Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

Abstract

We consider symmetric operators of the form S := A ⊗ I T + I H ⊗ T where A is symmetric and T = T * is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet ΠS for S * preserving the tensor structure. The corresponding γ-field and Weyl function are expressed by means of the γ-field and Weyl function corresponding to the boundary triplet ΠA for A * and the spectral measure of T . Applications to 1-D Schrödinger and Dirac operators are given. A model of electron transport through a quantum dot assisted by cavity photons is proposed. In this model the boundary operator is chosen to be the well-known Jaynes-Cumming operator which is regarded as the Hamiltonian of the quantum dot.Mathematics Subject Classification: 47A80, 47B25, 81Q05, 81Q37

One-Dimensional Schrödinger Operators with Complex Potentials

“…Then by Proposition 6.7 zero belongs to the resolvent set of L a , L −1 a = G a , and D(L max ) = D(L a ) ⊕ Ker L max , (6.17) which can be restated as the decomposition (6.15). If λ ∈ C and V is replaced by V − λ then the new G → will be the resolvent at λ of L a , which proves the second assertion in (2). Finally, (6.16) is proved by a simple computation.…”

“…Proof. From Lemma 7.8 (2) and the reality of the boundary conditions at a we get W a (u, u) = 0, W a (v, v) = 0. (7.18)…”

“…If L possesses a realization L • with a non-empty resolvent set, then one of the following conditions holds: (1. 4) In the real case there is a strict correspondence between (1), (2) and (3) of (1.2) and (1), (2) and (3) of (1.4). In the complex case this correspondence holds if Conjecture 5.9 is true.…”

On a generalisation of Krein's example

“…There are related works by Boitsev, Neidhardt, and Popov [3] on tensor products of boundary triplets (with bounded operator L), Malamud and Neidhardt [15] for unitary equivalence and regularity properties of different self-adjoint realisations, Gesztesy, Weikard, and Zinchenko [5,6] for a general spectral theory of Schrödinger operators with bounded operator potentials, and Mogilevskii [17], see also the references therein. Moreover, when finishing this paper, the authors of the present paper have learned about the recent paper [2], where Boitsev, Brasche, Malamud, Neidhardt and Popov construct a boundary triplet for the adjoint of the symmetric operator T ⊗ id + id ⊗L with T being symmetric and L being self-adjoint. This generalises the situation of (1.3), where T = − d 2 / dt 2 on L 2 (R + ).…”

“…The focus in [2] is on self-adjoint extensions which do not respect the tensor structure (1.3) as models for quantum systems coupled to a reservoir. Note that in [15,2] one has to "regularise" the boundary triplet (i.e., one has to modify the boundary map and spectrally decompose L into bounded operators) in order to treat also unbounded operators L. In our approach, we can directly treat unbounded operators L without changing the boundary map or decomposing L. The special case of operators L with purely discrete spectrum has been treated e. g. in [21,Sec. 6.4] or in a slightly different setting in [20,Sec.…”