Open quantum dots: Resonances from perturbed symmetry and bound states in strong magnetic fields. (English) Zbl 0976.81006
An open quantum dot is a straight hard-wall channel with a potential well. If the potential depends only on the longitudinal variable, there are eigenvalues embedded in the continuous spectrum, which turn into resonances if a magnetic field is applied or the well is deformed. For the case where this symmetry breaking is weak, the perturbative expansion of the above resonances is given. The authors consider also the case of a strong magnetic field. Sufficient conditions are given for the discrete spectrum to be non-empty. In particular, if the dot potential is purely attractive, then the discrete spectrum is non-empty for any magnetic field.
Reviewer: Anatoly N.Kochubei (Kyïv)
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35Q40 | PDEs in connection with quantum mechanics |
References:
| [1] | Aguilar, J.; Combes, J.-M., Commun. Math. Phys., 22, 269-279 (1971) · Zbl 0219.47011 |
| [2] | Akis, R.; Vasilopoulos, P.; Debray, R., Phys. Rev., B56, 9594-9602 (1997) |
| [3] | A. Aslanyan, L. Parnowski and D. Vassiliev: Complex resonances in acoustic waveguides, mp_arc; A. Aslanyan, L. Parnowski and D. Vassiliev: Complex resonances in acoustic waveguides, mp_arc |
| [4] | Berggren, K.-F.; Ji, Zhen-li, Phys. Rev., B43, 4760-4764 (1991) |
| [5] | Berggren, K.-F.; Ji, Zhen-li, Phys. Scripta, T42, 141-148 (1992) |
| [6] | Blanckenbecler, R.; Goldberger, M. L.; Simon, B., Ann. Phys., 108, 69-78 (1977) |
| [7] | Briet, Ph.; Combes, J.-M.; Duclos, P., J. Math. Anal., 125, 90-99 (1987) · Zbl 0629.47043 |
| [8] | Bulla, W.; Gesztesy, F.; Renger, W.; Simon, B., Proc. Am. Math. Soc., 125, 1487-1495 (1997) · Zbl 0868.35080 |
| [9] | Davies, E. B.; Parnowski, L., J. Mech. Appl. Math., 51, 477-492 (1998) · Zbl 0908.76083 |
| [10] | Duclos, P.; Exner, P., Rev. Math. Phys., 7, 73-102 (1995) · Zbl 0837.35037 |
| [11] | Duclos, P.; Exner, P.; Meller, B., Helv. Phys. Acta, 71, 133-162 (1998) · Zbl 1080.35506 |
| [12] | Duclos, P.; Exner, P.; Šťovíček, P., Ann. Inst. Henri Poincaré, 62, 81-101 (1995) · Zbl 0838.35091 |
| [13] | Evans, D. V.; Levitin, D. M.; Vassiliev, D., J. Fluid Mech., 261, 21-31 (1994) · Zbl 0804.76075 |
| [14] | Exner, P.; Gawlista, R.; Šeba, P.; Tater, M., Ann. Phys., 252, 133-179 (1996) · Zbl 0891.47052 |
| [15] | Exner, P.; Šeba, P.; Šťovíček, P., Czech J. Phys., B39, 1181-1191 (1989) |
| [16] | Goldstone, J.; Jaffe, R. L., Phys. Rev., B45, 14100-14107 (1992) |
| [17] | Itoh, T.; Sano, N.; Yoshii, A., Phys. Rev., B45, 1431-1435 (1992) |
| [18] | Jensen, A., Local decay in time of solutions to Schrödinger equation with a dilation-analytic interaction, Manuscripta Mathematica, 25, 61-77 (1978) · Zbl 0397.35056 |
| [19] | Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer Heidelberg · Zbl 0148.12601 |
| [20] | Klaus, M., Ann. Phys., 108, 288-300 (1977) · Zbl 0427.47033 |
| [21] | Na, K. S.; Reichl, L. E., J. Stat. Phys., 92, 519-542 (1998) · Zbl 1027.78514 |
| [22] | Newton, R. G., J. Operator Theory, 10, 119-125 (1983) · Zbl 0527.35062 |
| [23] | Nakamura, A.; Nonoyama, S., Phys. Rev., B56, 9649-9656 (1997) |
| [24] | Nöckel, J. U., Phys. Rev., B46, 15348-15356 (1992) |
| [25] | Okiji, A.; Kasai, H.; Nakamura, A., Progr. Theor. Phys., 106, 209-224 (1991) |
| [26] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics, IV. Analysis of Operators (1978), Academic Press: Academic Press New York · Zbl 0401.47001 |
| [27] | Roitberg, I.; Vassiliev, D.; Weidl, T., Quarterly J. Mech. Appl. Math., 51, 1-13 (1998) · Zbl 0926.74033 |
| [28] | Simon, B., Ann. Phys., 97, 279-288 (1976) · Zbl 0325.35029 |
| [29] | Schult, R. L.; Ravenhall, D. G.; Wyld, H. W., Phys. Rev., B39, 5476-5479 (1989) |
| [30] | Schult, R. L.; Wyld, H. W.; Ravenhall, D. G., Phys. Rev., B41, 12760-12780 (1990) |
| [31] | Seto, N., Bargmann’s inequalities in spaces of an arbitrary dimension, Publ. RIMS, 9, 429-461 (1974) · Zbl 0276.35013 |
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.
