Arti¯cial interface conditions parametrized by a complex number 0 are introduced for 1D-Schr€ odinger operators. When this complex parameter equals the parameter 2 iR of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The e®ect of the arti¯cial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale ðh N Þ N 2N as h ! 0, according to 0 . Math. Models Methods Appl. Sci. 2011.21:541-618. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 02/08/15. For personal use only.:4Þ This can be viewed as a singular version of the black-box formalism of Ref. 56. In addition to the fact that this singular deformation is convenient for the original model with potential barriers presented with a discontinuous potential and with a nonlinear Adiabatic Evolution of 1D Shape Resonances 543 Math. Models Methods Appl. Sci. 2011.21:541-618. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 02/08/15. For personal use only.. The arti¯cial interface conditions parametrized by 0 are introduced in Sec. 2. With these new interface conditions ÀÁ is transformed into a non-self-adjoint operator conjugated with ÀÁ,The case with a potential is illustrated with numerical computations. . The functional analysis of the complex deformation parametrized with is done in Sec. 3. After introducing a Krein formula associated with the ð 0 ; Þ-dependent interface conditions, it mimics the standard approach to resonances summarized in Refs. 23 and 29 but things have to be reconsidered for we start from an already non-self-adjoint operator when 0 6 ¼ 0. Assumptions on the time-dependent variations of the potential which ensure the well-posedness of the dynamical systems are speci¯ed in the end of this section. . The small parameter problem modeling quantum wells in a semiclassical island is introduced in Sec. 4. Accurate exponential decay estimates are presented for the spectral problems reduced to ða; bÞ making use of the fact that our operators are proportional to Àh 2 Á outside ½a; b. Adiabatic Evolution of 1D Shape Resonances 545 Math. Models Methods Appl. Sci. 2011.21:541-618. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 02/08/15. For personal use only.

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Dedicated to the memory of P. Duclos. AbstractThis paper is concerned with a linearized version of the quantum transport problem where the Schrödinger-Poisson operator is replaced by a non-autonomous Hamiltonian, slowly varying in time. We consider an explicitly solvable system where a semiclassical island is described by a flat potential barrier, while a time dependent 'delta' interaction is used as a model for a single quantum well. Introducing, in addition to the complex deformation, a further modification formed by artificial interface conditions, we give a reduced equation for the adiabatic evolution of the sheet density of charges accumulating around the interaction point.