}_{\delta}>0$ such that, for all $R>\tilde{K}_{\delta}$, $a\in\Omega$ with $R|a|<R_{0}$, and $n\in\{1,\dots,n_{0}\}$,

[TABLE]

Proof.

Estimate (3.21) follows from Lemma 3.2. From Lemma 3.3 it follows that the frequency $\mathcal{N}$ is bounded in $r=R|a|$ provided $R$ is sufficiently large; hence $E(\varphi_{n}^{a},R|a|,\lambda_{n}^{a},A_{a})$ is uniformly estimated by $H(\varphi_{n}^{a},R|a|)$, so that (3.21) and (3.1)–(3.2) yield (3.20). Estimate (3.22) can be proved combining (3.20), (3.21) with the Poincaré inequality (3.1). We refer to [1, Lemma 5.8] for more details in a related problem. ∎

Lemma 3.7**.**

For $a\in\Omega$ let $\varphi_{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})$ be a solution of (1.14)–(1.15). For some fixed $\delta\in(0,\sqrt{\mu_{1}}/2)$, let $K_{\delta}>\Upsilon_{\delta}$ be as in Lemma 3.4. Then, for every $R>K_{\delta}$,

[TABLE]

Proof.

The proof follows from the boundedness of the frequency $\mathcal{N}(\varphi_{a},R|a|,\lambda_{a},A_{a})$ established in Lemma 3.4 and by its scaling properties. For $\delta\in(0,\sqrt{\mu_{1}}/2)$ fixed, let $K_{\delta}>\Upsilon_{\delta}$ and $r_{\delta}$ be as in Lemma 3.4, so that Lemma 3.4 yields that

[TABLE]

Then, by (3.1) and (3.2) it follows that

[TABLE]

Then (3.23) follows from (3.17). Estimates (3.24) and (3.25) follow from (3.23) and the Poincaré type inequalities (3.1) and (3.2). ∎

Remark 3.8**.**

Let us consider the blow-up family

[TABLE]

with $K_{\delta}>\Upsilon_{\delta}$ being as in Lemma 3.4 for some fixed $\delta\in(0,\sqrt{\mu_{1}}/2)$. From Lemma 3.7 it follows that, for every $p\in{\mathbb{S}}^{1}$ fixed, $r_{\delta}>0$ as in Lemma 3.4, and $R>K_{\delta}$, the blow-up family $\{\tilde{\varphi}_{a}\,:\,a=|a|p,\,R|a|<r_{\delta}\}$ is bounded in $H^{1,p}(D_{R},\mathbb{C})$.

4. Estimate on $\lambda_{0}-\lambda_{a}$

Respectively, the Courant-Fisher characterization for $\lambda_{a}$ and $\lambda_{0}$ gives

[TABLE]

and

[TABLE]

Before proceeding, we find useful to recall the following technical result which is proved in [1, Lemma 6.1] and concerns the maximum of quadratic forms depending on the pole $a\to 0$.

Lemma 4.1**.**

For every $a\in\Omega$, let us consider a quadratic form

[TABLE]

with $M_{j,n}(a)\in\mathbb{C}$ such that $M_{j,n}(a)=\overline{M_{n,j}(a)}$. Let us assume that there exist $\gamma\in(0,+\infty)$, $a\mapsto\sigma(a)\in\mathbb{R}$ with $\sigma(a)\geq 0$ and $\sigma(a)=O(|a|^{2\gamma})$ as $|a|\to 0^{+}$, and $a\mapsto\mu(a)\in\mathbb{R}$ with $\mu(a)=O(1)$ as $|a|\to 0^{+}$, such that the coefficients $M_{j,n}(a)$ satisfy the following conditions:

1. (i)

$M_{n_{0},n_{0}}(a)=\sigma(a)\mu(a)$; 2. (ii)

for all $j<n_{0}$, $M_{j,j}(a)\to M_{j}$ as $|a|\to 0^{+}$ for some $M_{j}\in\mathbb{R}$, $M_{j}<0$; 3. (iii)

for all $j<n_{0}$, $M_{j,n_{0}}(a)=\overline{M_{n_{0},j}(a)}=O\left(|a|^{\gamma}\sqrt{\sigma(a)}\right)$ as $|a|\to 0^{+}$; 4. (iv)

for all $j,n<n_{0}$ with $j\neq n$, $M_{j,n}(a)=O(|a|^{2\gamma})$ as $|a|\to 0^{+}$; 5. (v)

there exists $M\in\mathbb{N}$ such that $|a|^{(2+M)\gamma}=o(\sigma(a))$ as $|a|\to 0^{+}$.

Then

[TABLE]

where \|z\|=\|(z_{1},z_{2},\dots,z_{n_{0}})\|=\big{(}\sum_{j=1}^{n_{0}}|z_{j}|^{2}\big{)}^{1/2}.

4.1. Construction of the test functions using $\varphi_{n}^{0}$

Let $R_{0}$ be as in (3.4). For every $R>1$, $a\in\Omega$ with $|a|<R_{0}/R$ and $1\leq n\leq n_{0}$ we define

[TABLE]

where

[TABLE]

and $w_{n,R,a}^{int}$ is the unique solution to the minimization problem

[TABLE]

We notice that $w_{n,R,a}^{ext}$ and $w_{n,R,a}^{int}$ respectively solve

[TABLE]

and

[TABLE]

As a consequence of Proposition 2.1 we have that, for every $R>1$, $a\in\Omega$ such that $R|a|<R_{0}$, and $1\leq n\leq n_{0}$,

[TABLE]

Using the above estimates (4.3) and the Dirichlet principle (see the proof of [1, Lemma 6.2] for details in the case of half-integer circulation), we obtain that, for every $R>2$ and $1\leq n\leq n_{0}$,

[TABLE]

The above estimates can be made more precise in the case $n=n_{0}$ in view of (1.9): for every $R>2$ and $a\in\Omega$ with $R|a|<R_{0}$

[TABLE]

and consequently, in view of the Dirichlet principle,

[TABLE]

with $k$ as in (1.8). Furthermore, defining

[TABLE]

for all $R>2$ and $a\in\Omega$ such that $R|a|<R_{0}$, (1.9) implies that

[TABLE]

where $\psi_{k}$ is defined in (1.20).

4.2. Estimate of the Rayleigh quotient for $\lambda_{a}$

Lemma 4.2**.**

There exists $\mathfrak{c}\in\mathbb{R}$ such that

[TABLE]

where $k$ is as in (1.8).

Proof.

The proof follows the lines of [1, Lemma 6.7] and [4, Lemma 7.2]. Let us fix $R>2$. By proceeding with a Gram-Schmidt process we define

[TABLE]

where

[TABLE]

and

[TABLE]

From (4.3), (4.4) and an induction argument it follows that, for all $\ell,n$ such that $1\leq n\leq n_{0}-1$ and $n+1\leq\ell\leq n_{0}$,

[TABLE]

as $|a|\to 0$. Morever, from (4.5) and (4.6) we have that

[TABLE]

and

[TABLE]

Since $\text{dim}(\text{span}\{w_{1,R,a},\ldots,w_{n_{0},R,a}\})=n_{0}$, we have that also $\text{dim}(\text{span}\{\tilde{w}_{1,a},\ldots,\tilde{w}_{n_{0},a}\})=n_{0}$, and hence from (4.1) we deduce that

[TABLE]

which leads to

[TABLE]

where $p_{n,j}^{a}=\int_{\Omega}(i\nabla+A_{a})\tilde{w}_{n,a}\cdot\overline{(i\nabla+A_{a})\tilde{w}_{j,a}}\,dx-\lambda_{0}\delta_{nj}$, with $\delta_{nj}=1$ if $n=j$ and $\delta_{nj}=0$ otherwise. Using the estimates above we can now estimate $p_{n,j}^{a}$. First, using (4.5), (4.6), and (4.10)

[TABLE]

Next (4.3), (4.4) and (4.9) provide for $n<n_{0}$

[TABLE]

as $|a|\to 0$. Using (4.3), (4.4), (4.5), (4.6), (4.9) and (4.11), we have that, for all $n<n_{0}$,

[TABLE]

while the same estimates imply that, for all $n\neq\ell<n_{0}$,

[TABLE]

Therefore, the quadratic form in (4.12) satisfies the hypothesis of Lemma 4.1 with $\sigma(a)=|a|^{2|\alpha-k|}$, $\gamma=\sqrt{\mu_{1}}$, $M_{j}=\lambda_{j}^{0}-\lambda_{0}<0$ for $j<n_{0}$ and $M\in\mathbb{N}$ such that $(2+M)\sqrt{\mu_{1}}>2|\alpha-k|$, so that

[TABLE]

The proof is thereby complete. ∎

We notice that Lemma 4.2 does not give any information about the sign of the constant $\mathfrak{c}$.

4.3. Construction of the test functions using $\varphi_{n}^{a}$

Let $R_{0}$ be as in (3.4), $R>1$ and $|a|<\frac{R_{0}}{R}$. For every $1\leq n\leq n_{0}$ we define

[TABLE]

where

[TABLE]

and $v_{n,R,a}^{int}$ is the unique solution to the minimization problem

[TABLE]

We notice that $v_{n,R,a}^{ext}$ and $v_{n,R,a}^{int}$ respectively solve

[TABLE]

and

[TABLE]

The energy estimates obtained in Lemmas 3.6 and 3.7 imply the following estimates for the functions $v_{n,R,a}^{int}$.

Lemma 4.3**.**

For $\delta\in(0,\sqrt{\mu_{1}}/2)$ fixed, let $\tilde{K}_{\delta}$ be as in Lemma 3.6 and $R_{0}$ be as in (3.4). Let $R>\max\{2,\tilde{K}_{\delta}\}$ and $1\leq n\leq n_{0}$ be fixed. For every $a\in\Omega$ with $|a|<R_{0}/R$, let $v_{n,R,a}^{int}$ be defined as in (4.13). Then

[TABLE]

as $|a|\to 0^{+}$.

Proof.

The proof follows by combining the Dirichlet principle, a suitable cutting-off procedure, and Lemma 3.6 (see the proof of [1, Lemma 6.2] for details in the case of half-integer circulation). ∎

Lemma 4.4**.**

For $R>\max\{2,K_{\delta}\}$ fixed, with $K_{\delta}$ being as in Lemma 3.4, let $v_{n_{0},R,a}^{int}$ be defined as in (4.13). Then

[TABLE]

as $|a|\to 0^{+}$.

Proof.

The proof follows from the estimates of Lemma 3.7, a suitable cutting-off procedure, and the Dirichlet principle (see (4.13)). ∎

Remark 4.5**.**

For all $R>2$ and $a\in\Omega$ with $|a|<R_{0}/R$ we consider the blow-up family

[TABLE]

with $K_{\delta}$ as in Lemma 3.4 for some fixed $\delta\in(0,\sqrt{\mu_{1}}/2)$. From Lemma 4.4 it follows that, for every $p\in{\mathbb{S}}^{1}$ fixed, $r_{\delta}>0$ as in Lemma 3.4, and $R>\max\{K_{\delta},2\}$, the family of functions $\{Z_{a}^{R}\,:\,a=|a|p\in\Omega,\,|a|<r_{\delta}/R\}$ is bounded in $H^{1,0}(D_{R},\mathbb{C})$.

4.4. Estimate of the Rayleigh quotient for $\lambda_{0}$

An estimate from above for the limit eigenvalue $\lambda_{0}$ in terms of the approximating eigenvalue $\lambda_{a}$ can be obtained by choosing as test functions in (4.2) an orthonormal family constructed starting from the functions $\{v_{n,R,a}\}_{n=1,\dots,n_{0}}$, as done in the following.

Lemma 4.6**.**

For $\delta\in(0,\frac{1}{2}\sqrt{\mu_{1}})$ fixed, let $r_{\delta},K_{\delta}$ be as in Lemma 3.4 and $\alpha_{r_{\delta}}$ be as in (3.5). Then there exists ${\mathfrak{d}}_{\delta}>0$ such that

[TABLE]

for all $a\in\Omega$ such that |a|<\min\big{\{}\frac{r_{\delta}}{K_{\delta}},\alpha_{r_{\delta}}\big{\}}.

Proof.

In view of (3.18) it is enough to prove that $\lambda_{0}-\lambda_{a}\leq O(H(\varphi_{a},K_{\delta}|a|))$ as $|a|\to 0^{+}$.

Let us fix $R>\max\{2,K_{\delta},\tilde{K}_{\delta}\}$, with $\tilde{K}_{\delta}$ as in Lemma 3.6. As in the proof of Lemma 4.2, we use a Gram-Schmidt process, that is we define

[TABLE]

where

[TABLE]

and

[TABLE]

From (3.22), (4.15) and an induction argument it follows that, for every $1\leq n\leq n_{0}-1$ and $n+1\leq\ell\leq n_{0}$,

[TABLE]

as $|a|\to 0$. Moreover, from (3.25) and (4.17), we have that

[TABLE]

and, for $1\leq n\leq n_{0}-1$,

[TABLE]

Since $\text{dim}(\text{span}\{v_{1,R,a},\ldots,v_{n_{0},R,a}\})=n_{0}$, we have that also $\text{dim}(\text{span}\{\tilde{v}_{1,a},\ldots,\tilde{v}_{n_{0},a}\})=n_{0}$, and hence from (4.2) we deduce that

[TABLE]

which leads to

[TABLE]

where $q_{n,j}^{a}=\int_{\Omega}(i\nabla+A_{0})\tilde{v}_{n,a}\cdot\overline{(i\nabla+A_{0})\tilde{v}_{j,a}}\,dx-\lambda_{a}\delta_{nj}$. Using the results above we can now estimate $q_{n,j}^{a}$. First, using (4.16), (3.23), and (4.20)

[TABLE]

as $|a|\to 0^{+}$. Next (4.15), (3.20), (4.19), and the fact that $\lambda_{n}^{a}\to\lambda_{n}^{0}$ as $|a|\to 0$, provide, for $n<n_{0}$,

[TABLE]

as $|a|\to 0$. Now, using (3.20), (3.23), (4.15), (4.16), (4.19), (4.20), and (4.21), we prove that, for all $n<n_{0}$,

[TABLE]

while the same estimates imply that, for all $n\neq\ell<n_{0}$,

[TABLE]

Therefore, the quadratic form in (4.22) satisfies the hypothesis of Lemma 4.1 with $\gamma=\sqrt{\mu_{1}}-\delta$, $\sigma(a)=H(\varphi_{a},K_{\delta}|a|)=O(|a|^{2\gamma})$ (by (3.19)), $M_{j}=\lambda_{j}^{0}-\lambda_{0}<0$ and $M$ any natural number such that $M>2(|\alpha-k|-\sqrt{\mu_{1}}+2\delta)/(\sqrt{\mu_{1}}-\delta)$ by Corollary 3.5. Therefore the right hand side in (4.22) satisfies

[TABLE]

as $|a|\to 0^{+}$. Then the conclusion follows from (4.22). ∎

4.5. Energy estimates

Corollary 4.7**.**

For $\delta\in(0,\frac{1}{2}\sqrt{\mu_{1}})$ fixed, let $K_{\delta}$ be as in Lemma 3.4. Then

1. (i)

$|\lambda_{0}-\lambda_{a}|=O(1)\max\{H(\varphi_{a},K_{\delta}|a|),|a|^{2|\alpha-k|}\}$* as $a\to 0$;* 2. (ii)

|\lambda_{0}-\lambda_{a}|=O\Big{(}(H(\varphi_{a},K_{\delta}|a|))^{\frac{|\alpha-k|}{|\alpha-k|+\delta}}\Big{)}* as $a\to 0$.*

Proof.

Estimate (i) is a direct consequence of Lemmas 4.2 and 4.6. Corollary 3.5 implies that |a|^{2|\alpha-k|}=O\big{(}(H(\varphi_{a},K_{\delta}|a|))^{\frac{|\alpha-k|}{|\alpha-k|+\delta}}\big{)} as $a\to 0$, so that (ii) follows from (i). ∎

5. Blow-up analysis

In order to obtain a more precise estimate of the order of vanishing of the eigenvalue variation $|\lambda_{0}-\lambda_{a}|$ than Corollary 4.7, we have now to compare the order of $H(\varphi_{a},K_{\delta}|a|)$ with $|a|^{2|\alpha-k|}$. We observe that the estimates obtained so far (in particular Corollary 3.5) are not enough to decide what is the dominant term among $H(\varphi_{a},K_{\delta}|a|)$ and $|a|^{2|\alpha-k|}$. To this aim, our next step is a blow-up analysis for scaled eigenfunctions (3.26) along a fixed direction $p\in{\mathbb{S}}^{1}$. In order to identify the limit profile of the blow-up family (3.26), the following energy estimate of the difference between approximating and limit scaled eigenfunctions plays a crucial role.

Let ${\mathcal{D}}^{1,2}_{0}(\mathbb{R}^{2},\mathbb{C})$ be the completion of $C^{\infty}_{\rm c}(\mathbb{R}^{2}\setminus\{0\},\mathbb{C})$ with respect to the magnetic Dirichlet norm

[TABLE]

Theorem 5.1** (Energy estimates for eigenfunction variation).**

Let $p\in{\mathbb{S}}^{1}$ be fixed. For some fixed $\delta\in(0,\sqrt{\mu_{1}}/2)$, let $K_{\delta}>\Upsilon_{\delta}$ be as in Lemma 3.4. For every $R>\max\{2,K_{\delta}\}$ and $a=|a|p\in\Omega$ such that $|a|<R_{0}/R$, let $v_{n_{0},R,a}$ be as in $\S$ 4.3. Then

[TABLE]

where $C>0$ is independent of $a,R,p$,

[TABLE]

and, for $p$ and $R$ fixed,

[TABLE]

as $|a|\to 0^{+}$.

Proof.

The proof exploits the invertibility of the differential of the function $F$ defined below, in the spirit of [4, Theorem 8.2] and [1, Theorem 7.2]. Let

[TABLE]

In the above definition, $(H^{1,0}_{0,\mathbb{R}}(\Omega,\mathbb{C}))^{\star}$ is the real dual space of $H^{1,0}_{0,\mathbb{R}}(\Omega,\mathbb{C})=H^{1,0}_{0}(\Omega,\mathbb{C})$, which is here meant as a vector space over $\mathbb{R}$ endowed with the norm

[TABLE]

and $(i\nabla+A_{0})^{2}\varphi-\lambda\varphi\in(H^{1,0}_{0,\mathbb{R}}(\Omega,\mathbb{C}))^{\star}$ acts as

[TABLE]

for all $\varphi\in H^{1,0}_{0}(\Omega,\mathbb{C})$. It is easy to prove that the function $F$ is Fréchet-differentiable at $(\lambda_{0},\varphi_{0})$, with differential $dF(\lambda_{0},\varphi_{0})\in\mathcal{L}(\mathbb{C}\times H^{1,0}_{0}(\Omega,\mathbb{C}),\mathbb{R}\times\mathbb{R}\times(H^{1,0}_{0,\mathbb{R}}(\Omega,\mathbb{C}))^{*})$ given by

[TABLE]

for every $(\lambda,\varphi)\in\mathbb{C}\times H^{1,0}_{0}(\Omega,\mathbb{C})$. From the simplicity assumption (1.5) it follows that $dF(\lambda_{0},\varphi_{0})$ is invertible, see [1, Lemma 7.1] for details.

From the definition of $v_{n_{0},R,a}$, (1.17), (3.19), (3.23), (4.5), and (4.16) it follows that

[TABLE]

as $|a|\to 0$, so that $v_{n_{0},R,a}\to\varphi_{0}$ in $H^{1}_{0}(\Omega,\mathbb{C})$ as $|a|\to 0^{+}$. Then, from the invertibility of $dF(\lambda_{0},\varphi_{0})$ we have that

[TABLE]

as $|a|\to 0^{+}$. We denote

[TABLE]

where

[TABLE]

In view of (4.16), (3.23), and Corollary 4.7 we have that

[TABLE]

as $|a|\to 0^{+}$. The normalization condition for the phase in (1.15) together with (4.17), (4.5), and (3.25) yield

[TABLE]

as $|a|\to 0^{+}$.

For every $a\in\Omega$, the map

[TABLE]

is an isometry of $\mathcal{D}^{1,2}_{0}(\mathbb{R}^{2},\mathbb{C})$.

Since $H^{1,0}_{0}(\Omega,\mathbb{C})$ is continuously embedded into $\mathcal{D}^{1,2}_{0}(\mathbb{R}^{2},\mathbb{C})$ by trivial extension outside $\Omega$ and $\|u\|_{\mathcal{D}^{1,2}_{0}(\mathbb{R}^{2},\mathbb{C})}=\|u\|_{H^{1,0}_{0}(\Omega,\mathbb{C})}$ for every $u\in H^{1,0}_{0}(\Omega,\mathbb{C})$, we have that

[TABLE]

For every $\varphi\in\mathcal{D}^{1,2}_{0}(\mathbb{R}^{2},\mathbb{C})$ we have that

[TABLE]

From scaling and integration by parts we have that, letting $\tilde{\varphi}_{a}$ be defined in (3.26),

[TABLE]

being $\nu=\frac{x}{|x|}$ the outer unit normal vector. In a similar way we have that, defining $Z_{a}^{R}$ as in (4.18) and using (4.14),

[TABLE]

Combining (5.4), (5.5), (5.6), (5.7), and recalling that $\mathcal{T}_{a}$ is an isometry of $\mathcal{D}^{1,2}_{0}(\mathbb{R}^{2},\mathbb{C})$, we obtain that

[TABLE]

where

[TABLE]

From Remarks 3.8 and 4.5 it follows that, for $R>\max\{2,K_{\delta}\}$ and $p\in{\mathbb{S}}^{1}$ fixed,

[TABLE]

so that, for $p$ and $R$ fixed, $h(p,a,R)=O(1)$ as $a\to 0$. Moreover Remark 4.5 implies that, for $R>\max\{2,K_{\delta}\}$ and $p\in{\mathbb{S}}^{1}$ fixed,

[TABLE]

Hence the conclusion follows from (5.1), (5.2), (5.3), and (5.8). ∎

The previous theorem allows estimating the energy variation of scaled eigenfunctions and improving the results of Corollary 3.5 as follows.

Corollary 5.2**.**

Let $p\in{\mathbb{S}}^{1}$ be fixed. Then

• $(i)$

$|a|^{2|\alpha-k|}=O(H(\varphi_{a},K_{\delta}|a|))$* as $a=|a|p\to 0$;*

• $(ii)$

letting $\tilde{\varphi}_{a}$ and $W_{a}$ be as in (3.26) and (4.7), for every $R>\max\{2,K_{\delta}\}$ there holds

[TABLE]

Proof.

Estimate (5.9) follows from scaling and Theorem 5.1. From (5.9) it follows that

[TABLE]

as $a=|a|p\to 0$. From Remark 3.8 and (4.8), the above estimate implies (i). ∎

In the following lemma we prove the existence and uniqueness of the function $\Psi_{p}$ satisfying (1.18) and (1.19), which will turn out to be the limit of the blowed-up family (3.26) as $a\to 0$ along the fixed direction $p\in{\mathbb{S}}^{1}$.

Lemma 5.3**.**

Let $p\in{\mathbb{S}}^{1}$. There exists a unique $\Psi_{p}\in H^{1,p}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$ satisfying (1.18) and (1.19).

Proof.

Let $\eta$ be a smooth cut-off function such that $\eta\equiv 0$ in $D_{1}$ and $\eta\equiv 1$ in $\mathbb{R}^{2}\setminus D_{R}$ for some $R>1$. Recalling the definition of $\psi_{k}$ (1.20), we have

[TABLE]

Here $\mathcal{D}^{1,2}_{p}(\mathbb{R}^{2},\mathbb{C})$ is the completion of $C^{\infty}_{c}(\mathbb{R}^{2}\setminus\{0\},\mathbb{C})$ with respect to

[TABLE]

By the Lax-Milgram’s Theorem, there exists a unique $g\in\mathcal{D}^{1,2}_{p}(\mathbb{R}^{2},\mathbb{C})$ which solves

[TABLE]

Then, $\Psi_{p}=g+\eta e^{i\alpha(\theta_{p}-\theta_{0}^{p})}\psi_{k}$ satisfies (1.18) and (1.19), so that the existence is proved.

The uniqueness follows from the fact that, if $\Psi_{p}^{1}$, $\Psi_{p}^{2}\in H^{1,p}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$ satisfy (1.18) and (1.19), then

[TABLE]

and

[TABLE]

which, in view of the Hardy inequality (1.3), implies that

[TABLE]

and hence that $\Psi_{p}^{1}-\Psi_{p}^{2}\in\mathcal{D}^{1,2}_{p}(\mathbb{R}^{2},\mathbb{C})$. Therefore we can test equation (5.10) with $\Psi_{p}^{1}-\Psi_{p}^{2}$ thus concluding that

[TABLE]

which implies that $\Psi_{p}^{1}\equiv\Psi_{p}^{2}$. ∎

We are now in position to prove that the scaled eigenfunctions (3.26) converge to a multiple of $\Psi_{p}$ as $a=|a|p\to 0$.

Lemma 5.4**.**

Let $p\in{\mathbb{S}}^{1}$ and $\delta\in(0,\sqrt{\mu_{1}}/2)$ be fixed and let $K_{\delta}>\Upsilon_{\delta}$ be as in Lemma 3.4. For $a=|a|p\in\Omega$ let $\tilde{\varphi}_{a}$ be as in (3.26). Then

[TABLE]

in $H^{1,p}(D_{R},\mathbb{C})$ for every $R>1$ and in $C^{2}_{\rm loc}(\mathbb{R}^{2}\setminus\{p\},\mathbb{C})$, where $\Psi_{p}$ is the function defined in Lemma 5.3. Moreover,

[TABLE]

Proof.

From Remark 3.8 and Corollary 5.2 it follows that, for every sequence $a_{n}=|a_{n}|p$ with $|a_{n}|\to 0$, there exist a subsequence $a_{n_{\ell}}$, $c\in[0,+\infty)$ and $\tilde{\Phi}\in H^{1,p}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$ such that

[TABLE]

for every $R>1$. Passing to the limit in the equation satisfied by $\tilde{\varphi}_{a}$, i.e. $(i\nabla+A_{p})^{2}\tilde{\varphi}_{a}=\lambda_{a}|a|^{2}\tilde{\varphi}_{a}$ in $\frac{1}{|a|}\Omega$, we obtain that $\tilde{\Phi}$ satisfies

[TABLE]

Moreover, by compact trace embeddings,

[TABLE]

so that $\tilde{\Phi}$ is not identically zero. Testing the equation for $\tilde{\varphi}_{a}$ with $\tilde{\varphi}_{a}$ itself, integrating by parts and exploiting the $C^{2}_{\rm loc}$-convergence of $\tilde{\varphi}_{a}$ in $\mathbb{R}^{2}\setminus\{p\}$ (which follows from classic elliptic estimates) we obtain that $\int_{D_{R}}|(i\nabla+A_{p})\tilde{\varphi}_{a_{n_{\ell}}}|^{2}\,dx\to\int_{D_{R}}|(i\nabla+A_{p})\tilde{\Phi}|^{2}\,dx$ as $\ell\to\infty$ for every $R>1$. Hence we conclude that, for all $R>1$, $\tilde{\varphi}_{a_{n_{\ell}}}\to\tilde{\Phi}$ strongly in $H^{1,p}(D_{R},\mathbb{C})$ as $\ell\to+\infty$.

By the strong $H^{1,p}_{\rm loc}(\mathbb{R}^{2},\mathbb{C})$-convergence and recalling (4.8), we can pass to the limit along $a_{n_{\ell}}$ in (5.9), to obtain

[TABLE]

This implies $c\neq 0$ (and hence $c>0$), otherwise we would have $\int_{\mathbb{R}^{2}\setminus D_{R}}|(i\nabla+A_{p})\tilde{\Phi}|^{2}\,dx<+\infty$, which together with (5.12) implies $\tilde{\Phi}\equiv 0$, thus contradicting (5.13).

Then Lemma 5.3 and (5.13) provide

[TABLE]

Since these limits depend neither on the sequence, nor on the subsequence, the proof is complete. ∎

Proof of Theorem 1.1.

Let $p\in\mathbb{S}^{1}$. From Corollary 4.7 part (i) and (5.11) we conclude that $\lambda_{0}-\lambda_{a}=O(|a|^{2|\alpha-k|})$ as $a=|a|p\to 0$. Since the function $a\mapsto\lambda_{a}$ is analytic in a neighborhood of [math], being $\lambda_{0}$ simple (see [16, Theorem 1.3]), and since $2|\alpha-k|$ is non-integer, we have that the Taylor polynomials of the function $\lambda_{0}-\lambda_{a}$ with center [math] and degree less or equal than $\lfloor 2|\alpha-k|\rfloor$ vanish, thus yielding the conclusion. ∎

Proof of Theorem 1.2.

It is a direct consequence of Lemma 5.4. ∎

6. Rate of convergence for eigenfunctions

Taking inspiration from [3], we now estimate the rate of convergence of the eigenfunctions. We then take into account the quantity

[TABLE]

and we split the argument in two different steps, the first considering the energy variation inside small disks of radius $R|a|$, the second considering the energy variation outside these disks.

Lemma 6.1**.**

Under the same assumptions as in Theorems 1.1 and 1.2, we have that, for every $p\in{\mathbb{S}}^{1}$ and $R>1$,

[TABLE]

where

[TABLE]

Moreover

[TABLE]

Proof.

We notice that, in view of (1.19), $\mathfrak{L}_{p}<+\infty$. The proof of (6.1) relies on a change of variables and on the convergences stated in (4.8) and in Theorem 1.2. We have that

[TABLE]

where $\Gamma_{p}$ is defined in (1.13). Indeed, suppose by contradiction that the above limit is zero. Since, for every $r_{1}>r_{2}>1$, $\Psi_{p}-e^{i\alpha(\theta_{p}-\theta_{0}^{p})}\psi_{k}\in H^{1,p}(D_{r_{1}}(p)\setminus D_{r_{2}}(p),\mathbb{C})$, the Hardy inequality (1.4) implies that $\Psi_{p}-e^{i\alpha(\theta_{p}-\theta_{0}^{p})}\psi_{k}\equiv 0$ in $\mathbb{R}^{2}\setminus D_{1}(p)$. Moreover, since (i\nabla+A_{p})^{2}\big{(}\Psi_{p}-e^{i\alpha(\theta_{p}-\theta_{0}^{p})}\psi_{k}\big{)}=~{}0 in $\mathbb{R}^{2}\setminus\Gamma_{p}$, a classical unique continuation principle (see e.g. [20]) implies that $\Psi_{p}-e^{i\alpha(\theta_{p}-\theta_{0}^{p})}\psi_{k}\equiv 0$ in $\mathbb{R}^{2}\setminus\Gamma_{p}$ necessarily. But this is impossible since, by (1.18) and classical elliptic estimates away from $p$, $\Psi_{p}$ is smooth in $\mathbb{R}^{2}\setminus\{p\}$, whereas $e^{i\alpha(\theta_{p}-\theta_{0}^{p})}\psi_{k}$ is discontinuous on $\Gamma_{p}\setminus\{0\}$ since it is the product of the continuous non-zero function $\psi_{k}$ and of the discontinuous function $e^{i\alpha(\theta_{p}-\theta_{0}^{p})}$ (see the definitions (1.11), (1.12) and (1.13)). ∎

Before addressing the energy variation outside the disk, it is worthwhile introducing a preliminary result. For all $R>2$ and $p\in\mathbb{S}^{1}$, let $z_{p,R}$ be the unique solution to

[TABLE]

From Lemma 5.4 it follows that the family of functions $Z_{a}^{R}$ introduced in (4.18) converges in $H^{1,0}(D_{R},\mathbb{C})$ to some multiple of $z_{p,R}$.

Lemma 6.2**.**

Let $p\in{\mathbb{S}}^{1}$ and $R>2$. For $a=|a|p\in\Omega$, let $Z_{a}^{R}$ be as in (4.18). Then

[TABLE]

in $H^{1,0}(D_{R},\mathbb{C})$, as $|a|\to 0^{+}$.

Proof.

Denote \gamma_{p,\delta}=\frac{\beta}{|\beta|}\Big{(}\frac{K_{\delta}}{\int_{\partial D_{K_{\delta}}}|\Psi_{p}|^{2}ds}\Big{)}^{\!1/2}. By (4.14) and (6.2) we have that $Z_{a}^{R}-\gamma_{p,\delta}z_{p,R}$ solves

[TABLE]

For $R>2$, let $\eta_{R}\,:\,\mathbb{R}^{2}\to\mathbb{R}$ be a smooth cut-off function such that

[TABLE]

Then, by the Dirichlet principle and Lemma 5.4,

[TABLE]

as $a=|a|p\to 0$. Finally, the Hardy type inequality (1.3) allows us to conclude. ∎

Lemma 6.3**.**

Let $\varphi_{0}\in H^{1,0}_{0}(\Omega,\mathbb{C})$ be a solution to (1.7) satisfying (1.5). Let $p\in\mathbb{S}^{1}$. For $a=|a|p\in\Omega$, let $\varphi_{a}\in H^{1,a}_{0}(\Omega,\mathbb{C})$ satisfy (1.14)–(1.15). Then, for all $R>\max\{2,K_{\delta}\}$,

[TABLE]

where $\lim_{a=|a|p\to 0}G(p,a,R)=G(p,R)$ for some $G(p,R)$ such that

[TABLE]

Proof.

Let $R>\max\{2,K_{\delta}\}$. From Theorem 5.1 and (5.11) we have that

[TABLE]

where $g(p,a,R)=o(1)$ as $|a|\to 0^{+}$ and

[TABLE]

where \gamma_{p,\delta}=\frac{\beta}{|\beta|}\Big{(}\frac{K_{\delta}}{\int_{\partial D_{K_{\delta}}}|\Psi_{p}|^{2}ds}\Big{)}^{\!1/2} and the constant ${\rm const}>0$ is independent of $a$. From Lemmas 5.4 and 6.2 we have that

[TABLE]

in $H^{-1/2}(\partial D_{R})$ as $a=|a|p\to 0$. Therefore $h(p,a,R)\leq f(p,a,R)$ with

[TABLE]

To complete the proof is then enough to show that

[TABLE]

Using an integration by parts we can rewrite

[TABLE]

which implies

[TABLE]

The first term in the right hand side of (6.6) goes to zero as $R\to+\infty$ because of (1.19). To estimate the second term, we consider a test function $\eta_{R}$ satisfying (6.3) and the additional property $|\nabla\eta_{R}|\leq 4/R$ in $D_{R}\setminus D_{R/2}$. Recalling that $\psi_{k}-z_{p,R}$ satisfies $(i\nabla+A_{0})^{2}(\psi_{k}-z_{p,R})=0$ in $D_{R}$ with the boundary condition $\psi_{k}-z_{p,R}=\psi_{k}-e^{i\alpha(\theta_{0}^{p}-\theta_{p})}\Psi_{p}$ on $\partial D_{R}$, the Dirichlet principle and the Hardy inequality (1.4) provide

[TABLE]

which goes to zero again thanks to (1.19). Therefore we have obtained (6.5) and the proof is complete. ∎

Proof of Theorem 1.3.

Let $p\in{\mathbb{S}}^{1}$ and $\varepsilon>0$. From Lemma 6.1 and (6.4) there exists some $R_{0}>\max\{2,K_{\delta}\}$ sufficiently large such that

[TABLE]

Moreover, again from Lemmas 6.3 and 6.1 there exists $\rho>0$ (depending on $p$, $\varepsilon$, and $R_{0}$) such that, if $a=|a|p$ and $|a|<\rho$, then

[TABLE]

and

[TABLE]

Therefore, taking into account Lemma 6.3, we have that, for all $a=|a|p$ with $|a|<\rho$,

[TABLE]

thus concluding the proof. ∎

Acknowledgments. The authors are partially supported by the project ERC Advanced Grant 2013 n. 339958 : “Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT”. They also acknowledge the support of the projects MIS F.4508.14 (FNRS) & ARC AUWB-2012-12/17-ULB1- IAPAS for a research visit at Université Libre de Bruxelles, where part of this work has been achieved. L. Abatangelo and V. Felli are partially supported by the PRIN2015 grant “Variational methods, with applications to problems in mathematical physics and geometry” and by the 2017-GNAMPA project “Stabilità e analisi spettrale per problemi alle derivate parziali”. Finally, the authors would like to thank Susanna Terracini for her encouragement and for fruitful discussions.