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Spectral analysis of dissipative Schrödinger operators

  • B. P. Allahverdiev (a1) and Ahmet Canoǧlu (a1)

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Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

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1Allahverdiev, B. P.. On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl's limit-circle case. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 242–57; English translation: Math. USSR Izv. 36 (1991), 247–62.
2Lax, P. D. and Phillips, R. S.. Scattering Theory (New York: Academic Press, 1967).
3Sz.-Nagy, Béla and Foias, Ciprian. Analyse harmonique des operateurs de l'espace de Hibert (Paris: Masson, and Budapest: Akad. Kiadò, 1967; English translation: Amsterdam: North-Holland, and Budapest: Akad. Koadò, 1970).
4Naimark, N. A.. Linear Differential Operators, 2nd edn (Moscow: ‘Nauka’, 1969; English translation of 1st edn, Parts I, II: New York: Ungar, 1967, 1968).
5Pavlov, B. S.. Spectral theory of nonselfadjoint differential operators. In Proceedings of the International Congress on Mathematics (Warsaw, 1983), Vol.2, 1011–25 (Warsaw: PWN, and Amsterdam: North-Holland, 1984).

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