Solving the Schroedinger equation using Smolyak interpolants.

Area of Science:

• Quantum mechanics
• Computational physics
• Numerical analysis

Background:

• Collocation methods are used to solve differential equations like the Schroedinger equation.
• Existing collocation methods often require solving generalized eigenvalue problems, which can be computationally intensive.
• Integrals are avoided in collocation methods, simplifying the solution process.

Purpose of the Study:

• To introduce a new collocation method for the Schroedinger equation.
• To overcome the disadvantage of generalized eigenvalue problems in previous collocation methods.
• To develop an efficient algorithm for computing energy levels and wavefunctions.

Main Methods:

• Combining Lagrange-like functions with a Smolyak interpolant.
• Developing an efficient algorithm leveraging grid structure for matrix-vector products.
• Utilizing collocation principles to solve the Schroedinger equation without generalized eigenvalue problems.

Main Results:

• A novel collocation method that does not require solving a generalized eigenvalue problem.
• An efficient algorithm for computing matrix-vector products for energy levels and wavefunctions.
• Demonstrated systematic convergence of energies with increased points and basis functions.

Conclusions:

• The new collocation method offers a significant improvement over existing techniques.
• The developed algorithm enhances computational efficiency for quantum mechanical problems.
• This method provides a robust and accurate approach for solving the Schroedinger equation.