Central Research Theme:
The central theme is the study of electronic transport in models of solid state and mesoscopic physics for which delicate quantum interference phenomena are dominant. Physical materials in this regime are disordered media such as alloys and doped semiconductors, quasicrystals, quantum Hall samples and mesoscopic systems. To understand the quantum transport therein is not only of physical interest, but also of potential technological importance. A close contact to the physics community is therefore necessary, but moreover the models lead naturally to the use of various mathematical methods which are of interest of their own. The basic objects of mathematical study are Schroedinger operators (discrete and continuous elliptic partial differential operators). Typically the translation invariance is broken due to random or almost-periodic potentials or the presence of exterior magnetic fields. While the spectral characteristics of such operators were studied in many works since the early 80's, the physically more relevant transport properties are often less well understood. Many questions linked to the strong and weak localization regime, to anomalous transport, to various metal-insulator transitions as well as the importance of electron-phonon and electron-electron interactions and dissipation remain open. Next follows a list of the main results of the H. Schulz-Baldes, with references given here. Results of other authors are not discussed.

Integer quantum Hall effect:
Apart from being an exhausive review, the article [BES] contains the first rigorous localization criterium for the quantum Hall regime, namely it is shown that if the Fermi energy varies in a spectral region with bounded mean square displacement, the Kubo Hall conductivity is constant, quantized and given by a non-commutative Chern number. Then this criterium is verified for discrete magnetic operators [BES] and shwon to lead to stability of the quantum Hall phase in the Fermi energy - disorder parameter plane [RiS]. The works [SKR,KRS,KS1,KS2] develop a different apprach to the integer quantum Hall effect by proving that the edge currents are quantized and calculated via a new index associated to a non-commutative winding number. Under a (physically unreasonable) gap condition, this winding number and the above Chern number are proven to be equal. This results from Bott periodicity in form of a duality theorem for pairings of K-theory with cyclic cohomology.

Perturbative results for products of random matrices:
Quasi-one-dimensional quantum systems (like wires, nanotubes, materials with a prefered orientation) can efficiently be describes by means of transfer matrices. Impurities in such materials lead to fine interference effects. Often one is interested in a regime where there is a small parameter like, for example, the strength of a disordered potential. Then one would like to study various physically interesting quantities perturbatively in the small parameter. For strictly one-dimensional systems, results concern the Lyapunov exponent, the rotation number (linked to the density of states) as well as the variance in the central limit theorem [JSS,SSS]. For a quasi-one-dimensional system given by an Anderson model on a strip, it is possible to calculate the smallest Lyapunov exponent perturbatively [S5]. This fits very well with numerical results [RS]. These results are also of relevence when studying quantum transport (see below).

Lower bounds on anomalous transport:
Certain one-dimensional almost-periodic operators have singular continuous spectral measures and since the pioneering work of Guarneri it is known that its degree of continuity, notably its Hausdorff dimension, gives a lower bound on quantum transport and wave packet spreading. In [GS2] is shown how bad scaling properties of the spectral measure (Hausdorff dimension strictly smaller than the packing dimension) allow to considerably improve the general lower bounds on adequate time scales. By now, various models are known for which this is important. Motivated by [BaS] discussed below, a general multiscaling lower bound on quantum dynamics is proven in [GS3]. The work [BGS] proves a lower bound on phase-averaged quantum transport in terms of the multifractal dimensions of the density of states (and not as before the spectral measure depending strongly on the initial state) for a wide class of quasi-periodic operators containing the critical Harper operator. It is based on a phase space analysis and number-theoretic estimates.

Upper bounds on anomalous transport:
The work [GS1] proves for the first time the corresponding upper dynamical bounds (weak localization bounds) for a class of Jacobi matrices constructed by inverse theory. These bounds are expressed in terms of the dimensionality of the spectral measures as well as properties of the eigenfunctions. They were refined to multiscaling upper bounds in [BaS] incorporating the multifractal dimensions for the first time in a rigorous study of quantum transport. These results combined allowed to construct a 3-dimensional tight-binding operator having absolutely continuous spectrum, but subdiffusive transport as slow as imposed by Guarneri's inequality [BS2].

Quantum transport in disordered systems:
A random polymer model is a one-dimensional Jacobi matrix constructed by random juxtaposition of two finite building blocs. If the associated transfert matrices commute at a so-called critical energy, it is proven in [JSS] that the localization length diverges there and that this leads to overdiffusive quantum transport. The technical estimates are on the large deviations of the random dynamical system given by the Pruefer phases. They are quantitative and believed to be optimal. Another random model with non-trivial quantum transport is the free Anderson model, a generalization of the Wegner n-orbital model, for which the quantum motion is proven to be diffusive in [SB1].

Derivation of Kubo formula and importance of dissipation:
In [SB2] the Kubo formula for the conductivity is derived by modelling the interactions of the electrons with the rest of the solid with a time-dependent stochastic process. This leads to a quantum Boltzmann equation and a mathematically well-defined Kubo formula. For the non-dissipative Hall conductivity it agrees with formulas obtained by adiabatics, but for the dissipative direct conductivity the zero-dissipation limit is finite and non-zero only if the quantum transport is diffusive, an insight implicitely contained in the Einstein relation. If the quantum transport is anomalous, the direct conductivity behaves according to the Anomalous Drude Formula [S1,SB1].