1. By Supervisor Urbashi Satpathi Dr. Prosenjit Singha De0

2. Experimental background Motivation PLDOS, Injectivity, Emissivity, Injectance, Emitance, LDOS, DOS Paradox and its practical implication Experimental background Motivation PLDOS, Injectivity, Emissivity, Injectance, Emitance, LDOS, DOS Paradox and its practical implication

3. Schematic description of experimental set up [ R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky and Hadas Shtrikman, Nature 385, 417 (1997) ]

4. Collector voltage, V CB

7. Analyticity Hilbert Transform relates the amplitude and argument

8. What information we can get from phase shift ?

9. Apparently does not follow Friedel sum rule (FSR) However if carefully seen w.r.t Fano resonance (FR) c an be understood from FSR  Besides there is a paradox at FR that can have tremendous practical implication.

10. Larmor precision time (LPT) Injectivity

11. Why injectivity is physical?

12. Local density of states Density of states This is an exact expression.

13. In semi-classical limit Hence,, is FSR (semi classical). [ M. Büttiker, Pramana Journal of Physics 58, 241 (2002) ]

14. and, is semi classical injectance. [ C. R. Leavens and G. C. Aers, Phys. Rev. B 39, 1202 (1989), E. H. Hauge, J. P. Falck, and T. A. Fjeldly, Phys. Rev. B 36, 4203 (1987) ]

15. Incident wave packet Scattered wave packet

16. i.e. in semi classical case, density of states is related to energy derivative of scattering phase shift. Considering no reflected part (E>>V), and no dispersion of wave packet, is stationary phase approximation.

17. The confinement potential, The scattering potential, is symmetric in x-direction

18. The Schrödinger equation of motion in the defect region is, In the no defect region,, where, and,, is the energy of incidence.

19. For symmetric potentials, For, where, w and, At resonance,

20. The potential at X, Internal wave function Modes of the quantum wire Injectance from wave function is,

21. , where,, and

22. Injectance from wave function is, Semi classical injectance is,,

23. and

29. There is a paradox at Fano resonance The semi classical injectivity gets exact at FR Useful for experimentalists

30. 1.Leggett's conjecture for a mesoscopic ring P. Singha Deo Phys. Rev. B {\bf 53}, 15447 (1996). 2. Nature of eigenstates in a mesoscopic ring coupled to a side branch. P. A. Sreeram and P. Singha Deo Physica B {\bf 228}, 345(1996. 3.Phase of Aharonov-Bohm oscillation in conductance of mesoscopic systems. P. Singha Deo and A. M. Jayannavar. Mod. Phys. Lett. B {\bf 10}, 787 (1996). 4.Phase of Aharonov-Bohm oscillations: effect of channel mixing and Fano resonances. P. Singha Deo Solid St. Communication {\bf 107}, 69 (1998). 5.Phase slips in Aharonov-Bohm oscillations P. Singha Deo Proceedings of International Workshop on $``$Novelphysics in low dimensional electron systems", organized byMax-Planck-Institut Fur Physik Komplexer Systeme, Germanyin August, 1997.\\Physica E {\bf 1}, 301 (1997). 6.Novel interference effects and a new Quantum phase in mesoscopicsystems P. Singha Deo and A. M. Jayannavar, Pramana Journal of Physics, {\bf 56}, 439 (2001). Proceedings of the Winter Institute on Foundations of Quantum Theoryand Quantum Optics, at S.N. Bose Centre,Calcutta, in January 2000. 7.Electron correlation effects in the presence of non-symmetry dictated nodes P. Singha Deo Pramana Journal of Physics, {\bf 58}, 195 (2002) 8.Scattering phase shifts in quasi-one-dimension P. Singha Deo, Swarnali Bandopadhyay and Sourin Das International Journ. of Mod. Phys. B, {\bf 16}, 2247 (2002) 9.Friedel sum rule for a single-channel quantum wire Swarnali Bandopadhyay and P. Singha Deo Phys. Rev. B {\bf 68} 113301 (2003) 10. Larmor precession time, Wigner delay time and the local density of states in a quantum wire. P. Singha Deo International Journal of Modern Physics B, {\bf 19}, 899 (2005) 11. Charge fluctuations in coupled systems: ring coupled to a wire or ring P. Singha Deo, P. Koskinen, M. Manninen Phys. Rev. B {\bf 72}, 155332 (2005). 12. Importance of individual scattering matrix elements at Fano resonances. P. Singha Deo} and M. Manninen Journal of physics: condensed matter {\bf 18}, 5313 (2006). 13. Nondispersive backscattering in quantum wires P. Singha Deo Phys. Rev. B {\bf 75}, 235330 (2007) 14. Friedel sum rule at Fano resonances P Singha Deo J. Phys.: Condens. Matter {\bf 21} (2009) 285303. 15. Quantum capacitance: a microscopic derivation S. Mukherjee, M. Manninen and P. Singha Deo Physica E (in press). 16. Injectivity and a paradox U. Satpathy and P. Singha Deo International journal of modern physics (in press).