×

zbMATH — the first resource for mathematics

Non-self-adjoint graphs. (English) Zbl 1312.34068
Laplace operators with non-self-adjoint boundary conditions are considered on a finite metric graph. Spectral properties are provided, similarity transforms are studied, and existence of a Riesz basis is established.

MSC:
34B45 Boundary value problems on graphs and networks for ordinary differential equations
34L05 General spectral theory of ordinary differential operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
47A10 Spectrum, resolvent
47B44 Linear accretive operators, dissipative operators, etc.
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Agranovich, M. S., On series in root vectors of operators defined by forms with a selfadjoint principal part, Funktsional. Anal. i Prilozhen.. Funct. Anal. Appl., 28 28, 3, 151-167, (1994)
[2] Albeverio, S.; Fei, S.-M.; Kurasov, P., Point interactions: \(\mathcal{PT}\)-Hermiticity and reality of the spectrum, Lett. Math. Phys., 59, 3, 227-242, (2002) · Zbl 1053.81026
[3] Albeverio, S.; Kuzhel, S., One-dimensional Schr\"odinger operators with \(\mathcal{P}\)-symmetric zero-range potentials, J. Phys. A, 38, 22, 4975-4988, (2005) · Zbl 1070.81048
[4] [Astudillo] M. Astudillo. \newblockPseudo-Hermitian Laplace operators on star-graphs: real spectrum and self-adjointness. \newblock Master thesis, Lund University, 2008.
[5] Bender, C. M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys., 70, 6, 947-1018, (2007)
[6] Berkolaiko, G.; Kuchment, P., Introduction to quantum graphs, Mathematical Surveys and Monographs 186, xiv+270 pp., (2013), American Mathematical Society, Providence, RI · Zbl 1318.81005
[7] Birkhoff, G. D., On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc., 9, 2, 219-231, (1908) · JFM 39.0386.01
[8] Birkhoff, G. D., Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc., 9, 4, 373-395, (1908) · JFM 39.0386.02
[9] Blank, J.; Exner, P.; Havl\'\i \vcek, M., Hilbert space operators in quantum physics, Theoretical and Mathematical Physics, xviii+664 pp., (2008), Springer, New York; AIP Press, New York · Zbl 1163.47060
[10] Bolte, J.; Endres, S., The trace formula for quantum graphs with general self adjoint boundary conditions, Ann. Henri Poincar\'e, 10, 1, 189-223, (2009) · Zbl 1207.81028
[11] Borisov, D.; Krej\vci\vr\'\i k, D., \(\mathcal{P}\mathcal{T}\)-symmetric waveguides, Integral Equations Operator Theory, 62, 4, 489-515, (2008) · Zbl 1178.35141
[12] van Casteren, J. A., Operators similar to unitary or selfadjoint ones, Pacific J. Math., 104, 1, 241-255, (1983) · Zbl 0457.47002
[13] \bibDSIIIbook author=Dunford, N., author=Schwartz, J. T., title=Linear operators. Part III: Spectral operators, pages=i-xx and 1925–2592, publisher=Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, date=1971, note=With the assistance of William G. Bade and Robert G. Bartle; Pure and Applied Mathematics, Vol. VII, review=\MR 0412888 (54 #1009),
[14] Edmunds, D. E.; Evans, W. D., Spectral theory and differential operators, Oxford Mathematical Monographs, xviii+574 pp., (1987), The Clarendon Press, Oxford University Press, New York · Zbl 0664.47014
[15] Kiselev, A. V.; Faddeev, M. M., On the similarity problem for nonselfadjoint operators with absolutely continuous spectrum, Funktsional. Anal. i Prilozhen.. Funct. Anal. Appl., 34 34, 2, 143-145, (2000) · Zbl 0985.47017
[16] [Guseinov] G. Sh. Guseinov, \newblockOn the concept of spectral singularities, \newblockPramana 73(3):587–603, 2009.
[17] Harmer, M., Hermitian symplectic geometry and extension theory, J. Phys. A, 33, 50, 9193-9203, (2000) · Zbl 0983.53051
[18] [AccQG] A. Hussein, \newblockMaximal quasi–accretive Laplacians on finite metric graphs, \newblockJ. Evol. Equ., DOI 10.1007/s00028-014-0224-8, 2014.
[19] [IndefiniteQG] A. Hussein, \newblockSign-indefinite second order differential operators on finite metric graphs, \newblock Preprint: arXiv:1211.4144, 2013 (to appear in Rev. Math. Phys.).
[20] Kato, T., Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, xix+592 pp., (1966), Springer-Verlag New York, Inc., New York
[21] Kiselev, A. V., Similarity problem for non-self-adjoint extensions of symmetric operators. Methods of spectral analysis in mathematical physics, Oper. Theory Adv. Appl. 186, 267-283, (2009), Birkh\"auser Verlag, Basel · Zbl 1176.47013
[22] Kochan, D.; Krej\vci\vr\'\i k, D.; Nov\'ak, R.; Siegl, P., The Pauli equation with complex boundary conditions, J. Phys. A, 45, 44, 444019, 14 pp., (2012) · Zbl 1263.81166
[23] Kostrykin, V.; Schrader, R., Kirchhoff’s rule for quantum wires, J. Phys. A, 32, 4, 595-630, (1999) · Zbl 0928.34066
[24] Kostrykin, V.; Schrader, R., Laplacians on metric graphs: eigenvalues, resolvents and semigroups. Quantum graphs and their applications, Contemp. Math. 415, 201-225, (2006), Amer. Math. Soc., Providence, RI · Zbl 1122.34066
[25] [VKScattering] V. Kostrykin and R. Schrader, \newblockThe inverse scattering problem for metric graphs and the traveling salesman problem, \newblock Preprint: arXiv:math-ph/0603010v1, 2006.
[26] Kostrykin, V.; Potthoff, J.; Schrader, R., Heat kernels on metric graphs and a trace formula. Adventures in mathematical physics, Contemp. Math. 447, 175-198, (2007), Amer. Math. Soc., Providence, RI · Zbl 1155.34017
[27] Kostrykin, V.; Potthoff, J.; Schrader, R., Contraction semigroups on metric graphs. Analysis on graphs and its applications, Proc. Sympos. Pure Math. 77, 423-458, (2008), Amer. Math. Soc., Providence, RI · Zbl 1165.47029
[28] Kostrykin, V.; Potthoff, J.; Schrader, R., Brownian motions on metric graphs, J. Math. Phys., 53, 9, 095206, 36 pp., (2012) · Zbl 1293.60080
[29] Krej\vci\vr\'\i k, D.; B\'\i la, H.; Znojil, M., Closed formula for the metric in the Hilbert space of a \(\mathcal{PT}\)-symmetric model, J. Phys. A, 39, 32, 10143-10153, (2006) · Zbl 1117.81058
[30] Krej\vci\vr\'\i k, D., Calculation of the metric in the Hilbert space of a \(\mathcal{PT}\)-symmetric model via the spectral theorem, J. Phys. A, 41, 24, 244012, 6 pp., (2008) · Zbl 1142.81008
[31] Krej\vci\vr\'\i k, D.; Siegl, P., \(\mathcal{PT}\)-symmetric models in curved manifolds, J. Phys. A, 43, 48, 485204, 30 pp., (2010) · Zbl 1207.81030
[32] Krej\vci\vr\'\i k, D.; Siegl, P.; Zelezn\'y, J., On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Anal. Oper. Theory, 8, 1, 255-281, (2014) · Zbl 1410.34086
[33] Kuchment, P., Graph models for waves in thin structures, Waves Random Media, 12, 4, R1-R24, (2002) · Zbl 1063.35525
[34] Kuchment, P., Quantum graphs. I. Some basic structures, Waves Random Media, 14, 1, S107-S128, (2004) · Zbl 1063.81058
[35] [Kurasov2] P. Kurasov, \newblockQuantum graphs: Spectral theory and inverse problems, \newblock to appear in Journal of Spectral Theory.
[36] Kuzhel, S.; Trunk, C., On a class of \(J\)-self-adjoint operators with empty resolvent set, J. Math. Anal. Appl., 379, 1, 272-289, (2011) · Zbl 1220.47053
[37] Malamud, M. M., A criterion for a closed operator to be similar to a selfadjoint operator, Ukrain. Mat. Zh., 37, 1, 49-56, 134, (1985)
[38] Markus, A. S., Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs 71, iv+250 pp., (1988), American Mathematical Society, Providence, RI · Zbl 0678.47005
[39] Miha\u \i lov, V. P., On Riesz bases in \({\mathcal{L}}_{2}(0,\,1)\), Dokl. Akad. Nauk SSSR, 144, 981-984, (1962)
[40] Mostafazadeh, A., Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys., 7, 7, 1191-1306, (2010) · Zbl 1208.81095
[41] Naboko, S. N., Conditions for similarity to unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen., 18, 1, 16-27, (1984)
[42] Reed, M.; Simon, B., Methods of modern mathematical physics. I, xv+400 pp., (1980), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York
[43] Scholtz, F. G.; Geyer, H. B.; Hahne, F. J. W., Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Physics, 213, 1, 74-101, (1992) · Zbl 0749.47041
[44] [PetrPhD] P. Siegl, \newblockSyst\`eme quantitique non-Hermitiens, repr\'esentations ind\'ecomposables et quantification par \'etats coh\'erents, \newblock PhD thesis, Universit\'e Paris Diderot and Czech Technical University in Prague, 2011.
[45] [Znojil] M. Znojil, \newblockQuantum star-graph analogues of PT-symmetric square wells, \newblockCan. J. Phys. 90:1287–1293, 2012.
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.