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Register a new userSUSY Method for the Three-Dimensional Schrodinger Equation with Effective Mass
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The three-dimensional Schrodinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.
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New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.
In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrodinger equation (NLS) with partial harmonic potential begin{equation*}tag{NLS} ipartial_t u + left(Delta_{mathbb{R}^3 }-x^2 right) u = |u|^2 u, quad u|_{t=0} = u_0. end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrodinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies.
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We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrodinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high energy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultra-violet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.
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