We propose a criterion of transversality and disjointness for the Friedrichs and Krein extensions of a nonnegative symmetric operator in terms of the vectors {φj, j ϵ 𝕁} that form a Riesz basis of the defect subspace. The criterion is applied to the Friedrichs and Krein extensions of the minimal Schrödinger operator Ad with point potentials. We also present a new proof of the fact that the Friedrichs extension of the operator Ad is a free Hamiltonian.
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References
S. Albeverio, F Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, New York (1988).
S. Albeverio, A. Kostenko, and M. Malamud, “Spectral theory of semibounded Sturm–Liouville operators with local interactions on a discrete set,” J. Math. Phys., 51, No. 10, 24 (2010).
Yu. Arlinskiĭ and S. Belyi, “Non-negative self-adjoint extensions in rigged Hilbert space,” Oper. Theory: Adv. Appl., 236, 11–41 (2013).
Yu. Arlinskiĭ and Yu. Kovalev, “Factorizations of nonnegative symmetric operators,” Meth. Funct. Anal. Top., 19, No. 3, 211–226 (2013).
Yu. Arlinskii and E. Tsekanovskii, “M. Krein research on semi-bounded operators, its contemporary developments, and applications,” Oper. Theory: Adv. Appl., 190, 65–112 (2009).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1966).
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
F. A. Berezin and L. D. Faddeev, “A remark on the Schr¨odinger equation with singular potential,” Dokl. Akad. Nauk SSSR, 137, No. 5, 1011–1014 (1967).
V. A. Derkach and M. M. Malamud, “The extension theory of Hermitian operators and the moment problem,” J. Math. Sci., 73, 141–242 (1995).
I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1965).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1966).
Yu. G. Kovalev, “1D nonnegative Schrödinger operators with point interactions,” Mat. Stud., 39, No. 2, 150–163 (2013).
Yu. G. Kovalev, “On the theory of nonnegative Hamiltonians in a plane and in a space,” Ukr. Mat. Visn., 11, No. 2, 203–226 (2014).
V. D. Koshmanenko, Singular Bilinear Forms in the Theory of Perturbations of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1993)
A. Kostenko and M. Malamud, “1-D Schrödinger operators with local point interactions on a discrete set,” J. Different. Equat., No. 249, 253–304 (2010).
M. G. Krein, “Theory of self-adjoint expansions of semibounded Hermitean operators and its applications,” Mat. Sb., 20, No. 3, 431–495 (1947).
M. M. Malamud and K. Schmüdgen, “Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions,” J. Funct. Anal., 263, 3144–3194 (2012).
V. A. Mikhailets, “Spectral properties of the one-dimensional Schrödinger operator with point intersections,” Repts Math. Phys., 36, No. 2/3, 495–500 (1995).
R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York (1980).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 4, pp. 495–505, April, 2018.
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Kovalev, Y.G. On the Criteria of Transversality and Disjointness of Nonnegative Self-Adjoint Extensions of Nonnegative Symmetric Operators. Ukr Math J 70, 568–580 (2018). https://doi.org/10.1007/s11253-018-1517-9
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DOI: https://doi.org/10.1007/s11253-018-1517-9