## Magnetic spectral bounds on starlike plane domains.(English)Zbl 1319.35130

Let $$\Omega\subset {\mathbb R}^2$$ be starlike domain with area $$A$$ defined as $\Omega = \{ re^{i\theta} : 0 \leq r < R(\theta) \},$ where the radius function $$R(\cdot)$$ is positive, $$2\pi$$-periodic, and Lipschitz continuous. The authors consider the magnetic Laplacian $H=\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2$ on $$\Omega$$ under either Dirichlet or Neumann boundary conditions. The constant $$\beta$$ represents the magnetic flux through the domain. Let $$\lambda_j$$ denote the $$j$$th eigenvalue of the operator $$H$$. The main result of the paper under review says that $\sum\limits_{j=1}^n \Phi(\lambda_j A/G)$ is maximal when $$\Omega$$ is a disk centered at the origin, where $$\Phi:{\mathbb R}_+\to {\mathbb R}$$ is concave and increasing; $$G$$ is the following scale-invariant geometric factor measuring the deviation of the domain from roundness: $G=\max\left\{1+\frac{1}{2\pi} \int\limits_0^{2\pi} (\log R)'(\theta)^2 d\theta; \frac{2\pi}{A^2}\int\limits_\Omega |x|^2 dx \right\}$ with $$G \geq 1$$ for all domains and $$G=1$$ for disks.
A similar result is proved for the magnetic Pauli operator with Dirichlet boundary conditions.

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs 35J20 Variational methods for second-order elliptic equations

DLMF
Full Text:

### References:

 [1] Y. Aharonov and A. Casher, Ground state of a spin-1 / 2 charged particle in a two-dimensional magnetic field. Phys. Rev. A19 (1979) 2461-2462. [2] M.S. Ashbaugh and R.D. Benguria, Isoperimetric inequalities for eigenvalues of the Laplacian, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. In Vol. 76. Proc. of Sympos. Pure Math. Amer. Math. Soc. Providence, RI (2007) 105-139. · Zbl 1221.35261 [3] C. Bandle, Isoperimetric Inequalities and Applications. In Vol. 7 of Monogr. Stud. Math. Pitman (Advanced Publishing Program), Boston, Mass. (1980). · Zbl 0436.35063 [4] R.D. Benguria and H. Linde, Isoperimetric inequalities for eigenvalues of the LaPlace operator. Fourth Summer School in Analysis and Mathematical Physics. In Vol. 476 of Contemp. Math. Amer. Math. Soc. Providence, RI (2008) 1-40. · Zbl 1162.35056 [5] V. Bonnaillie-Noël, M. Dauge, D. Martin and G. Vial, Computations of the first eigenpairs for the Schrödinger operator with magnetic field. Comput. Methods Appl. Mech. Engrg.196 (2007) 3841-3858. · Zbl 1173.81300 [6] L. Erdös, Rayleigh-type isoperimetric inequality with a homogeneous magnetic field. Calc. Var. Partial Differ. Eqs.4 (1996) 283-292. · Zbl 0846.35094 [7] L. Erdös, Recent developments in quantum mechanics with magnetic fields, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. In Vol. 76, Part 1. Proc. of Sympos. Pure Math. Amer. Math. Soc. Providence, RI (2007) 401-428. [8] L. Erdös, M. Loss and V. Vougalter, Diamagnetic behavior of sums of Dirichlet eigenvalues. Ann. Inst. Fourier, Grenoble50 (2000) 891-907. · Zbl 0957.35104 [9] S. Fournais and B. Helffer, Spectral Methods in Surface Superconductivity. In Vol. 77 of Progr. Nonlin. Differ. Eq. Appl. Birkhäuser Boston, Inc., Boston, MA (2010). · Zbl 1256.35001 [10] R.L. Frank, A. Laptev and S. Molchanov, Eigenvalue estimates for magnetic Schrödinger operators in domains. Proc. Amer. Math. Soc.136 (2008) 4245-4255. · Zbl 1186.35119 [11] R.L. Frank, M. Loss and T. Weidl, Pólya’s conjecture in the presence of a constant magnetic field. J. Eur. Math. Soc. (JEMS)11 (2009) 1365-1383. · Zbl 1179.35205 [12] B. Helffer and A. Morame, Magnetic bottles in connection with superconductivity. J. Funct. Anal.185 (2001) 604-680. · Zbl 1078.81023 [13] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers Math. Birkhäuser Verlag, Basel (2006). · Zbl 1109.35081 [14] D. Henry, Perturbation of the Boundary in Boundary-value Problems of Partial Differential Equations. With editorial assistance from Jack Hale and Antônio Luiz Pereira. In Vol. 318 of London Math. Soc. Lect. Note Series. Cambridge University Press, Cambridge (2005). · Zbl 1170.35300 [15] S. Kesavan, Symmetrization and Applications. In Vol. 3 of Series in Analysis. World Scientific Publishing Co., Hackensack, NJ (2006). · Zbl 1110.35002 [16] A. Laptev and T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions. Acta Math.184 (2000) 87-111. · Zbl 1142.35531 [17] R.S. Laugesen, J. Liang and A. Roy, Sums of magnetic eigenvalues are maximal on rotationally symmetric domains. Ann. Henri Poincaré13 (2012) 731-750. · Zbl 1242.81082 [18] R.S. Laugesen and B.A. Siudeja, Sharp spectral bounds on starlike domains. J. Spectral Theory4 (2014) 309-347. · Zbl 1296.35099 [19] J.M. Luttinger, Generalized isoperimetric inequalities. J. Math. Phys.14 (1973) 586-593, 1444-1447, 1448-1450. · Zbl 0261.52006 [20] NIST Digital Library of Mathematical Functions. Release 1.0.8 of 2014-04-25. Available at . · Zbl 1019.65001 [21] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. In Vol. 27 of Ann. Math. Stud. Princeton University Press, Princeton, N.J. (1951). · Zbl 0044.38301 [22] D. Saint-James, Etude du champ critique H_{c_{3}} dans une geometrie cylindrique. Phys. Lett.15 (1965) 13-15. [23] S. Son, Spectral Problems on Triangles And Disks: Extremizers and Ground States. Ph.D. thesis, University of Illinois at Urbana-Champaign (2014). Available at . [24] J.W. Strutt (Lord Rayleigh), The Theory of Sound. In Vol. 1, 2nd edition. Macmillan and Co., London (1894).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.