We consider an ideal Fermi gas confined to a geometric structure consisting of a central region -the sample -connected to several infinitely extended ends -the reservoirs. Under physically reasonable assumptions on the propagation properties of the one-particle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated Landauer-Büttiker formula.An operator on H is a linear map A : D → H , where D is a subspace of H . We say that D is the domain of A which we denote by Dom(A). A is densely defined if its domain is dense in H . The range and the kernel of A are the subspaces RanThe graph of an operator A on H is the the subspaceAn operator A is completely characterized by its graph. Moreover, a subspace G ⊂ H × H is the graph of an operator iff 〈0, v〉 ∈ G implies v = 0. If A and B are two operators such that Gr (A) ⊂ Gr (B ) we say that B is an extension of A and we write A ⊂ B . An operator A is closed if its graph is closed in H ×H , and this is the case iff Dom(A), equipped with the graph norm of A, is a Banach space. If A is both closed and bijective, then Gr (A −1 ) = KGr (A) and thus A −1 is also closed. If the closure Gr (A) cl of the graph of A in H × H is a graph we say that A is closable and we define its closure as the operator A cl such that Gr (A cl ) = Gr (A) cl . It is clear that A cl is the smallest closed extension of A, that is to say that if B is closed and A ⊂ B , then A cl ⊂ B . An operator A is densely defined iff J(Gr (A) ⊥ ) is a graph. In this case, the adjoint of A is the operator A * defined by Gr (A * ) = J(Gr (A) ⊥ ). A * is closed and its domain is given by(A * u, v) = (u, Av) holds for all 〈u, v〉 ∈ Dom(A * ) × Dom(A), in particular Ker (A * ) = Ran (A) ⊥ .A is closable iff A * is densely defined. In this case A cl = A * * and A cl * = A * . An operator A is bounded if there exists a constant C such that Gr (A) ⊂ {〈u, v〉 | v ≤ C u }. One easily verifies that A is continuous as a map from Dom(A) to H iff it is bounded. A bounded operator is obviously closable and its closure coincide with its unique continuous extension to the closure of Dom(A). In particular a bounded densely defined operator A has a unique continuous extension A cl with domain Dom(A cl ) = H . A cl is closed and bounded. The collection of all bounded operators with domain H is denoted by B(H ). It is a Banach algebra (actually a C * -algebra, see Section 2.2) with norm A ≡ sup u∈H , u =1Au .By the closed graph theorem, an operator A with domain H is bounded iff it is closed. If A is bounded and densely defined, then Dom(A * ) = H and A * is bounded. Furthermore, A * = A and A * A = A 2 .
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