×

zbMATH — the first resource for mathematics

Spectral boundary-value problems for the Dirac system with a singular potential. (English. Russian original) Zbl 1072.35158
St. Petersbg. Math. J. 16, No. 1, 25-57 (2004); translation from Algebra Anal. 16, No. 1, 33-69 (2004).
The boundary-value problem with two real parameters \(\lambda\) and \(\mu\) for the Dirac equation \[ \begin{matrix} (\mathcal{D}+V)w(x)-\lambda w(x)=f(x),\quad x\in\Omega\subset R^3,\quad \mathcal{D}=\sum\limits_1^3\alpha_jD_j+\beta\\ Bu:=i\sigma(\nu(x))v^+(x)-\mu u^+(x)-g(x),\qquad x\in\Gamma=\partial\Omega \end{matrix} \] is considered. Here \(D_j=-i\partial_j,\) \(w=\left(\begin{matrix} u\\ v\end{matrix}\right),\) \(\alpha_j=\left(\begin{matrix} 0 & \sigma_j\\ \sigma_j & 0\end{matrix}\right)\) are Hermitian \((4\times 4)\)-matrices, \(\sigma_j\) are the Pauli matrices, \(\beta=\alpha_4\) is the diagonal \((4\times 4)\)-matrix with the main diagonal \((1, 1, -1, -1)\), \(u^+(x)=u(x)|_\Gamma\) , \(V(x)\) is a Hermitian potential, which may have a Coulomb-like singularity. Let the values of the parameters \(\lambda\) and \(\mu\) be such that the problem has a unique solution and let \(f=0\). The problem consists in the finding \(u^+\) in terms of \(g\). The \(R\)-matrix is the operator mapping \(g\) to \(u^+\). Two approaches to the \(R\)-matrix construction are stated at \(f=0\), \(g=0\):
(I) The problem with \(\lambda\) as a spectral parameter and fixed \(\mu\).
(II) The problem with \(\mu\) as a spectral parameter and fixed \(\lambda\).
The authors investigate spectral properties of both problems and use them for the \(R\)-matrix construction.
MSC:
35Q40 PDEs in connection with quantum mechanics
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009
[2] R. Szmytkowski, Metoda R-macierzy dla róvnan Schrödingera i Diraca, Politechnika Gdanska, Gdansk, 1999.
[3] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. · Zbl 0051.28802
[4] M. S. Agranovich, Spectral problems for the Dirac system with a spectral parameter in the local boundary conditions, Funktsional. Anal. i Prilozhen. 35 (2001), no. 3, 1 – 18, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 35 (2001), no. 3, 161 – 175. · Zbl 1035.81017 · doi:10.1023/A:1012368826639 · doi.org
[5] Victor Ivriĭ, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Mathematics, vol. 1100, Springer-Verlag, Berlin, 1984.
[6] M. Š. Birman and M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Trudy Moskov. Mat. Obšč. 27 (1972), 3 – 52; ibid. 28 (1973), 3 – 34 (Russian). · Zbl 0251.35075
[7] Bernd Thaller, The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. · Zbl 0765.47023
[8] H. Kalf, U.-W. Schmincke, J. Walter, and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Springer, Berlin, 1975, pp. 182 – 226. Lecture Notes in Math., Vol. 448. · Zbl 0311.47021
[9] G. Nenciu, Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms, Comm. Math. Phys. 48 (1976), no. 3, 235 – 247. · Zbl 0349.47014
[10] G. V. Rozenbljum, Distribution of the discrete spectrum of singular differential operators, Izv. Vysš. Učebn. Zaved. Matematika 1(164) (1976), 75 – 86 (Russian).
[11] Gen Nakamura and Tetsuo Tsuchida, Uniqueness for an inverse boundary value problem for Dirac operators, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1327 – 1369. · Zbl 0952.35152 · doi:10.1080/03605300008821551 · doi.org
[12] Jean Dolbeault, Maria J. Esteban, and Eric Séré, On the eigenvalues of operators with gaps. Application to Dirac operators, J. Funct. Anal. 174 (2000), no. 1, 208 – 226. · Zbl 0982.47006 · doi:10.1006/jfan.1999.3542 · doi.org
[13] Volker Vogelsang, Remark on essential selfadjointness of Dirac operators with Coulomb potentials, Math. Z. 196 (1987), no. 4, 517 – 521. · Zbl 0617.35127 · doi:10.1007/BF01160893 · doi.org
[14] F. J. Narcowich, Mathematical theory of the \? matrix. I. The eigenvalue problem, J. Mathematical Phys. 15 (1974), 1626 – 1634. , https://doi.org/10.1063/1.1666517 Francis J. Narcowich, The mathematical theory of the \? matrix. II. The \? matrix and its properties, J. Mathematical Phys. 15 (1974), 1635 – 1642. · Zbl 0295.35067 · doi:10.1063/1.1666518 · doi.org
[15] M. S. Agranovich, Spectral problems for second-order strongly elliptic systems in domains with smooth and nonsmooth boundaries, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 3 – 78 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 5, 847 – 920. · Zbl 1057.35019 · doi:10.1070/RM2002v057n05ABEH000552 · doi.org
[16] Peter Hamacher and Jürgen Hinze, Finite-volume variational method for the Dirac equation, Phys. Rev. A (3) 44 (1991), no. 3, 1705 – 1711. · doi:10.1103/PhysRevA.44.1705 · doi.org
[17] Upke-Walther Schmincke, Essential selfadjointness of Dirac operators with a strongly singular potential, Math. Z. 126 (1972), 71 – 81. · Zbl 0248.35091 · doi:10.1007/BF01580357 · doi.org
[18] Upke-Walther Schmincke, Essential selfadjointness of Dirac operators with a strongly singular potential, Math. Z. 126 (1972), 71 – 81. · Zbl 0248.35091 · doi:10.1007/BF01580357 · doi.org
[19] M. S. Agranovich, Elliptic boundary problems, Partial differential equations, IX, Encyclopaedia Math. Sci., vol. 79, Springer, Berlin, 1997, pp. 1 – 144, 275 – 281. Translated from the Russian by the author. · Zbl 0880.35001 · doi:10.1007/978-3-662-06721-5_1 · doi.org
[20] Rainer Wüst, A convergence theorem for selfadjoint operators applicable to Dirac operators with cutoff potentials, Math. Z. 131 (1973), 339 – 349. · Zbl 0274.47008 · doi:10.1007/BF01174908 · doi.org
[21] Rainer Wüst, Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials, Math. Z. 141 (1975), 93 – 98. · Zbl 0311.47020 · doi:10.1007/BF01236987 · doi.org
[22] Rainer Wüst, Dirac operations with strongly singular potentials. Distinguished self-adjoint extensions constructed with a spectral gap theorem and cut-off potentials, Math. Z. 152 (1977), no. 3, 259 – 271. · Zbl 0361.35051 · doi:10.1007/BF01488968 · doi.org
[23] M. Klaus and R. Wüst, Characterization and uniqueness of distinguished selfadjoint extensions of Dirac operators, Comm. Math. Phys. 64 (1978/79), no. 2, 171 – 176. · Zbl 0408.47022
[24] M. Klaus and R. Wüst, Spectral properties of Dirac operators with singular potentials, J. Math. Anal. Appl. 72 (1979), no. 1, 206 – 214. · Zbl 0423.47014 · doi:10.1016/0022-247X(79)90284-1 · doi.org
[25] Upke-Walther Schmincke, Distinguished selfadjoint extensions of Dirac operators, Math. Z. 129 (1972), 335 – 349. · Zbl 0252.35062 · doi:10.1007/BF01181622 · doi.org
[26] Интеграл\(^{\приме}\)ные операторы в пространствах суммируемых функций, Издат. ”Наука”, Мосцощ, 1966 (Руссиан). · Zbl 0145.39703
[27] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
[28] P. Grisvard, Caractérisation de quelques espaces d’interpolation, Arch. Rational Mech. Anal. 25 (1967), 40 – 63 (French). · Zbl 0187.05901 · doi:10.1007/BF00281421 · doi.org
[29] R. Seeley, Interpolation in \?^\? with boundary conditions, Studia Math. 44 (1972), 47 – 60. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. · Zbl 0237.46041
[30] Обобщенные функции и уравнения в свертках, Физматлит ”Наука”, Мосцощ, 1994 (Руссиан, щитх Руссиан суммары). · Zbl 0837.46028
[31] M. Š. Birman, Scattering problems for differential operators with constant coefficients, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 1 – 16 (Russian). · Zbl 0196.16101
[32] Введение в теорию линейных несамосопряженных операторов в гил\(^{\приме}\)бертовом пространстве, Издат. ”Наука”, Мосцощ, 1965 (Руссиан). · Zbl 0138.07803
[33] Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. · Zbl 0647.47052
[34] Volker Vogelsang, Selfadjoint extensions of Dirac operators for nonspherically symmetric potentials in Coulomb scattering, Integral Equations Operator Theory 10 (1987), no. 6, 841 – 858. · Zbl 0643.47028 · doi:10.1007/BF01196123 · doi.org
[35] G. Nenciu, Distinguished self-adjoint extension for Dirac operator with potential dominated by multicenter Coulomb potentials, Helv. Phys. Acta 50 (1977), no. 1, 1 – 3.
[36] M. Klaus, Dirac operators with several Coulomb singularities, Helv. Phys. Acta 53 (1980), no. 3, 463 – 482 (1981).
[37] Victor Ivrii, Microlocal analysis and precise spectral asymptotics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. · Zbl 0906.35003
[38] Представления группы вращени и группы Лоренца, их применения, Государств. Издат. Физ.-Мат. Лит., Мосцощ, 1958 (Руссиан). · Zbl 0108.22005
[39] V. Ya. Ivriĭ, Exact spectral asymptotics for elliptic operators acting in vector bundles, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 30 – 38, 96 (Russian).
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.