Abstract
We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a \(\mathcal {C}\)-selfadjoint operator generates a contraction \(C_0\)-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of complex symmetric, unbounded semigroups and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all \(\mathcal {C}\)-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a \(\mathcal {C}\)-selfadjoint, unbounded semigroup is not necessarily \(\mathcal {C}\)-selfadjoint.
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References
Bagchi, B.: Evolution operator for time-dependent non-hermitian hamiltonians. Lett. High Energy Phys. 1(3), 04–08 (2018)
Bender, C., Fring, A., Günther, U., Jones, H.: Quantum physics with non-Hermitian operators. J. Phys. A: Math. Theor. 45(44), 440301 (2012)
Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \({{\cal{P}\cal{T}}}\)-symmetry. Phys. Rev. Lett. 80(24), 5243 (1998)
de Leeuw, K.: On the adjoint semigroup and some problems in the theory of approximation. Math. Z. 73(3), 219–234 (1960)
de Morisson Faria, C.F., Fring, A.: Time evolution of non-Hermitian Hamiltonian systems. J. Phys. A: Math. Theor. 39(29), 9269 (2006)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)
Fattorini, H.O.: The Cauchy Problem. Encyclopedia of Mathematics and its Applications, vol. 18. Addison-Wesley Publishing Co., Reading (1983)
Feller, W.: Semi-groups of transformations in general weak topologies. Ann. Math. 57, 287–308 (1953)
Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and physical aspects of complex symmetric operators. J. Phys. A 47, 353001 (2014). 54 pp
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006)
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications II. Trans. Am. Math. Soc. 359, 3913–3931 (2007)
Hai, P.V.: Unbounded weighted composition operators on Fock space. to appear in Potential Anal. (2020). https://doi.org/10.1007/s11118-018-09757-5
Hai, P.V., Khoi, L.H.: Boundedness and compactness of weighted composition operators on Fock spaces \({\cal{F}}^p({\mathbb{C}})\). Acta Math. Vietnam. 41, 531–537 (2016)
Hai, P.V., Khoi, L.H.: Complex symmetry of weighted composition operators on the Fock space. J. Math. Anal. Appl. 433, 1757–1771 (2016)
Hai, P.V., Khoi, L.H.: Complex symmetric \({C}_0\)-semigroups on the Fock space. J. Math. Anal. Appl. 445, 1367–1389 (2017)
Hai, P.V., Putinar, M.: Complex symmetric differential operators on Fock space. J. Differ. Equ. 265, 4213–4250 (2018)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups, vol. 31. American Mathematical Society, Providence (1996)
Hughes, R.J.: Semigroups of unbounded linear operators in Banach space. Trans. Am. Math. Soc. 230, 113–145 (1977)
Izuchi, K.H.: Cyclic vectors in the Fock space over the complex plane. Proc. Am. Math. Soc. 133, 3627–3630 (2005)
Jafari, F., Tonev, T., Toneva, E., Yale, K.: Holomorphic flows, cocycles, and coboundaries. Michigan Math. J. 44(2), 239–253 (1997)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)
Le, T.: Normal and isometric weighted composition operators on the Fock space. Bull. Lond. Math. Soc. 46, 847–856 (2014)
Miao, Y.G., Xu, Z.M.: Investigation of non-Hermitian Hamiltonians in the Heisenberg picture. Phys. Lett. A 380(21), 1805–1810 (2016)
Partington, J.R.: Linear Operators and Linear Systems: an Analytical Approach to Control Theory, vol. 60. Cambridge University Press, Cambridge (2004)
Phillips, R.S.: The adjoint semi-group. Pac. J. Math 5(2), 269–283 (1955)
Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)
Scolarici, G., Solombrino, L.: Time evolution of non-Hermitian quantum systems and generalized master equations. Czech J. Phys. 56(9), 935–941 (2006)
Tamilselvan, K., Kanna, T., Khare, A.: A systematic construction of parity-time (\({{\cal{P}\cal{T}}}\))-symmetric and non-\({{\cal{P}\cal{T}}}\)-symmetric complex potentials from the solutions of various real nonlinear evolution equations. J. Phys. A: Math. Theor. 50(41), 415203 (2017)
Tucsnak, M., Weiss, G.: Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts: Basel Textbooks. Birkhäuser Verlag, Basel (2009)
Znojil, M.: Non-self-adjoint operators in quantum physics: ideas, people, and trends. In: Non-selfadjoint Operators in Quantum Physics, pp. 7–58. Wiley, Hoboken, NJ (2015)
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Hai, P.V., Putinar, M. Complex symmetric evolution equations. Anal.Math.Phys. 10, 14 (2020). https://doi.org/10.1007/s13324-020-00358-3
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DOI: https://doi.org/10.1007/s13324-020-00358-3