Letters in Mathematical Physics

, Volume 36, Issue 2, pp 177-187

First online:

Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators

  • Kimball A. MiltonAffiliated withDepartment of Physics and Astronomy, The University of OklahomaTheoretical Physics Group, Blackett Laboratory, Imperial College
  • , Rhiju DasAffiliated withOklahoma School of Science and Mathematics

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The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N 2, whereN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this Letter, we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian isH=p 2/2+λq 2k /2k. Construction of such matrix elements does not require solving the implicit equations of motion. Low-order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator groundstate gives a result for thek=2 anharmonic oscillator groundstate energy accurate to better than 1% while a two-state approximation reduces the error to less than 0.1%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.

Mathematics Subject Classifications (1991)

Primary: 81Q99 Secondary: 39A10

Key words

lattice field theory Hamiltonians finite-element method Gaussian knots quantum systems matrix elements