1. Recent Advances in the Theoretical Methods and Computational Schemes for Investigations of Resonances in Few-Body Atomic Systems Y. K. Ho Institute of Atomic and Molecular Sciences Academia Sinica, Taipei, Taiwan (APFB11, Seoul, Korea, August 22-26, 2011)

2. Outline of Presentation Introduction Stabilization Method Complex Absorbing Potential Method Complex-scaling for screened Coulomb (Yukawa) potentials Borromean binding for muonic molecular ions and H 2 + ions with screened Coulomb potentials

4. V A Mandelshtam, T. R. Ravuri and H.S. Taylor, Phys. Rev.Lett. 70 (1993) 1932

6. A Model Problem

7. A Model Problem Mandelshtam et al PRL 70(1993) 1932. A Spherical-box approach to resonances V(r)=V 0 r 2 exp(-r) with V 0 =7.5, φ n (r) =(2πR)  1/2 sin(nπr/R) Present results : Er =3.426,  =0.0254, 80-term basis functions Mandelshtam et al : Er =3.42,  =0.025, 80-term basis functions At the energy 3.426 a.u., a series of avoided crossing representing a resonance.

8. Fitting to the Lorentzian form yields resonance energy E r and a total width Γ, with E r = 3.426,  = 0.0254

11. Stabilization diagram for the 1 S e states of Si 12+ in plasmas for =10. Kar and Ho, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044007 For resonance calculations, we multiple by a scaling constant w.

12. Fig. 2. The best fitting (solid line) of the calculated density of states (circles) corresponding to 18 th energy level in the stabilization diagram for the lowest 1 S e states. Kar and Ho, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044007

13. The 2s 2 1 S e resonance energy E r of two-electron ions as a function of 1/Z and 1/D. Kar and Ho, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044007

14. The 2s 2 1 S e resonance width  of two-electron ions as a function of 1/Z and 1/D. Kar and Ho, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044007

15. Complex Absorbing Potential Method for resonances calculations

16. Sahoo and Ho, Chin, J. Phys. (1996)

17. Sahoo and Ho, Phys. Rev. B 69, 165323 (2004) Sahoo, Lin and Ho, Physica E 40, 3107 (2008) Quantum-confined hydrogenic impurity in a spherical quantum dot under the influence of parallel electric and magnetic fields

18. Potential energy profile for a hydrogenic impurity in a single quantum well of depth V 0 and radius R 0. V0V0

19. Potential energy profile for a hydrogenic impurity in a single quantum well of depth V 0 and radius R 0 subject to an external electric field F.

20. Quantum size effect on the field induced energy shift of the lowest 1s resonating state of confined hydrogen atom at F=0.1 a.u. * for different magnetic field strengths Sahoo and Ho, Phys. Rev. B 69, 165323 (2004) Sahoo, Lin and Ho, Physica E 40, 3107 (2008)

21. Quantum size effect on the Stark width of the lowest 1s resonating state of confined hydrogen atom at F=0.1 a.u. * for different magnetic field strengths. Sahoo and Ho, Phys. Rev. B 69, 165323 (2004) Sahoo, Lin and Ho, Physica E 40, 3107 (2008)

23. Y. K. Ho, Phys. Reports. 99, 1 (1983) and references therein.

24. Chakraborty and Ho, Chem. Phys. Lett. 438, 99 (2007) Euro. Phys. J. D 49, 59 (2008) Phys. Rev. A 77, 014502 (2008)

25. Chakraborty and Ho, Chem. Phys. Lett. 438, 99 (2007)

28. Chakraborty and Ho, Euro. Phys. J. D 49, 59 (2008)

30. Borromean binding in muonic molecular ions with screened Coulomb potentials

32. Ghoshal and Ho, J. Phys. B 43, 115007 (2010)

35. Three-Body Two-Body Three-Body Two-Body ECSCP SSCP ppµ System Ghoshal and Ho, J. Phys. B 43, 115007 (2010)

36. m ddµ system

37. Three-Body Two-Body Three-Body ECSCP SSCP ddµ system Ghoshal and Ho, J. Phys. B 43, 115007 (2010)

41. The bound 3 P o ( =0, J =1) and 3 P o ( =1, J =1) states of the molecular H 2 + ion in terms of the screening parameters  along with H(1s 2 S) threshold energies. Borromean window ↑ ↑ Kar and Ho, Chem. Phys. Letts 506, 282 (2011)

42. The bound 1 D e ( =0, J =2) and 1 D e ( =1, J =2) states of the molecular H 2 + ion in terms of the screening parameters  along with H(1s 2 S) threshold energies. Borromean window ↑↑ Kar and Ho, Chem. Phys. Letts. 506, 282 (2011)

43. System H 2 + ( 1 S e ) ( =0, J=0) 1.1878 1.1899 a 1.19 b 1.190 c 1.3734 1.373 (4) a 1.35 b 1.37 c H 2 + ( 1 S e ) ( =1, J=0) 1.190 c 1.331 c H 2 + ( 3 P o ) ( =0, J=1) 1.18781.3715 H 2 + ( 3 P o ) ( =1, J=1) 1.18781.3389 H 2 + ( 1 D e ) ( =0, J=2) 1.18781.3626 H 2 + ( 1 D e ) ( =1, J=2) 1.18781.3223 a Bertini et al, Phys. Rev. A 69, 042504 (2004) b Bressanini, Mella and Morosi, Phys. Rev. A 55 (1997) 200. c Ghoshal and Ho, J. Phys. B 43, 115007 (2010) Windows for Borromean binding for S, P, and D states of H 2 + Kar and Ho, Chem. Phys. Letts. 506, 282 (2011)

44. Acknowledgements The works are supported by the National Science Council of Taiwan, R.O.C. I am thankful to the following collaborators: Dr. S. Kar (Professor at the Center for Theoretical Atomic and Molecular Physics. Harbin Institute of Technology, Harbin, China) Dr. Sumana Chakraborty Dr. A. Ghoshal Thank you all for your attention.

45. The financial support from National Science Council of Taiwan is sincerely acknowledged