Magnetic Neumann Laplacian on a sharp cone

Abstract

This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture $\alpha$ and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit $\alpha \rightarrow 0$ and establish a full asymptotic expansion for the $n$-th eigenvalue and the $n$-th eigenfunction.

Appendices

Appendix 1: Spherical magnetic coordinates

In dilated spherical coordinates $(t,\theta ,\varphi )\in \mathcal {P}$ such that

$$\begin{aligned} (x,y,z)=\Phi (t,\theta ,\varphi )=\alpha ^{-1/2}(t\cos \theta \sin \alpha \varphi ,\ t\sin \theta \sin \alpha \varphi ,\ t\cos \alpha \varphi ), \end{aligned}$$

the magnetic potential reads

$$\begin{aligned} \varvec{\mathsf {A}}(t,\theta ,\varphi )&= \frac{\alpha ^{-1/2}t}{2}( \cos \alpha \varphi \ \sin \beta -\sin \theta \ \sin \alpha \varphi \ \cos \beta , \cos \theta \ \sin \alpha \varphi \ \cos \beta ,\\&-\cos \theta \ \sin \alpha \varphi \ \sin \beta )^\mathsf{T}. \end{aligned}$$

The Jacobian matrix associated with $\Phi$ is

$$\begin{aligned} D\Phi (t,\theta ,\varphi )=\alpha ^{-1/2}\begin{pmatrix} \cos \theta \sin \alpha \varphi &{}\quad -t\sin \theta \sin \alpha \varphi &{}\quad \alpha \ t\cos \theta \cos \alpha \varphi \\ \sin \theta \sin \alpha \varphi &{}\quad \cos \theta \sin \alpha \varphi &{}\quad \alpha \ t\sin \theta \cos \alpha \varphi \\ \cos \alpha \varphi &{}\quad 0&{}\quad -\alpha \ t\sin \alpha \varphi \end{pmatrix}. \end{aligned}$$

We can compute

$$\begin{aligned} (D\Phi )^{-1}(t,\theta ,\varphi )=\alpha ^{1/2}t^{-1}\begin{pmatrix} t\cos \theta \sin \alpha \varphi &{}\quad \sin \theta \sin \alpha \varphi &{}\quad \cos \alpha \varphi \\ -\sin \theta (\sin \alpha \varphi )^{-1}&{}\quad \cos \theta (\sin \alpha \varphi )^{-1}&{}\quad 0\\ \frac{1}{\alpha }\cos \theta \cos \alpha \varphi &{}\quad \frac{1}{\alpha }\sin \theta \cos \alpha \varphi &{}\quad -\frac{1}{\alpha }\sin \alpha \varphi \end{pmatrix}. \end{aligned}$$

Consequently, the metric becomes

$$\begin{aligned} G=(D\Phi )^{-1}\,{\ ^\mathsf{T}}(D\Phi )^{-1}=\alpha \begin{pmatrix} 1&{}\quad 0&{}\quad 0\\ 0&{}\quad ^{-2}(\sin \alpha \varphi )^{-2}&{}\quad 0\\ 0&{}\quad 0&{}\quad (\alpha t)^{-2} \end{pmatrix}. \end{aligned}$$

The change of variables leads to define the new magnetic potential

$$\begin{aligned} \tilde{\varvec{\mathsf {A}}}(t,\theta ,\varphi )&= {\ ^\mathsf{T}}D\Phi \ \varvec{\mathsf {A}}(t,\theta ,\varphi )\nonumber \\&= \alpha ^{-1}\frac{t^2}{2}\left( 0,\sin ^2\alpha \varphi \ \cos \beta -\cos \alpha \varphi \ \sin \alpha \varphi \ \sin \theta \ \sin \beta ,\cos \theta \ \sin \beta \right) \qquad \end{aligned}$$

(6.1)

$$\begin{aligned}&= \alpha ^{-1}\frac{t^2}{2}\left( 0,\sin ^2\alpha \varphi \ \cos \beta -\frac{1}{2}\sin 2\alpha \varphi \ \sin \theta \ \sin \beta ,\cos \theta \ \sin \beta \right) . \end{aligned}$$

(6.2)

Let $\psi$ be a function in the form domain $\mathrm{H}^1_{\varvec{\mathsf {A}}}(\mathcal {C}_{\alpha })$ of the Schrödinger operator $(-i\nabla +\varvec{\mathsf {A}})^2$ and $\tilde{\psi }(t,\theta ,\varphi ) = \alpha ^{-1/4}\psi (x,y,z)$ (where $\alpha ^{-1/4}$ is a normalization coefficient). The change of variables on the norm and quadratic form reads

$$\begin{aligned}&\!\!\!\Vert \psi \Vert ^2_{\mathrm{L}^2(\mathcal {C}_{\alpha })}=\int \limits _{\mathcal {P}}|\tilde{\psi }(t,\theta ,\varphi )|^2\, t^2\sin \alpha \varphi \, \mathrm{d}t \, \mathrm{d}\theta \, \mathrm{d}\varphi ,\\&\!\!\!\quad \int \limits _{\mathcal {C}_{\alpha }}\left| (-i\nabla +\varvec{\mathsf {A}})\psi (x,y,z)\right| ^2\, \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z\\&\!\!\!\quad =\int \limits _{\mathcal {P}} \langle G(-i\nabla _{t,\theta ,\varphi }+\tilde{\varvec{\mathsf {A}}})\tilde{\psi }, (-i\nabla _{t,\theta ,\varphi }+\tilde{\varvec{\mathsf {A}}})\tilde{\psi }\rangle \ t^2\sin \alpha \varphi \, \mathrm{d}t \, \mathrm{d}\theta \, \mathrm{d}\varphi \\&\!\!\!\quad = \alpha \int \limits _{\mathcal {P}} \Bigg (|\partial _{t}\tilde{\psi }|^2 +\frac{1}{t^2\sin ^2\alpha \varphi }\left| \left( -i\partial _{\theta }+\frac{t^2\sin ^2\alpha \varphi \ \cos \beta }{2\alpha }-\frac{t^2\sin 2\alpha \varphi \ \sin \theta \ \sin \beta }{4\alpha }\right) \tilde{\psi }\right| ^2\\&\!\!\!\qquad +\frac{1}{\alpha ^2t^2}\left| \left( -i\partial _{\varphi }+\frac{t^2}{2}\cos \theta \ \sin \beta \right) \tilde{\psi }\right| ^2\Bigg ) \, t^2\sin \alpha \varphi \, \mathrm{d}t \, \mathrm{d}\theta \, \mathrm{d}\varphi . \end{aligned}$$

Appendix 2: Model operators

Proposition 7.1

Let ${\mathfrak H}_{\omega }$ be defined on $\mathrm{L}^2(\mathbb R_{+},t^2\, \mathrm{d}t)$ by

$$\begin{aligned} {\mathfrak H}_{\omega } = -\frac{1}{t^{2}}\partial _{t}t^2\partial _{t}+t^2+\frac{\omega ^2}{t^2}. \end{aligned}$$

The eigenpairs of ${\mathfrak H}_{\omega }$ are $(\mathfrak l_{n}^\omega ,\mathfrak f_{n}^\omega )_{n\ge 1}$ given by

$$\begin{aligned} \mathfrak l_{n}^\omega =4n-2+\sqrt{1+4\omega ^2},\qquad \mathfrak f_{n}(t)=P^\omega _{n}(t^2)\ \mathrm{e}^{-t^2/2}, \end{aligned}$$

with $P^\omega _{n}$ a polynomial function of degree $n-1$.

Corollary 7.2

For $c>0$ the eigenpairs of the operator

$$\begin{aligned} \tilde{\mathfrak H}= -\frac{1}{t^{2}}\partial _{t}t^2\partial _{t}+c t^2, \end{aligned}$$

defined on $\mathrm{L}^2(\mathbb R_{+},t^2\, \mathrm{d}t)$ are given by

$$\begin{aligned} \mathfrak l_{n}=c^{1/2}(4n-1),\qquad \mathfrak f_{n}(t)=c^{1/4}\mathfrak f_{n}^0(c^{1/4}t)=c^{1/4}P_{n}^0(c^{1/4}t)\ \mathrm{e}^{-c^{1/2} t^2/2}. \end{aligned}$$