Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions. (English) Zbl 1534.35342
Summary: We discuss the spectral properties of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported by a compact smooth surface. It turns out that the criticality of the interaction may result in a new interval of essential spectrum. The position and the length of the interval are explicitly controlled by the coupling constants and the principal curvatures of the surface. This effect is completely new compared to lower-dimensional critical situations or special geometries considered up to now, in which only a single new point in the essential spectrum was observed.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 47A10 | Spectrum, resolvent |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 53A05 | Surfaces in Euclidean and related spaces |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 78A30 | Electro- and magnetostatics |
Keywords:
Dirac operator; pseudodifferential operators; essential spectrum; principal curvature; transmission condition; boundary integral operatorReferences:
| [1] | 10.1016/0022-1236(83)90078-2 · Zbl 0528.58035 · doi:10.1016/0022-1236(83)90078-2 |
| [2] | 10.1090/chel/350 · doi:10.1090/chel/350 |
| [3] | 10.1137/20M1323576 · Zbl 1485.35419 · doi:10.1137/20M1323576 |
| [4] | 10.1016/j.matpur.2013.12.006 · Zbl 1297.81083 · doi:10.1016/j.matpur.2013.12.006 |
| [5] | 10.1137/14097759X · Zbl 1314.81083 · doi:10.1137/14097759X |
| [6] | 10.1007/s00220-015-2481-y · Zbl 1341.81023 · doi:10.1007/s00220-015-2481-y |
| [7] | 10.4171/JST/289 · Zbl 1437.35595 · doi:10.4171/JST/289 |
| [8] | 10.1007/s40509-019-00186-6 · Zbl 1423.81072 · doi:10.1007/s40509-019-00186-6 |
| [9] | 10.1016/j.jfa.2020.108700 · Zbl 1446.35155 · doi:10.1016/j.jfa.2020.108700 |
| [10] | 10.1142/S0129055X23500368 · Zbl 1536.81052 · doi:10.1142/S0129055X23500368 |
| [11] | 10.1007/s11005-022-01544-z · Zbl 1490.81073 · doi:10.1007/s11005-022-01544-z |
| [12] | ; Brackx, F.; De Schepper, H., The Hilbert transform on a smooth closed hypersurface, Cubo, 10, 2, 83, (2008) · Zbl 1167.30029 |
| [13] | ; do Carmo, Manfredo P., Differential geometry of curves and surfaces, (1976) · Zbl 0326.53001 |
| [14] | 10.1017/CBO9780511894541 · doi:10.1017/CBO9780511894541 |
| [15] | 10.1063/1.528469 · Zbl 0694.46053 · doi:10.1063/1.528469 |
| [16] | 10.1007/BF01361859 · Zbl 0361.35050 · doi:10.1007/BF01361859 |
| [17] | 10.4007/annals.2014.179.2.6 · Zbl 1297.49079 · doi:10.4007/annals.2014.179.2.6 |
| [18] | 10.2140/apde.2018.11.705 · Zbl 1508.81832 · doi:10.2140/apde.2018.11.705 |
| [19] | ; McLean, William, Strongly elliptic systems and boundary integral equations, (2000) · Zbl 0948.35001 |
| [20] | 10.1016/j.aim.2022.108547 · Zbl 07567808 · doi:10.1016/j.aim.2022.108547 |
| [21] | ; Miyanishi, Y.; Rozenblum, G., Eigenvalues of the Neumann-Poincaré operator in dimension 3 : Weyl’s law and geometry, Algebra i Analiz, 31, 2, 248, (2019) · Zbl 1513.47041 |
| [22] | 10.1093/imrn/rnz341 · Zbl 1473.35100 · doi:10.1093/imrn/rnz341 |
| [23] | 10.1007/s00220-019-03642-x · Zbl 1445.35273 · doi:10.1007/s00220-019-03642-x |
| [24] | 10.1007/978-3-030-60453-0_5 · doi:10.1007/978-3-030-60453-0_5 |
| [25] | 10.5565/PUBLMAT6221804 · Zbl 06918953 · doi:10.5565/PUBLMAT6221804 |
| [26] | 10.1080/17476933.2020.1851211 · Zbl 1491.35150 · doi:10.1080/17476933.2020.1851211 |
| [27] | 10.1007/978-1-4757-4187-2 · doi:10.1007/978-1-4757-4187-2 |
| [28] | 10.1090/surv/081 · doi:10.1090/surv/081 |
| [29] | 10.1142/9781848161122 · doi:10.1142/9781848161122 |
| [30] | 10.2140/apde.2020.13.1521 · Zbl 1452.35026 · doi:10.2140/apde.2020.13.1521 |
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.
