Geodesic
Pauli operators and the $\overline\partial$-Neumann problem
Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 165-171 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We apply methods from complex analysis, in particular the $\overline{\partial}$-Neumann operator, to study spectral properties of Pauli operators. For this purpose we consider the weighted $\overline{\partial}$-complex on $\mathbb{C}^n$ with a plurisubharmonic weight function. The Pauli operators appear at the beginning and at the end of the weighted $\overline{\partial}$-complex. We use the spectral properties of the corresponding $\overline{\partial}$-Neumann operator to answer the question when the Pauli operators are with compact resolvent. It is also of importance to know whether the related Bergman space of entire functions is of infinite dimension. The main results are formulated in terms of the properties of the Levi matrix of the weight function. If the weight function is decoupled, one gets additional informations. Finally, we point out that a corresponding Dirac operator fails to be with compact resolvent.
Keywords: $\overline{\partial} $-Neumann problem, Pauli operators, Schrödinger operators, compactness. 
@article{UFA_2017_9_3_a16,
     author = {F. Haslinger},
     title = {Pauli operators and the $\overline\partial${-Neumann} problem},
     journal = {Ufa mathematical journal},
     pages = {165--171},
     year = {2017},
     volume = {9},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a16/}
}
TY  - JOUR
AU  - F. Haslinger
TI  - Pauli operators and the $\overline\partial$-Neumann problem
JO  - Ufa mathematical journal
PY  - 2017
SP  - 165
EP  - 171
VL  - 9
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a16/
LA  - en
ID  - UFA_2017_9_3_a16
ER  - 
%0 Journal Article
%A F. Haslinger
%T Pauli operators and the $\overline\partial$-Neumann problem
%J Ufa mathematical journal
%D 2017
%P 165-171
%V 9
%N 3
%U http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a16/
%G en
%F UFA_2017_9_3_a16
F. Haslinger. Pauli operators and the $\overline\partial$-Neumann problem. Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 165-171. http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a16/

[1] F. Berger, F. Haslinger, “On some spectral properties of the weighted $\overline{\partial}$-Neumann operator”, J. Math. Kyoto Univ. (to appear)

[2] M. Christ, “On the $\overline{\partial} $ equation in weighted $L^2$ norms in $\mathbb{C}^1$”, J. Geom. Anal., 1:3 (1991), 193–230 | DOI | MR | Zbl

[3] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger operators with applications to quantum mechanics and global geometry, Springer, Berlin, 1987 | MR

[4] E. B. Davies, Spectral theory and differential operators, Cambridge studies in advanced mathematics, 42, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[5] F. Haslinger, The $\overline{\partial}$-Neumann problem and Schrödinger operators, de Gruyter Expositions in Mathematics, 59, Walter De Gruyter, Berlin, 2014 | MR

[6] F. Haslinger, B. Helffer, “Compactness of the solution operator to $\overline{\partial} $ in weighted $L^ 2$-spaces”, J. Funct. Anal., 243:2 (2007), 679–697 | DOI | MR | Zbl

[7] G. Rozenblum, N. Shirokov, “Infiniteness of zero modes for the Pauli operator with singular magnetic field”, J. Func. Anal., 233:1 (2006), 135–172 | DOI | MR | Zbl

[8] I. Shigekawa, “Spectral properties of Schrödinger operators with magnetic fields for a spin $ 1/2 $ particle”, J. Funct. Anal., 101:2 (1991), 255–285 | DOI | MR | Zbl

[9] E. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, 1993 | MR | Zbl

[10] B. Thaller, The Dirac Equation, Springer, Berlin, 1991 | MR | Zbl