Research  

1. Approximation of the Schrödinger Equation of 2D Materials:

The Hamiltonian of the Schrodinger Equation Hψ = Eψ is H = -∂²/∂x² - ∂²/∂z² + V(x, z), where the potential based on Kronig Penny Model is given by: V(x, z) = Σᵢ Σₗₙ V₀ⁱ δₑ(x + bᵢ - nᵢ) exp(-(z + Llᵢ)²ηᵢ) and Gaussian function defined by δₑ(x) = 1 / √(2π) e⁻(x² / 2ε²).

 Observations: If the centers of the Gaussian functions are taken as [0, 0.4, 0.6], there are three isolated bands below the Fermi level, each separated by a band gap. However, we obtain one isolated and two entangled bands if the centers are taken at one-third ratios, i.e., [0, 0.33, 0.66]. Since all these bands are truncated within a certain value, we observe band folding in the given range.
Acknowledgements: This research was done under Prof. Daniel Massatt- Louisiana State University.

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2. Research in Numerical Analysis: "Study of Quadrature Methods”, the nodes and corresponding weights of Gauss Legendre & Gauss Chebyshev were determined using direct solving of non-homogeneous equations and through orthogonal polynomials of Legendre and Chebyshev polynomials under the supervision of Dr. J. Sucharitha, Professor, Osmania University, Hyderabad, India, 2013.