Irreducibility of the Bloch variety for finite-range Schrödinger operators. (English) Zbl 1507.81093
Summary: We study the Bloch variety of discrete Schrödinger operators associated with a complex periodic potential and a general finite-range interaction, showing that the Bloch variety is irreducible for a wide class of lattice geometries in arbitrary dimension. Examples include the triangular lattice and the extended Harper lattice.
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 30H30 | Bloch spaces |
| 33E15 | Other wave functions |
| 39A23 | Periodic solutions of difference equations |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
References:
| [1] | Ando, K.; Isozaki, H.; Morioka, H., Spectral properties of Schrödinger operators on perturbed lattices, Ann. Henri Poincaré, 17, 8, 2103-2171 (2016) · Zbl 1347.81044 |
| [2] | Bättig, D., A toroidal compactification of the two dimensional Bloch-manifold (1988), ETH: ETH Zurich, PhD thesis |
| [3] | Bättig, D., A directional compactification of the complex Fermi surface and isospectrality, (Séminaire sur les Équations aux Dérivées Partielles, 1989-1990 (1990), École Polytech.: École Polytech. Palaiseau), pp. Exp. No. IV, 11 · Zbl 0716.35082 |
| [4] | Bättig, D., A toroidal compactification of the Fermi surface for the discrete Schrödinger operator, Comment. Math. Helv., 67, 1, 1-16 (1992) · Zbl 0812.58068 |
| [5] | Bättig, D.; Knörrer, H.; Trubowitz, E., A directional compactification of the complex Fermi surface, Compos. Math., 79, 2, 205-229 (1991) · Zbl 0754.58038 |
| [6] | Do, N.; Kuchment, P.; Sottile, F., Generic properties of dispersion relations for discrete periodic operators, J. Math. Phys., 61, 10, Article 103502 pp. (2020) · Zbl 1454.81060 |
| [7] | Faust, M.; Sottile, F., Critical points of discrete periodic operators · Zbl 07861433 |
| [8] | Fisher, L.; Li, W.; Shipman, S. P., Reducible Fermi surface for multi-layer quantum graphs including stacked graphene, Commun. Math. Phys., 385, 3, 1499-1534 (2021) · Zbl 1468.81052 |
| [9] | Gieseker, D.; Knörrer, H.; Trubowitz, E., An overview of the geometry of algebraic Fermi curves, (Algebraic Geometry: Sundance 1988. Algebraic Geometry: Sundance 1988, Contemp. Math., vol. 116 (1991), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 19-46 · Zbl 0751.14021 |
| [10] | Gieseker, D.; Knörrer, H.; Trubowitz, E., The Geometry of Algebraic Fermi Curves, Perspectives in Mathematics, vol. 14 (1993), Academic Press, Inc.: Academic Press, Inc. Boston, MA · Zbl 0778.14011 |
| [11] | Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0367.14001 |
| [12] | Isozaki, H.; Morioka, H., A Rellich type theorem for discrete Schrödinger operators, Inverse Probl. Imaging, 8, 2, 475-489 (2014) · Zbl 1322.47033 |
| [13] | Knörrer, H.; Trubowitz, E., A directional compactification of the complex Bloch variety, Comment. Math. Helv., 65, 1, 114-149 (1990) · Zbl 0723.32006 |
| [14] | Kuchment, P., An overview of periodic elliptic operators, Bull. Am. Math. Soc. (N.S.), 53, 3, 343-414 (2016) · Zbl 1346.35170 |
| [15] | Kuchment, P.; Vainberg, B., On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials, Commun. Partial Differ. Equ., 25, 9-10, 1809-1826 (2000) · Zbl 0960.35077 |
| [16] | Kuchment, P.; Vainberg, B., On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators, Commun. Math. Phys., 268, 3, 673-686 (2006) · Zbl 1125.39021 |
| [17] | Li, W.; Shipman, S. P., Irreducibility of the Fermi surface for planar periodic graph operators, Lett. Math. Phys., 110, 9, 2543-2572 (2020) · Zbl 1502.39010 |
| [18] | W. Liu, Fermi isospectrality of discrete periodic Schrödinger operators with separable potentials on \(\mathbb{Z}^2\), Preprint. · Zbl 1529.81066 |
| [19] | W. Liu, Fermi isospectrality for discrete periodic Schrödinger operators, Comm. Pure Appl. Math., to appear. · Zbl 1537.81086 |
| [20] | Liu, W., Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues, Geom. Funct. Anal., 32, 1, 1-30 (2022) · Zbl 1484.35169 |
| [21] | Liu, W., Topics on Fermi varieties of discrete periodic Schrödinger operators, J. Math. Phys., 63, 2, Article 023503 pp. (2022) · Zbl 1507.35003 |
| [22] | Shaban, W.; Vainberg, B., Radiation conditions for the difference Schrödinger operators, Appl. Anal., 80, 3-4, 525-556 (2001) · Zbl 1026.47025 |
| [23] | Shipman, S. P., Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators, Commun. Math. Phys., 332, 2, 605-626 (2014) · Zbl 1301.39015 |
| [24] | Shipman, S. P., Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators, J. Spectr. Theory, 10, 1, 33-72 (2020) · Zbl 1505.47040 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
