Quantum Hamilton - Jacobi soluton for spectra of several one dimensional potentials with special properties

TL;DR

This thesis applies the quantum Hamilton-Jacobi formalism to analyze spectra of various one-dimensional potentials, including exactly solvable, quasi-exactly solvable, and PT-symmetric potentials, providing a straightforward method to find eigenvalues and eigenfunctions.

Contribution

It introduces a simple conjecture on singularities in the complex plane within the QHJ formalism to efficiently compute spectra of complex potentials.

Findings

Effective method for bound state and band edge eigenvalues

Applicable to ES, QES, and PT-symmetric potentials

Simplifies spectral analysis through singularity conjecture

Abstract

In this thesis the quantum Hamilton - Jacobi (QHJ) formalism is used for (i) potentials which exhibit different spectra for different ranges of the potential parameters, (ii) exactly solvable (ES) periodic potentials (iii) quasi - exactly solvable (QES) periodic potentials and (iv) the PT symmetric potentials (ES and QES). The QHJ formalism provides a simple and elegant method to obtain the bound state and the band edge eigenvalues and the eigenfunctions. For this purpose, a simple conjecture on the singularities of the logarithmic derivative of the wave function in the complex plane is made and used in a straight forward fashion to obtain the desired results.