Skip to main content

A Sharp Upper Bound on the Spectral Gap for Graphene Quantum Dots

Abstract

The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected C3-domains with infinite mass boundary conditions. This bound is given in terms of a conformal variation, explicit geometric quantities and of the first eigenvalue for the disk. Its proof relies on the min-max principle applied to the squares of these Dirac operators. A suitable test function is constructed by means of a conformal map. This general upper bound involves the norm of the derivative of the underlying conformal map in the Hardy space \(\mathcal {H}^{2}(\mathbb {D})\). Then, we apply known estimates of this norm for convex and for nearly circular, star-shaped domains in order to get explicit geometric upper bounds on the eigenvalue. These bounds can be re-interpreted as reverse Faber-Krahn-type inequalities under adequate geometric constraints.

This is a preview of subscription content, access via your institution.

References

  1. Agricola, I., Friedrich, T.: Upper bounds for the first eigenvalue of the Dirac operator on surfaces. J. Geom. Phys. 30, 1–22 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  2. Akhmerov, A.R., Beenakker, C.W.J.: Boundary conditions for Dirac fermions on a terminated honeycomb lattice. Phys. Rev. B 77, 085423 (2008)

    ADS  Article  Google Scholar 

  3. Antunes, P., Freitas, P.: New bounds for the principal Dirichlet eigenvalue of planar regions. Exp. Math. 15, 333–342 (2006)

    MathSciNet  Article  Google Scholar 

  4. Arrizabalaga, N., Le Treust, L., Mas, A., Raymond, N.: The MIT Bag Model as an infinite mass limit, arXiv:1808.09746

  5. Barbaroux, J.-M., Cornean, H., Le Treust, L., Stockmeyer, E.: Resolvent convergence to Dirac operators on planar domains. arXiv:1810.02957

  6. Bär, C.: Lower eigenvalue estimates for Dirac operator. Math. Ann. 293, 39–46 (1992)

    MathSciNet  Article  Google Scholar 

  7. Bär, C.: Extrinsic bounds for eigenvalues of the Dirac operator. Ann. Glob. Anal. Geom. 16, 573–596 (1998)

    MathSciNet  Article  Google Scholar 

  8. Benguria, R., Fournais, S., Stockmeyer, E., Van Den Bosch, H.: Self-adjointness of two-dimensional Dirac operators on domains. Ann. Henri Poincaré 18, 1371–1383 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  9. Benguria, R., Fournais, S., Stockmeyer, E., Van Den Bosch, H.: Spectral gaps of Dirac operators describing graphene quantum dots. Math. Phys. Anal. Geom. 20, 11 (2017)

    MathSciNet  Article  Google Scholar 

  10. Booß-Bavnbek, B., Wojciechowski, K.P.: Elliptic Boundary Problems for Dirac Operators. Mathematics: Theory & Applications. Birkhäuser, Boston (1993)

    Book  Google Scholar 

  11. Faber, G: Beweis, daß unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Münch Ber., 169–172 (1923)

  12. Freitas, P., Krejčiřík, D.: A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains. Proc. Amer. Math. Soc. 136, 2997–3006 (2008)

    MathSciNet  Article  Google Scholar 

  13. Gaier, D.: On conformal mapping of nearly circular regions. Pac. J. Math. 12, 149–162 (1962)

    MathSciNet  Article  Google Scholar 

  14. Garnett, J.B., Marshall, D.E.: Harmonic Measure. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  15. Grisvard, P.: Elliptic problems in non-smooth domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  16. Henrot, A.: Extremum problems for eigenvalues of elliptic operators. Basel, Birkhäuser (2006)

    Book  Google Scholar 

  17. Henrot, A.: Shape optimization and spectral theory. De Gruyter, Warsaw (2017)

    Book  Google Scholar 

  18. Hijazi, O., Montiel, S., Zhang, X.: Dirac operator on embedded hypersurfaces. Math. Res. Lett. 8, 195–208 (2001)

    MathSciNet  Article  Google Scholar 

  19. Kovalev, L.V.: Conformal contractions and lower bounds on the density of harmonic measure. Potential Anal. 46, 385–391 (2017)

    MathSciNet  Article  Google Scholar 

  20. Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)

    MathSciNet  Article  Google Scholar 

  21. Kythe, K.P.: Complex Analysis. Conformal Inequalities and the Bieberbach Conjecture. CRC Press, Boca Raton (2016)

    Book  Google Scholar 

  22. Le Treust, L., Ourmières-Bonafos, T.: Self-adjointness of Dirac operators with infinite mass boundary conditions in sectors. Ann. Henri Poincaré 19, 1465–1487 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  23. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  24. Moroianu, A., Ourmières-Bonafos, T., Pankrashkin, K.: Dirac operators on hypersurfaces as large mass limits. arXiv:1811.03340

  25. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

  26. Payne, L.E., Weinberger, H.F.: Some isoperimetric inequalities for membrane frequencies and torsional rigidity. J. Math. Anal. Appl. 2, 210–216 (1961)

    MathSciNet  Article  Google Scholar 

  27. Pólya, G.: Two more inequalities between physical and geometrical quantities. J. Indian Math. Soc. (N.S.) 24, 413–419 (1960)

    MathSciNet  MATH  Google Scholar 

  28. Pólya, G., Szegő, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)

    Google Scholar 

  29. Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)

    Book  Google Scholar 

  30. Raulot, S.: The Hijazi inequality on manifolds with boundary. J. Geom. Phys. 56, 2189–2202 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  31. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 4: Analysis of Operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  32. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  33. Schmidt, K.M.: A remark on boundary value problems for the Dirac operator. Q. J. Math. Oxf. Ser. (2) 46, 509–516 (1995)

    MathSciNet  Article  Google Scholar 

  34. Stockmeyer, E., Vugalter, S.: Infinite mass boundary conditions for Dirac operators, J. Spectr. Theory, arXiv:1603.09657 (in press)

  35. Szegő, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3, 343–356 (1954)

    MathSciNet  Google Scholar 

  36. Van Den Bosch, H.: Spectrum of Graphene Quantum Dots. PhD Thesis, Santiago (2017)

Download references

Acknowledgments

The authors have benefited a lot from fruitful discussions with Leonid Kovalev, Konstantin Pankrashkin, and Loïc Le Treust. TOB is grateful for the stimulating research stay and the hospitality of the Nuclear Physics Institute of Czech Republic where this project has been initiated. VL is grateful for the possibility to have in 2018 two inspiring research stays at the University Paris-Sud where a large part of the work was done.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vladimir Lotoreichik.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

VL is supported by the grant No. 17-01706S of the Czech Science Foundation (GAČR) and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

TOB is supported by the ANR “Défi des autres savoirs (DS10) 2017” programm, reference ANR-17-CE29-0004, project molQED and by the PHC Barrande 40614XA funded by the French Ministry of Foreign Affairs and the French Ministry of Higher Education, Research and Innovation.

Appendices

Appendix: The Massless Dirac Operator with Infinite Mass Boundary Conditions on a Disk

The goal of this Appendix is to prove Proposition 8. Namely, we are aiming to characterize the principal eigenvalue \(\mu _{\mathbb {D}}\) and the associated eigenfunctions for the self-adjoint operator \(\mathsf {D}_{\mathbb {D}}\) on the unit disk

$$ \mathbb{D} = \{x\in\mathbb{R}^{2}\colon |x| < 1\}. $$

The material of this appendix is essentially known (see for instance [36, App. D]). However, we recall it here for the sake of completeness.

A.1 The Representation of the Operator \(\mathsf {D}_{\mathbb {D}}\) in Polar Coordinates

First, we introduce the polar coordinates (r,𝜃) on the disk \(\mathbb {D}\). They are related to the Cartesian coordinates x = (x1,x2) via the identities

$$ x(r,\theta) = \left( \begin{array}{ccc} x_{1}(r,\theta)\\ x_{2}(r,\theta) \end{array}\right)\!,\!\quad\text{where}\!\quad x_{1} = x_{1}(r,\theta) = r\cos\theta,\!\quad x_{2} = x_{2}(r,\theta) = r\sin\theta, $$

for all \(r \in \mathbb {I} := (0,1)\) and \(\theta \in \mathbb {T}\). Further, we consider the moving frame (erad,eang) associated with the polar coordinates

$$ \mathbf{e}_{\text{rad}}(\theta) = \frac{\mathsf{d} x}{\mathsf{d} r} = \left( \begin{array}{ccc} \cos\theta\\ \sin\theta \end{array}\right) \quad\text{and}\quad \mathbf{e}_{\text{ang}}(\theta) = \frac{\mathsf{d} \mathbf{e}_{\text{rad}}}{\mathsf{d}\theta} = \left( \begin{array}{ccc} -\sin\theta\\ \cos\theta \end{array}\right). $$

The Hilbert space \(L^{2}_{\text {cyl}}(\mathbb {D},\mathbb {C}^{2}) := L^{2}(\mathbb {I}\times \mathbb {T},\mathbb {C}^{2};r\mathsf {d} r \mathsf {d} \theta )\) can be viewed as the tensor product \({L^{2}_{r}}(\mathbb {I})\otimes L^{2}(\mathbb {T},\mathbb {C}^{2})\), where \({L^{2}_{r}}(\mathbb {I}) = L^{2}(\mathbb {I};r\mathsf {d} r)\). Let us consider the unitary transform

$$ V \colon L^{2}(\mathbb{D},\mathbb{C}^{2}) \rightarrow L^{2}_{\text{cyl}}(\mathbb{D},\mathbb{C}^{2}),\qquad (Vv)(r,\theta) = u\left( r\cos\theta,r\sin\theta\right), $$

and introduce the cylindrical Sobolev space by

$$ H^{1}_{\text{cyl}}(\mathbb{D},\mathbb{C}^{2}) \!:=\! V\!\left( \!H^{1}(\mathbb{D},\mathbb{C}^{2})\!\right) = \left\{v \!\in\! L^{2}_{\text{cyl}}(\mathbb{D},\mathbb{C}^{2}) \colon \partial_{r} v, r^{-1}(\partial_{\theta} v) \!\in\! L^{2}_{\text{cyl}}(\mathbb{D},\mathbb{C}^{2})\right\} $$

We consider the operator acting in the Hilbert space \(L^{2}_{\text {cyl}}(\mathbb {D},\mathbb {C}^{2})\) defined as

$$ \widetilde{\mathsf{D}_\mathbb{D}} := V \mathsf{D}_\mathbb{D} V^{-1},\quad \text{dom}\left( \widetilde{\mathsf{D}_\mathbb{D}}\right) = V\left( \text{dom}\left( {\mathsf{D}_\mathbb{D}}\right)\right). $$
(1)

Now, let us compute the action of \(\widetilde {\mathsf {D}_{\mathbb {D}}}\) on a function \(v\in \text {dom}\left (\widetilde {\mathsf {D}_{\mathbb {D}}}\right )\). Notice that there exists \(u\in \text {dom}\left ({\mathsf {D}_{\mathbb {D}}}\right )\) such that v = V u and the partial derivatives of v with respect to the polar variables (r,𝜃) can be expressed through those of u with respect to the Cartesian variables (x1,x2) via the standard relations (for x = x(r,𝜃))

$$ \begin{array}{lll} (\partial_{r} v)(r,\theta) & = \sin\theta(\partial_{2} u)(x) + \cos\theta(\partial_{1} u)(x), \\ r^{-1}(\partial_{\theta} v)(r,\theta) & = \cos\theta(\partial_{2} u)(x) - \sin\theta(\partial_{1} u)(x), \end{array} $$

and the other way round

$$ \begin{array}{lll} (\partial_{1} u)(x) & = \cos\theta (\partial_{r} v)(r,\theta) - \sin\theta\frac{(\partial_{\theta} v)(r,\theta)}{r},\\ (\partial_{2} u)(x) &= \sin\theta (\partial_{r} v)(r,\theta) + \cos\theta\frac{(\partial_{\theta} v)(r,\theta)}{r}. \end{array} $$

Using the latter formulæ we can express the action of the differential expression −i(σ ⋅∇) in polar coordinates as follows (for x = x(r,𝜃))

$$ \begin{array}{lll} (-\mathsf{i}(\sigma\cdot\nabla) u)(x) & = -\mathsf{i}\left( \begin{array}{ccc} \partial_{1} u_{2}(x) - \mathsf{i} \partial_{2} u_{2}(x)\\ \partial_{1} u_{1}(x) + \mathsf{i} \partial_{2} u_{1}(x) \end{array}\right)\\ & = -\mathsf{i}\left( \begin{array}{ccc} \text{e}^{-\mathsf{i}\theta} (\partial_{r} v_{2})(r,\theta) - \mathsf{i} \text{e}^{-\mathsf{i}\theta}r^{-1}(\partial_{\theta} v_{2})(r,\theta) \\ \text{e}^{\mathsf{i}\theta} (\partial_{r} v_{1})(r,\theta) + \mathsf{i} \text{e}^{\mathsf{i}\theta}r^{-1}(\partial_{\theta} v_{1})(r,\theta) \end{array}\right). \end{array} $$

Note that a basic computation yields

$$ \sigma\cdot \mathbf{e}_{\text{rad}} = \cos\theta\sigma_{1} + \sin\theta\sigma_{2} = \left( \begin{array}{ccc} 0 & \text{e}^{-\mathsf{i}\theta}\\ \text{e}^{\mathsf{i}\theta} &0 \end{array}\right). $$
(2)

Hence, the operator \(\widetilde {\mathsf {D}_{\mathbb {D}}}\) acts as

$$ \begin{array}{@{}rcl@{}} \widetilde{\mathsf{D}_\mathbb{D}} v &=& -\mathsf{i}(\sigma\cdot \mathbf{e}_{\text{rad}}) \left( \partial_{r} v + \frac{v - \sigma_{3} {\mathsf K} v}{2r}\right),\\ \text{dom}\left( \widetilde{\mathsf{D}_\mathbb{D}}\right) & =& \left\{ v\in H_{\text{cyl}}^{1}(\mathbb{D},\mathbb{C}^{2})\colon v_{2}(1,\theta) = \mathsf{i} \text{e}^{\mathsf{i}\theta} v_{1}(1,\theta)\right\}, \end{array} $$
(3)

where K is the spin-orbit operator in the Hilbert space \(L^{2}(\mathbb {T};\mathbb {C}^{2})\) defined as

$$ {\mathsf K} = -2\mathsf{i}\partial_{\theta} + \sigma_{3}, \qquad \text{dom}\left( {\mathsf K}\right) = H^{1}(\mathbb{T},\mathbb{C}^{2}). $$
(4)

Let us investigate the spectral properties of the spin-orbit operator K.

Proposition 1

Let the operator K be as in (30). Then the following hold.

  1. (i)

    K is self-adjoint and has a compact resolvent.

  2. (ii)

    \(\mathsf {Sp}\left ({\mathsf K}\right ) = \{2k + 1\}_{k\in \mathbb {Z}}\) and \(\mathcal {F}_{k} := \ker \left ({\mathsf K} - (2k + 1)\right ) = \mathsf {span} (\phi _{k}^{+}, \phi _{k}^{-})\) , where

    $$ \phi_{k}^{+} = \frac1{\sqrt{2\pi}} \left( \begin{array}{ccc} \text{e}^{\mathsf{i} k\theta}\\0 \end{array}\right) \quad\text{and}\quad \phi_{k}^{-} = \frac1{\sqrt{2\pi}} \left( \begin{array}{ccc} 0\\ \text{e}^{\mathsf{i} (k + 1)\theta} \end{array}\right). $$
  3. (iii)

    \((\sigma \cdot \mathbf {e}_{\text {rad}}) \phi _{k}^{\pm } = \phi _{k}^{\mp }\) and \(\sigma _{3} \phi _{k}^{\pm } = \pm \phi _{k}^{\pm }\).

Proof

(i) The operator K is clearly self-adjoint in \(L^{2}(\mathbb {T},\mathbb {C}^{2})\), because adding the matrix σ3 can be viewed as a symmetric bounded perturbation of an unbounded self-adjoint momentum operator \(H^{1}(\mathbb {T},\mathbb {C}^{2})\ni \phi \mapsto -\mathsf {i} \phi ^{\prime }\) in the Hilbert space \(L^{2}(\mathbb {T},\mathbb {C}^{2})\). As \(\text {dom}\left (\mathsf {K}\right ) = H^{1}(\mathbb {T},\mathbb {C}^{2})\) is compactly embedded into \(L^{2}(\mathbb {T},\mathbb {C}^{2})\) the resolvent of K is compact.(ii) Let ϕ = (ϕ+,ϕ)∈dom (K) and \(\lambda \in \mathbb {R}\) be such that Kϕ = λϕ. The eigenvalue equation on ϕ reads as follows

$$ (\phi^{\pm})^{\prime} = \frac{\mathsf{i}}{2} \left( \lambda \mp 1\right)\phi^{\pm}. $$

The generic solution of the above system of differential equations is given by

$$ \phi^{\pm}(\theta) = A_{\pm} \exp\left( \mathsf{i} \tfrac{\lambda\mp 1}2 \theta\right), \qquad A_{\pm}\in\mathbb{C}. $$

Hence, the periodic boundary condition ϕ±(0) = ϕ±(2π) implies that the eigenvalues of K are exhausted by λ = 2k + 1 for \(k\in \mathbb {Z}\) and that \(\{\phi _{k}^{+},\phi _{k}^{-}\}\) is a basis of \(\mathcal {F}_{k}\).(iii) These algebraic relations are obtained via basic matrix calculus using (28). □

We are now ready to introduce subspaces of \(\text {dom}\left (\widetilde {\mathsf {D}_{\mathbb {D}}}\right )\) that are invariant under its action. The analysis of \(\widetilde {\mathsf {D}_{\mathbb {D}}}\) reduces to the study of its restrictions to each invariant subspace.

Proposition 2

There holds

$$ L^{2}_{\text{cyl}}\left( \mathbb{D}, \mathbb{C}^{2}\right)\simeq {L^{2}_{r}}(\mathbb{I})\otimes L^{2}(\mathbb{T},\mathbb{C}^{2}) = \oplus_{k\in\mathbb{Z}} {\mathcal E}_{k}, $$

where \({\mathcal E}_{k} = {L^{2}_{r}}(\mathbb {I})\otimes \mathcal {F}_{k}\) and \({L^{2}_{r}}(\mathbb {I}):= L^{2}(\mathbb {I};r\mathsf {d} r)\). Moreover, the following hold true.

  1. (i)

    For any \(k\in \mathbb {Z}\),

    $$ d_{k} u := \widetilde{\mathsf{D}_\mathbb{D}} u,\qquad \text{dom}\left( d_{k}\right) := \text{dom}\left( \widetilde{\mathsf{D}_\mathbb{D}}\right) \cap {\mathcal E}_{k} $$

    is a well-defined self-adjoint operator in the Hilbert space \({\mathcal E}_{k}\).

  2. (ii)

    For any \(k \in \mathbb {Z}\), the operator dk is unitarily equivalent to the operator dk in the Hilbert space \({L^{2}_{r}}(\mathbb {I},\mathbb {C}^{2})\) defined as

    $$ \begin{array}{@{}rcl@{}} \mathbf{d}_{k} \!\!&=&\!\! \left( \!\begin{array}{ccc} 0 & -\mathsf{i} \frac{\mathsf{d}}{\mathsf{d} r} - \mathsf{i}\frac{k + 1}{r}\\ -\mathsf{i}\frac{\mathsf{d}}{\mathsf{d} r} + \mathsf{i} \frac{k}r &0 \end{array}\!\right)\!,\\ {} \text{dom}{}\left( \mathbf{d}_{k}\right) \!\!&=&\!\! \left\{{\kern-.5pt}u = (u_{+}{\kern-.5pt},{\kern-.5pt}u_{-}{\kern-.5pt}){\kern-.5pt} \colon{\kern-.5pt} u_{\pm}{\kern-.5pt},{\kern-.5pt} u_{\pm}^{\prime}{\kern-.5pt},{\kern-.5pt} \tfrac{ku_{+}}{r}{\kern-.5pt},{\kern-.5pt} \tfrac{(k{\kern-.5pt}+{\kern-.5pt}1)u_{-}}{r} \!\in\! {L^{2}_{r}}(\mathbb{I}){\kern-.5pt},{\kern-.5pt} u_{-}(1) = \mathsf{i} u_{+}{\kern-.5pt}({\kern-.5pt}1{\kern-.5pt}){}\right\}\!. \end{array} $$
    (5)
  3. (iii)

    \(\mathsf {Sp}\left (\mathsf {D}_{\mathbb {D}}\right ) = \mathsf {Sp}\left (\widetilde {\mathsf {D}_{\mathbb {D}}}\right ) = \bigcup _{k\in \mathbb {Z}}\mathsf {Sp}\left (\mathbf {d}_{k}\right )\).

Proof

(i) Let us check that dk is well defined. Pick a function \(u\in \text {dom}\left (\widetilde {\mathsf {D}_{\mathbb {D}}}\right )\cap \mathcal {E}_{k}\). By definition, u writes as

$$ u(r,\theta) = u_{+}(r)\phi_{k}^{+}(\theta) + u_{-}(r)\phi_{k}^{-}(\theta), $$

and, since \(u\in H_{\text {cyl}}^{1}(\mathbb {D},\mathbb {C}^{2})\), we have \(u_{\pm },u_{\pm }^{\prime },\frac {k}ru_{+},\frac {k + 1}r u_{-}\in {L_{r}^{2}}(\mathbb {I})\). Applying the differential expression obtained in (29), we get

$$ \begin{array}{lll} (\widetilde{\mathsf{D}_\mathbb{D}} u)(r,\theta) & = -\mathsf{i}(\sigma\cdot \mathbf{e}_{\text{rad}}) \left( \partial_{r} v + \frac{v - \sigma_{3} {\mathsf K} v}{2r}\right)u(r,\theta)\\ &=\! \left[-\mathsf{i} u^{\prime}_{-}(r)\! -\! \frac{\mathsf{i}(k + 1)}{r} u_{-}(r)\right]\!\phi_{k}^{+}(\theta)\! +\! \left[-\mathsf{i} u_{+}^{\prime}(r) + \frac{\mathsf{i} k}r u_{+}(r)\right]\!\phi_{k}^{-}(\theta). \end{array} $$
(6)

It yields \(\widetilde {\mathsf {D}_{\mathbb {D}}} \left (\text {dom}\left (\mathsf {D}_{\mathbb {D}}\right )\cap {\mathcal E}_{k}\right ) \subset {\mathcal E}_{k}\). It is now an easy exercise to show that dk is self-adjoint.

(ii) Let us introduce the unitary transform

$$ W_{k} \colon {\mathcal E}_{k} \!\rightarrow\! {L_{r}^{2}}(\mathbb{I},\mathbb{C}^{2}),\quad (W_{k}u)(r) = \left( (u(r,\cdot),\phi_{k}^{+})_{L^{2}(\mathbb{T},\mathbb{C}^{2})}, (u(r,\cdot),\phi_{k}^{-})_{L^{2}(\mathbb{T},\mathbb{C}^{2})}\right)^{\top}. $$

For \(u\in {\mathcal E}_{k}\) it is clear that we have \(\|W_{k}u\|_{{L_{r}^{2}}(\mathbb {I},\mathbb {C}^{2})} = \|u\|_{L^{2}_{\text {cyl}}(\mathbb {D},\mathbb {C}^{2})}\) and we observe that

$$ \mathbf{d}_{k} = W_{k}d_{k} W_{k}^{-1},\qquad \text{dom}\left( \mathbf{d}_{k}\right) = W_{k}\left( \text{dom}\left( d_{k}\right)\right). $$

(iii) The first equality is a consequence of (27), while the second one is an application of [31, Theorem XIII.85]. □

A.2 Eigenstructure of the Disk

Before describing the eigenstructure of the disk recall that C denotes the charge conjugation operator introduced in (14). It is not difficult to see that C is anti-unitary and maps dom (dk) onto dom (d−(k+ 1)) for all \(k\in \mathbb {Z}\). Furthermore, a computation yields

$$ C \mathbf{d}_{-(k + 1)}C = - \mathbf{d}_{k}. $$
(7)

In particular, C2 = 12, which also reads C− 1 = C. Combined with (33) and as the spectrum of dk is discrete one immediately observes that

$$ \mathsf{Sp}\left( \mathbf{d}_{k}\right) = - \mathsf{Sp}\left( \mathbf{d}_{-(k + 1)}\right). $$
(8)

Hence, we can restrict ourselves to \(k\geqslant 0\).

Lemma 3

Let \(k\in \mathbb {N}_{0}\) . Let d k be the self-adjoint operator defined in ( 31 ). Then for all \(k\in \mathbb {N}\) the following hold.

  1. (i)

    dom (dk) ⊂dom (d0)

  2. (ii)

    \(\|\mathbf {d}_{k} u\|_{{L^{2}_{r}}(\mathbb {I};\mathbb {C}^{2})}^{2} \geqslant \|\mathbf {d}_{0} u\|_{{L^{2}_{r}}(\mathbb {I};\mathbb {C}^{2})}^{2}\) forall u ∈dom (dk).

Proof

Let \(k\in \mathbb {N}\) and u = (u+,u)∈dom (dk). It is clear that u ∈dom (d0) and that for integrability reasons u(0) = 0. Hence, we have

$$ \begin{array}{lll} \left\|u_{+}^{\prime} - \tfrac{k}r u_{+}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} & = \left\|u_{+}^{\prime}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} - 2k\Re\left( u_{+}^{\prime},\tfrac{1}{r}u_{+}\right)_{{L^{2}_{r}}(\mathbb{I})} + k^{2} \left\| \tfrac{1}{r} u_{+}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2}\\ & \ge \left\|u_{+}^{\prime}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} - 2k{\Re{\int}_{0}^{1}} u_{+}^{\prime}\overline{u_{+}}\mathsf{d} r\\ & = \left\|u_{+}^{\prime}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} - k{{\int}_{0}^{1}}(|u_{+}|^{2})^{\prime}\mathsf{d} r\\ & = \left\|u_{+}^{\prime}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} - k|u_{+}(1)|^{2}. \end{array} $$
(9)

Analogously, we get

$$ \begin{array}{lll} \left\|u_{-}^{\prime} + \tfrac{k + 1}r u_{-}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} & \ge \left\| u_{-}^{\prime} + \tfrac{1}r u_{-} \right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} + 2k\Re\left( u_{-}^{\prime},\tfrac{1}{r}u_{-}\right)_{{L^{2}_{r}}(\mathbb{I})}\\ & = \left\| u_{-}^{\prime} + \tfrac{1}r u_{-} \right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} + k{{\int}_{0}^{1}}(|u_{-}|^{2})^{\prime}\mathsf{d} r\\ & = \left\| u_{-}^{\prime} + \tfrac{1}r u_{-} \right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} + k|u_{-}(1)|^{2} \end{array} $$
(10)

Combining (35) and (36) with the boundary condition u(1) = iu+(1) we get

$$ \begin{array}{lll} \|\mathbf{d}_{k}u\|^{2}_{{L^{2}_{r}}(\mathbb{I},\mathbb{C}^{2})} & \ge \left\| u_{+}^{\prime}\right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} + \left\| u_{-}^{\prime} + \tfrac{1}r u_{-} \right\|_{{L^{2}_{r}}(\mathbb{I})}^{2} + k\left( |u_{-}(1)|^{2} - |u_{+}(1)|^{2}\right)\\ & = \|\mathbf{d}_{0}u\|_{{L^{2}_{r}}(\mathbb{I},\mathbb{C}^{2})}^{2} + k\left( |u_{-}(1)|^{2} - |u_{+}(1)|^{2}\right) = \|\mathbf{d}_{0} u\|_{{L^{2}_{r}}(\mathbb{I},\mathbb{C}^{2})}^{2}. \end{array} $$

Now, we have all the tools to prove Proposition 8.

Proof of Proposition 8

As a direct consequence of Lemma 34 and the min-max principle, we obtain that

$$ \mu_{1}({\mathbf{d}_{k}^{2}}) \geqslant \mu_{1}({\mathbf{d}_{0}^{2}}) = \mu_{\mathbb{D}}^{2}. $$

Thus, by Proposition 33 (iii) and Eq. 34, in order to investigate the first eigenvalue of \(\mathsf {D}_{\mathbb {D}}\), we only have to focus on the operator d0.

Let μ > 0 be an eigenvalue of d0 and u be an associated eigenfunction. In particular, \(u = (u_{+},u_{-})^{\top } \in \text {dom}\left ({\mathbf {d}_{0}^{2}}\right )\) and we have

$$ 0 = (\mathbf{d}_{0} + \mu)(\mathbf{d}_{0} - \mu) u = \left( \begin{array}{ccc} -u_{+}^{\prime\prime} - \frac{u_{+}^{\prime}}{r} - \mu^{2} \\ -u_{-}^{\prime\prime} - \frac{u_{-}^{\prime}}{r} + \frac{u_{-}}{r^{2}} - \mu^{2} \end{array}\right) . $$

Hence, we obtain

$$ u_{+}(r) = a_{+}J_{0}(\mu r) + b_{+}Y_{0}(\mu r) \quad\text{and}\quad u_{-}(r) = a_{-} J_{1}(\mu r) + b_{-} Y_{1}(\mu r), $$

with some constants \(a_{\pm },b_{\pm } \in \mathbb {C}\) and where Jν and Yν (ν = 0, 1) denote the Bessel function of the first kind of order ν and the Bessel function of the second kind of order ν, respectively. Taking into account that

$$ \lim_{r\rightarrow 0^{+}} r^{2} |Y_{0}^{\prime}(r)|^{2} {=} \frac{4}{\pi^{2}},\quad \lim_{r\rightarrow 0^{+}}r^{4} |Y_{1}^{\prime}(r)|^{2} {=}\frac{4}{\pi^{2}}, $$

(see [25, §10.7(i)]), the condition u ∈dom (d0) implies b± = 0 or, in other words,

$$u_{+}(r) = a_{+} J_{0}(\mu r) \quad\text{and}\quad u_{-}(r) = a_{-} J_{1}(\mu r). $$

Now, as u satisfies the eigenvalue equation d0u = μu we get \(u_{+}^{\prime } = \mathsf {i}\mu u_{-}\) and the identity

$$ a_{-}\mu J_{1}(\mu r) = \mathsf{i} \mu a_{+} J_{1}(\mu r) $$

holds for all \(r\in \mathbb {I}\). In particular, we obtain a = ia+ which gives

$$ u = a_{+} \left( \begin{array}{ccc} J_{0}(\mu r)\\ \mathsf{i} J_{1}(\mu r) \end{array}\right). $$
(11)

Now, the boundary condition u(1) = iu+(1) reads as

$$ J_{0}(\mu) = J_{1}(\mu), $$
(12)

which gives the eigenvalue equation, whose first positive root is the principal eigenvalue of \(\mathsf {D}_{\mathbb {D}}\). An eigenfunction of \(\widetilde {\mathsf {D}_{\mathbb {D}}}\) corresponding to the eigenvalue \(\mu _{\mathbb {D}}\) is given in polar coordinates by

$$ w(r,\theta) = u\otimes \phi_{0}^{-} = \frac1{\sqrt{2\pi}}\left( \begin{array}{ccc}J_{0}(r\mu_{\mathbb{D}} )\\\mathsf{i} \text{e}^{\mathsf{i} \theta}J_{1}(r\mu_{\mathbb{D}}) \end{array}\right) $$

where \(\phi _{0}^{-}\) is as in Proposition 32 (ii), u is as in (37) (with a+ = 1) and \(\mu _{\mathbb {D}}\) is the smallest positive root of (38). □

Appendix B: Proof of Proposition 27

  1. Step 1.

    For any zΩ, the map \(\overline {\Omega } \ni y \to |y-z|\) is continuous. Hence, the maps defined as

    $$ r_{\mathrm{i}} := \overline{\Omega}\ni y \mapsto \inf_{z\in\partial{\Omega}}|y - z|, \qquad r_{\mathrm{o}} := \overline{\Omega}\ni y \mapsto \sup_{z\in\partial{\Omega}}|y-z|, $$

    are continuous on \(\overline {\Omega }\) as well and they attain their upper and lower bounds. In particular, there exist \(y_{\mathrm {i}}, y_{\mathrm {o}} \in \overline {\Omega }\) such that

    $$ r_{\mathrm{i}}(y_{\mathrm{i}})= \max_{y\in\overline{\Omega}}r_{\mathrm{i}}(y), \qquad r_{\mathrm{o}}(y_{\mathrm{o}})= \min_{y\in\overline{\Omega}}r_{\mathrm{o}}(y). $$
  2. Step 2.

    Assume that Ω has an axis of symmetry Λ. By Step 1 there exist \(y_{\mathrm {i}},y_{\mathrm {o}} \in \overline {\Omega }\) such that ri(yi) = supy∈Ωri(y) and ro(yo) = infy∈Ωro(y). Our aim is to show that yi,yo can be both chosen in Λ. Let us suppose that yi,yo∉Λ and define the reflection \(\mathcal {R}_{{\Lambda } }\colon \overline {\Omega }\rightarrow \overline {\Omega }\) with respect to Λ. Remark that \(\mathcal {R}_{{\Lambda } } y_{\mathrm {i}}\) and \(\mathcal {R}_{{\Lambda } } y_{\mathrm {o}}\) also satisfy \(r_{\mathrm {i}}(\mathcal {R}_{{\Lambda } } y_{\mathrm {i}}) = \max _{y\in \overline {\Omega }}r_{\mathrm {i}}(y)\) and \(r_{\mathrm {o}}(\mathcal {R}_{{\Lambda } } y_{\mathrm {o}}) = \min _{y\in \overline {\Omega }}r_{\mathrm {o}}(y)\). Set \(\widetilde {y}_{\mathrm {i}} := \frac 12 y_{\mathrm {i}} + \frac 12 \mathcal {R}_{{\Lambda } } y_{\mathrm {i}}\) and \(\widetilde {y}_{\mathrm {o}} := \frac 12 y_{\mathrm {o}} + \frac 12 \mathcal {R}_{{\Lambda } } y_{\mathrm {o}}\). As Ω is convex we have \(\widetilde {y}_{\mathrm {i}}, \widetilde {y}_{\mathrm {o}} \in \overline {\Omega }\). Also by convexity of Ω, we get

    $$ \frac12 \mathbb{D}_{r_{\mathrm{i}}(y_{\mathrm{i}})}(y_{\mathrm{i}}) + \frac12\mathbb{D}_{r_{\mathrm{i}}(y_{\mathrm{i}})}(\mathcal{R}_{{\Lambda} } y_{\mathrm{i}}) = \mathbb{D}_{r_{\mathrm{i}}(y_{\mathrm{i}})}(\widetilde{y}_{\mathrm{i}}) \subset {\Omega}, $$
    (13)

    where \(\mathbb {D}_{r}(y)\) denotes the disk of radius r > 0 centred at \(y\in \mathbb {R}^{2}\). Now, (13) implies \(r_{\mathrm {i}}(\widetilde {y}_{\mathrm {i}}) \geqslant r_{\mathrm {i}}(y_{\mathrm {i}})\) and we obtain \(r_{\mathrm {i}}(\widetilde {y}_{\mathrm {i}}) = \max _{y\in \overline {\Omega }} r_{\mathrm {i}}(y)\).

    Similarly, by convexity of Ω, we get

    $$ \frac12 \mathbb{D}_{r_{\mathrm{o}}(y_{\mathrm{o}})}(y_{\mathrm{o}}) + \frac12\mathbb{D}_{r_{\mathrm{o}}(y_{\mathrm{o}})}(\mathcal{R}_{{\Lambda} } y_{\mathrm{o}}) = \mathbb{D}_{r_{\mathrm{o}}(y_{\mathrm{o}})}(\widetilde{y}_{\mathrm{o}}) \supset {\Omega}. $$

    In particular, \(r_{\mathrm {o}}(\widetilde {y}_{\mathrm {o}}) \leqslant \min _{y\in \overline {\Omega }}r_{\mathrm {o}}(y)\) and we have equality in this inequality.

  3. Step 3.

    Suppose now that Ω has two axes of symmetry Λ1 and Λ2. Let yΛ ∈Ω be the unique point of intersection of these axes. Thanks to Steps 1 and 2 for all y ∈Ω we necessarily have ri(y) ≤ ri(yΛ) and \(r_{\mathrm {o}}(y) \geqslant r_{\mathrm {o}}(y_{{\Lambda } })\). Next, define the function

    $$ \mathcal{G}(r_{1},r_{2}) := \left( \frac{|{\Omega}|+ \pi {r_{1}^{2}}}{2\pi}\right)^{\frac12} \exp\left( 2(r_{\mathrm{c}} -r_{2}){\Phi}(r_{1},r_{\mathrm{c}})\right),\quad r_{1} < r_{2}, r_{\mathrm{c}} < r_{2}. $$

    Remark that \(\mathcal {G}\) is a non-decreasing function of r1 whereas it is a non-increasing function of r2. Now, we have

    $$ \mathcal{F}_{\mathrm{c}}({\Omega} -y) = \mathcal{G}\left( r_{\mathrm{i}}(y),r_{\mathrm{o}}(y)\right) \leqslant \mathcal{G}\left( r_{\mathrm{i}}(y_{\mathrm{i}}),r_{\mathrm{o}}(y_{\mathrm{o}})\right) = \mathcal{F}_{\mathrm{c}}({\Omega} - y_{{\Lambda} }). $$

    Hence, \(\mathcal {F}_{\mathrm {c}}^{\star }({\Omega }) =\sup _{y\in {\Omega }} \mathcal {F}_{\mathrm {c}}({\Omega }-y)= \mathcal {F}_{\mathrm {c}}({\Omega } -y_{{\Lambda } })\), by which the proof is concluded.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lotoreichik, V., Ourmières-Bonafos, T. A Sharp Upper Bound on the Spectral Gap for Graphene Quantum Dots. Math Phys Anal Geom 22, 13 (2019). https://doi.org/10.1007/s11040-019-9310-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-019-9310-z

Keywords

  • Dirac operator
  • Infinite mass boundary conditions
  • Lowest eigenvalue
  • Shape optimization

Mathematics Subject Classification (2010)

  • 35P15
  • 58J50