## Approximation to Expectation Value of Non-Selfadjoint Operator in Anharmonic Exponential Oscillator System with Use of Two-Sided Rayleigh Quotient

 AUTHOR(S): Semra Bayat Ozdemir, Metin Demiralp TITLE Approximation to Expectation Value of Non-Selfadjoint Operator in Anharmonic Exponential Oscillator System with Use of Two-Sided Rayleigh Quotient PDF KEYWORDS Anharmonic Exponential Oscillator, Differential-Difference Equation, Variational Approximation, Two-sided Rayleigh (Ostrowski) Quotient ABSTRACT This work focuses on the derivation of a solution methodology to exponential anharmonic oscillator system based on the expectation values. A differential-difference equation is constructed from the second derivative of expectation value of the exponential analytic function. Rearranging the equation gives us an eigenvalue problem. But the derived operator is not self-adjoint. This makes it impossible to use classical definition of Rayleigh Quotient for expectation value of the operator. Two-sided Rayleigh (or Ostrowski) Quotient considers both left and right eigenfunction of the non-selfadjoint operator and this definition of expectation value gives the energy value of the corresponding system. For the approximate eigenfunctions, the energy value is approximate. We construct the final expectation value equation for the operator. But the optimization process for finding the minimum approximate energy value isn’t analytically solvable. To validate the methodology, harmonic oscillator is studied at the end and an acceptable result is found with elementary approximations of eigenfuctions. Improvement of the approximations and solution to exponential anharmonic oscillator system are left as future work. Cite this paper Semra Bayat Ozdemir, Metin Demiralp. (2016) Approximation to Expectation Value of Non-Selfadjoint Operator in Anharmonic Exponential Oscillator System with Use of Two-Sided Rayleigh Quotient. International Journal of Theoretical and Applied Mechanics, 1, 170-175