Auxiliary fields as a tool for computing analytical solutions of the Schrödinger equation

Abstract

We propose a new method to obtain approximate solutions for the Schrödinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows in many cases to find analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which improve a lot the corresponding formulae that can be found in literature.

Bounds for Hamiltonians with arbitrary kinetic parts

“…Let us note |ν 0 , ρ 0 an eigenstate of the nonrelativistic Hamiltonian H NR given by (7) with ν = ν 0 = p 0 /T ′ (p 0 ) and ρ = ρ 0 = V ′ (r 0 )/(|λ| r λ−1 0 ). Such a state is an approximation of the corresponding eigenstate of H 9 .…”

The Quantum N-Body Problem and the Auxiliary Field Method: New Developments

“…, α x 2 + β, … A good accuracy can be obtained with a proper choice of P(x) [1][2][3][4]. Using the comparison theorem [5], it can be shown that E(r 0 ) is an upper (lower) bound of the exact eigenvalues ifṼ (r,…”

“…Within this theory, quarks are assumed to be in the fundamental representation of the colour group SU(N c ), with N c → ∞ and the constraint α S N c = α 0 ∼ O (1). Its main feature is the suppression of internal quark loops, provided the number of flavours is finite.…”

“…The original Hamiltonian is then recovered. The idea of the AFM is to replace the auxiliary fieldν by a real parameter ν [1,2]. In this case,Ṽ (r, ν) is a linear function of P(r ), andH (ν) is solvable for some particular P(r ): r 2 or −1/r for instance (with P(r ) = r , only S-states can be obtained analytically).…”

Charm and bottom baryon masses in the combined1/Ncand1/mQexpansion versus the quark model

“…To this aim, the auxiliary field technique will be used in order to transform the Hamiltonian (17) into an analytically solvable one [22,23].…”