M. A. Topinka, B. J. Leroy, S. E. Shaw, E. J. Heller, R. M. Westervelt et al., Imaging Coherent Electron Flow from a Quantum Point Contact, Science, vol.289, issue.5488, pp.2892323-2326, 2000.
DOI : 10.1126/science.289.5488.2323

D. K. Ferry, S. M. Goodnick, and J. Bird, Transport in Nanostructures, 2009.

A. Baumgartner, T. Ihn, K. Ensslin, K. Maranowski, and A. C. Gossard, Quantum Hall effect transition in scanning gate experiments, Physical Review B, vol.76, issue.8, p.85316, 2007.
DOI : 10.1103/PhysRevB.76.085316

M. A. Eriksson, R. G. Beck, M. Topinka, J. A. Katine, R. M. Westervelt et al., Cryogenic scanning probe characterization of semiconductor nanostructures, Applied Physics Letters, vol.69, issue.5, pp.671-673, 1996.
DOI : 10.1063/1.117801

H. Sellier, B. Hackens, M. G. Pala, F. Martins, S. Baltazar et al., On the imaging of electron transport in semiconductor quantum structures by scanning-gate microscopy: successes and limitations, Semiconductor Science and Technology, vol.26, issue.6, p.64008, 2011.
DOI : 10.1088/0268-1242/26/6/064008

URL : https://hal.archives-ouvertes.fr/hal-00709523

M. A. Topinka, B. J. Leroy, R. M. Westervelt, S. E. Shaw, R. Fleischmann et al., Coherent branched flow in a two-dimensional electron gas, Nature, vol.15, issue.6825, pp.410183-186, 2001.
DOI : 10.1038/35065553

M. A. Topinka, B. J. Leroy, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard, Imaging coherent electron wave flow in a two-dimensional electron gas, Physica E: Low-dimensional Systems and Nanostructures Proceedings of the Fourteenth International Conference on the Electronic Properties of Two-Dimensional Systems, pp.678-683, 2002.
DOI : 10.1016/S1386-9477(01)00375-7

B. J. Leroy, M. A. Topinka, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard, Imaging electron density in a two-dimensional electron gas, Applied Physics Letters, vol.80, issue.23, pp.4431-4433, 2002.
DOI : 10.1063/1.1484548

B. J. Leroy, M. A. Topinka, A. C. Bleszynski, R. M. Westervelt, S. E. Shaw et al., Imaging coherent electron flow in a two-dimensional electron gas, 5th International Conference on noncontact AFM in, pp.134-139, 2003.
DOI : 10.1016/S0169-4332(02)01493-9

. Gossard, Imaging electron interferometer, Phys. Rev. Lett, vol.94, p.126801, 2005.

N. Aoki, C. R. Da-cunha, R. Akis, D. K. Ferry, and Y. Ochiai, Scanning gate microscopy investigations on an InGaAs quantum point contact, Applied Physics Letters, vol.87, issue.22, p.223501, 2005.
DOI : 10.1063/1.2136408

K. E. Aidala, R. E. Parrott, T. Kramer, E. J. Heller, R. M. Westervelt et al., Imaging magnetic focusing of coherent electron waves, Nature Physics, vol.19, issue.7, pp.464-468, 2007.
DOI : 10.1021/nl0504585

M. P. Jura, M. A. Topinka, L. Urban, A. Yazdani, H. Shtrikman et al., Unexpected features of branched flow through high-mobility two-dimensional electron gases, Nature Physics, vol.54, issue.12, pp.841-845, 2007.
DOI : 10.1038/nphys756

M. P. Jura, M. A. Topinka, M. Grobis, L. N. Pfeiffer, K. W. West et al., Electron interferometer formed with a scanning probe tip and quantum point contact, Physical Review B, vol.80, issue.4, p.41303, 2009.
DOI : 10.1103/PhysRevB.80.041303

M. P. Jura, M. Grobis, M. A. Topinka, L. N. Pfeiffer, K. W. West et al., Spatially probed electron-electron scattering in a two-dimensional electron gas, Physical Review B, vol.82, issue.15, p.155328, 2010.
DOI : 10.1103/PhysRevB.82.155328

S. Schnez, C. Rössler, T. Ihn, K. Ensslin, C. Reichl et al., Imaging the lateral shift of a quantum point contact using scanning gate microscopy, Physical Review B, vol.84, issue.19, 2011.
DOI : 10.1103/PhysRevB.84.195322

N. Aoki, R. Brunner, A. M. Burke, R. Akis, R. Meisels et al., Direct Imaging of Electron States in Open Quantum Dots, Physical Review Letters, vol.108, issue.13, p.136804, 2012.
DOI : 10.1103/PhysRevLett.108.136804

A. A. Kozikov, C. Rössler, T. Ihn, K. Ensslin, C. Reichl et al., Interference of electrons in backscattering through a quantum point contact, New Journal of Physics, vol.15, issue.1, p.13056, 2013.
DOI : 10.1088/1367-2630/15/1/013056

A. Bachtold, M. S. Fuhrer, S. Plyasunov, M. Forero, E. H. Anderson et al., Scanned Probe Microscopy of Electronic Transport in Carbon Nanotubes, Physical Review Letters, vol.84, issue.26, pp.6082-6085, 2000.
DOI : 10.1103/PhysRevLett.84.6082

R. Crook, C. G. Smith, M. Y. Simmons, and D. A. Ritchie, Imaging electrostatic microconstrictions in long 1D wires, Physica E: Low-dimensional Systems and Nanostructures Proceedings of the Fourteenth International Conference on the Ele ctronic Properties of Two-Dimensional Systems, pp.695-698, 2002.
DOI : 10.1016/S1386-9477(01)00381-2

A. Pioda, S. Ki?in, T. Ihn, M. Sigrist, A. Fuhrer et al., Spatially Resolved Manipulation of Single Electrons in Quantum Dots Using a Scanned Probe, Physical Review Letters, vol.93, issue.21, p.216801, 2004.
DOI : 10.1103/PhysRevLett.93.216801

P. Fallahi, A. C. Bleszynski, R. M. Westervelt, J. Huang, J. D. Walls et al., Imaging a Single-Electron Quantum Dot, Nano Letters, vol.5, issue.2, pp.223-226, 2005.
DOI : 10.1021/nl048405v

M. T. Woodside and P. L. Mceuen, Scanned Probe Imaging of Single-Electron Charge States in Nanotube Quantum Dots, Science, vol.296, issue.5570, pp.1098-1101, 2002.
DOI : 10.1126/science.1069923

A. C. Bleszynski, F. A. Zwanenburg, R. M. Westervelt, A. L. Roest, E. P. Bakkers et al., Scanned Probe Imaging of Quantum Dots inside InAs Nanowires, Nano Letters, vol.7, issue.9, pp.2559-2562, 2007.
DOI : 10.1021/nl0621037

E. E. Boyd and R. M. Westervelt, Extracting the density profile of an electronic wave function in a quantum dot, Physical Review B, vol.84, issue.20, p.205308, 2011.
DOI : 10.1103/PhysRevB.84.205308

B. Hackens, F. Martins, T. Ouisse, H. Sellier, S. Bollaert et al., Imaging and controlling electron transport inside a quantum ring, Nature Physics, vol.2, issue.12, pp.826-830, 2006.
DOI : 10.1038/nature04594

URL : https://hal.archives-ouvertes.fr/hal-00167167

F. Martins, B. Hackens, M. G. Pala, T. Ouisse, H. Sellier et al., Imaging Electron Wave Functions Inside Open Quantum Rings, Physical Review Letters, vol.99, issue.13, p.136807, 2007.
DOI : 10.1103/PhysRevLett.99.136807

URL : https://hal.archives-ouvertes.fr/hal-00193197

M. G. Pala, B. Hackens, F. Martins, H. Sellier, V. Bayot et al., Local density of states in mesoscopic samples from scanning gate microscopy, Physical Review B, vol.77, issue.12, p.125310, 2008.
DOI : 10.1103/PhysRevB.77.125310

URL : https://hal.archives-ouvertes.fr/hal-00193195

N. Paradiso, S. Heun, S. Roddaro, L. N. Pfeiffer, K. W. West et al., Selective control of edge-channel trajectories by scanning gate microscopy. Physica E: Low-dimensional Systems and Nanostructures, 18th International Conference on Electron Properties of Two-Dimensional Systems, pp.1038-1041, 2010.

R. Crook, C. G. Smith, A. C. Graham, I. Farrer, H. E. Beere et al., Imaging Fractal Conductance Fluctuations and Scarred Wave Functions in a Quantum Billiard, Physical Review Letters, vol.91, issue.24, p.246803, 2003.
DOI : 10.1103/PhysRevLett.91.246803

M. R. Connolly, R. K. Puddy, D. Logoteta, P. Marconcini, M. Roy et al., Unraveling Quantum Hall Breakdown in Bilayer Graphene with Scanning Gate Microscopy, Nano Letters, vol.12, issue.11, pp.125448-5454, 2012.
DOI : 10.1021/nl3015395

H. Van-houten and C. Beenakker, Quantum Point Contacts, Physics Today, vol.49, issue.7, pp.22-27, 1996.
DOI : 10.1063/1.881503

B. J. Van-wees, H. Van-houten, C. W. Beenakker, J. G. Williamson, L. P. Kouwenhoven et al., Quantized conductance of point contacts in a two-dimensional electron gas, Physical Review Letters, vol.60, issue.9, pp.848-850, 1988.
DOI : 10.1103/PhysRevLett.60.848

N. Agrait, A. L. Yeyati, and J. M. Van-ruitenbeek, Quantum properties of atomic-sized conductors, Physics Reports, vol.377, issue.2-3, pp.81-279, 2003.
DOI : 10.1016/S0370-1573(02)00633-6

. Jones, One-dimensional transport and the quantisation of the ballistic resistance, Journal of Physics C: Solid State Physics, vol.21, issue.8, p.209, 1988.

M. Büttiker, Quantized transmission of a saddle-point constriction, Physical Review B, vol.41, issue.11, pp.7906-7909, 1990.
DOI : 10.1103/PhysRevB.41.7906

S. Datta, Electronic transport in mesoscopic systems, 1997.

H. Van-houten, C. W. Beenakker, and B. J. Van-wees, Chapter 2: Quantum Point Contacts, Semiconductors and Semimetals, vol.35, pp.9-112, 1992.
DOI : 10.1016/S0080-8784(08)62392-3

M. A. Topinka, R. M. Westervelt, and E. J. Heller, Imaging Electron Flow, Physics Today, vol.56, issue.12, pp.47-52, 2003.
DOI : 10.1063/1.1650228

S. E. Shaw, R. Fleischmann, and E. J. Heller, Quantum coherence beyond the thermal length, 2001.

S. E. Shaw, Propagation in Smooth Random Potentials, 2002.

G. Metalidis and P. Bruno, Green???s function technique for studying electron flow in two-dimensional mesoscopic samples, Physical Review B, vol.72, issue.23, p.235304, 2005.
DOI : 10.1103/PhysRevB.72.235304

A. Cresti, Microscopic current imaging in quantum point contact devices, Journal of Applied Physics, vol.100, issue.5, p.53711, 2006.
DOI : 10.1063/1.2345049

C. Gorini, R. A. Jalabert, W. Szewc, S. Tomsovic, and D. Weinmann, Theory of scanning gate microscopy, Physical Review B, vol.88, issue.3, p.35406, 2013.
DOI : 10.1103/PhysRevB.88.035406

URL : https://hal.archives-ouvertes.fr/hal-00797803

R. A. Jalabert, W. Szewc, S. Tomsovic, and D. Weinmann, What Is Measured in the Scanning Gate Microscopy of a Quantum Point Contact?, Physical Review Letters, vol.105, issue.16, p.166802, 2010.
DOI : 10.1103/PhysRevLett.105.166802

URL : https://hal.archives-ouvertes.fr/hal-00503005

A. Freyn, I. Kleftogiannis, and J. Pichard, Scanning Gate Microscopy of a Nanostructure Where Electrons Interact, Physical Review Letters, vol.100, issue.22, p.226802, 2008.
DOI : 10.1103/PhysRevLett.100.226802

URL : https://hal.archives-ouvertes.fr/hal-00218234

D. Weinmann, R. A. Jalabert, A. Freyn, G. Ingold, and J. Pichard, Detection of interaction-induced nonlocal effects using perfectly transmitting nanostructures, The European Physical Journal B, vol.66, issue.2, pp.239-244, 2008.
DOI : 10.1140/epjb/e2008-00403-7

URL : https://hal.archives-ouvertes.fr/hal-00265876

A. Abbout, Thermoelectric transport in quantum point contacts and chaotic cavities: Thermal effects and fluctuations, 2011.
URL : https://hal.archives-ouvertes.fr/tel-00793816

A. Abbout, G. Lemarié, and J. Pichard, Thermal Enhancement of Interference Effects in Quantum Point Contacts, Physical Review Letters, vol.106, issue.15, p.156810, 2011.
DOI : 10.1103/PhysRevLett.106.156810

URL : https://hal.archives-ouvertes.fr/hal-00537973

L. I. Schiff, Quantum mechanics, 1968.

E. W. Weisstein, Delta function. From MathWorld ? A Wolfram Web Resource

P. A. Mello and N. Kumar, Quantum Transport in Mesoscopic Systems: Complexity and Statistical Fluctuations. Mesoscopic Physics and Nanotechnology, 2004.
DOI : 10.1093/acprof:oso/9780198525820.001.0001

L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998.

P. Roman, Advanced quantum theory: an outline of the fundamental ideas. Addison-Wesley series in advanced physics, 1965.

P. A. Mello, P. Pereyra, and N. Kumar, Macroscopic approach to multichannel disordered conductors, Annals of Physics, vol.181, issue.2, pp.290-317, 1988.
DOI : 10.1016/0003-4916(88)90169-8

P. A. Mello and J. Pichard, Symmetries and parametrization of the transfer matrix in electronic quantum transport theory, Journal de Physique I, vol.1, issue.4, pp.493-513, 1991.
DOI : 10.1051/jp1:1991148

URL : https://hal.archives-ouvertes.fr/jpa-00246346

R. A. Jalabert and J. Pichard, Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles, Journal de Physique I, vol.5, issue.3, pp.287-324, 1995.
DOI : 10.1051/jp1:1995128

URL : https://hal.archives-ouvertes.fr/jpa-00247057

R. Landauer, Conductance from transmission: common sense points, Physica Scripta, vol.42, issue.T42, p.110, 1992.
DOI : 10.1088/0031-8949/1992/T42/020

M. Yosefin and M. Kaveh, Conduction in curvilinear constrictions: Generalization of the Landauer formula, Physical Review Letters, vol.64, issue.23, pp.2819-2822, 1990.
DOI : 10.1103/PhysRevLett.64.2819

H. A. Fertig and B. I. Halperin, Transmission coefficient of an electron through a saddle-point potential in a magnetic field, Physical Review B, vol.36, issue.15, pp.7969-7976, 1987.
DOI : 10.1103/PhysRevB.36.7969

D. S. Fisher and P. A. Lee, Relation between conductivity and transmission matrix, Physical Review B, vol.23, issue.12, pp.6851-6854, 1981.
DOI : 10.1103/PhysRevB.23.6851

V. Gasparian, T. Christen, and M. Büttiker, Partial densities of states, scattering matrices, and Green???s functions, Physical Review A, vol.54, issue.5, pp.4022-4031, 1996.
DOI : 10.1103/PhysRevA.54.4022

E. M. Lifshitz and L. D. Landau, Quantum mechanics. Non-relativistic Theory

H. Van-houten and C. Beenakker, Quantum Point Contacts, Physics Today, vol.49, issue.7, pp.22-27, 1996.
DOI : 10.1063/1.881503

R. I. Shekhter, G. B. Lesovik, and D. E. Khmelnitskii, Reflectionless quantum transport and fundamental ballistic-resistance steps in microscopic constrictions, JETP Letters, vol.48, pp.238-241, 1988.

A. Yacoby and Y. Imry, Quantization of the conductance of ballistic point contacts beyond the adiabatic approximation, Physical Review B, vol.41, issue.8, pp.5341-5350, 1990.
DOI : 10.1103/PhysRevB.41.5341

L. I. Glazman and M. Jonson, Cross-over to global adiabatics in 2D ballistic transport, Journal of Physics: Condensed Matter, vol.1, issue.32, p.5547, 1989.
DOI : 10.1088/0953-8984/1/32/027

M. C. Payne, Adiabaticity in quantum transport, Journal of Physics: Condensed Matter, vol.1, issue.30, p.4939, 1989.
DOI : 10.1088/0953-8984/1/30/007

A. Szafer and A. D. Stone, Theory of Quantum Conduction through a Constriction, Physical Review Letters, vol.62, issue.3, pp.300-303, 1989.
DOI : 10.1103/PhysRevLett.62.300

G. Kirczenow, Resonant conduction in ballistic quantum channels, Physical Review B, vol.39, issue.14, pp.10452-10455, 1989.
DOI : 10.1103/PhysRevB.39.10452

B. J. Van-wees, L. P. Kouwenhoven, E. M. Willems, C. J. Harmans, J. E. Mooij et al., Quantum ballistic and adiabatic electron transport studied with quantum point contacts, Physical Review B, vol.43, issue.15, pp.12431-12453, 1991.
DOI : 10.1103/PhysRevB.43.12431

E. W. Weisstein, Weber differential equations. From MathWorld ? A Wolfram Web Resource

J. N. Connor, On the analytical description of resonance tunnelling reactions, Molecular Physics, vol.15, issue.1, pp.37-46, 1968.
DOI : 10.1016/0003-4916(59)90025-9

P. A. Lee and D. S. Fisher, Anderson Localization in Two Dimensions, Physical Review Letters, vol.47, issue.12, pp.882-885, 1981.
DOI : 10.1103/PhysRevLett.47.882

C. W. Beenakker and H. Van-houten, Quantum Transport in Semiconductor Nanostructures. eprint arXiv:cond-mat, 2004.

P. E. Lindelof and M. Aagesen, Measured deviations from the saddle potential description of clean quantum point contacts, Journal of Physics: Condensed Matter, vol.20, issue.16, p.164207, 2008.
DOI : 10.1088/0953-8984/20/16/164207

M. A. Eriksson, R. G. Beck, M. A. Topinka, J. A. Katine, R. M. Westervelt et al., Effect of a charged scanned probe microscope tip on a subsurface electron gas, Superlattices and Microstructures, vol.20, issue.4, pp.435-440, 1996.
DOI : 10.1006/spmi.1996.0100

J. Vaní?ek and E. J. Heller, Uniform semiclassical wave function for coherent two-dimensional electron flow, Physical Review E, vol.67, issue.1, p.16211, 2003.
DOI : 10.1103/PhysRevE.67.016211

V. S. Vladimirov, Equations of mathematical physics. Pure and applied mathematics, 1971.