2007
DOI: 10.1002/mana.200610553
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Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

Abstract: We consider the spectral problem for the Schrödinger operator L on a quantum network, represented as a union of a finite number of quantum wells and finite or semi-infinite straight quantum wires connecting them to each other and to infinity. The absolutely continuous spectrum of the Schrödinger operator, with real piecewisecontinuous potential on the wells and constant potential on the wires, has a band structure defined by the geometry and the potentials on the semi-infinite wires. We split the Schrödinger o… Show more

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Cited by 31 publications
(47 citation statements)
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“…In what follows, we study second-order differential operators on metric graphs with matching conditions more general than those of δ−type, introduced above, namely, with the so-called weighted, or "Datta-Das Sarma", matching conditions, see [21,31,45]. In the case of differential expression (17) on the graph G ε , the corresponding modification is described as follows.…”
Section: Datta-das Sarma Conditionsmentioning
confidence: 99%
“…In what follows, we study second-order differential operators on metric graphs with matching conditions more general than those of δ−type, introduced above, namely, with the so-called weighted, or "Datta-Das Sarma", matching conditions, see [21,31,45]. In the case of differential expression (17) on the graph G ε , the corresponding modification is described as follows.…”
Section: Datta-das Sarma Conditionsmentioning
confidence: 99%
“…Theorem 1 and its corollaries supplement the very interesting recent investigations of matrix valued functions, cf. [5,6,13,16,18,26].…”
Section: Corollarymentioning
confidence: 99%
“…Theorem 2 Let the conditions (18) and ν < 1 hold. Then problem (22) has a solution y(t) satisfying the inequality…”
Section: Lemma 5 Let Condition (18) Hold Then the Periodic Problem (mentioning
confidence: 99%
“…In Section 4 we consider a so-called coupling of elliptic operators. Such couplings are of great interest in problems of mathematical physics, e.g., in the description of quantum networks; for more details and further references we refer the reader to the recent works [20,21,[44][45][46]. Suppose that R n , n > 1, is decomposed in a bounded domain Ω with smooth boundary C and the unbounded domain Ω = R n \Ω.…”
Section: Introductionmentioning
confidence: 99%