Wave Equations with Energy-Dependent Potentials

Abstract

We study wave equations with energy dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory.

Exact quantization formula for affine linearly energy-dependent potentials

“…While the construction of the quantization formula (21) is straightforward from a mathematical point of view, its actual derivation from physical concepts -as done for the conventional quantization formula [12] -is in general difficult, because in the presence of energy-dependent potentials the underlying Quantum Theory changes [7]. Nevertheless we will now demonstrate how the left hand side of our quantization formula (21) can be obtained without point transformations, but just from the actual Schrödinger equation.…”

“…In particular, existence of the usual L 2 -norm does not imply existence of the modified norm that has to be considered in the presence of an energy-dependent potential, see [7] for details. In order to study the normalizability of solutions to our particular equation (13), let us start from a solution φ of our conventional boundary value problem (4), (5) and assume conventional L 2 -normalizability of φ, that is, φ ∈ L 2 ( ).…”

“…In general, Schrödinger equations with energy-dependent potentials cannot be solved by conventional methods, only in very few cases the method can be adapted, such that it respects the energy dependence of the potential [8]. Furthermore, in the presence of such potentials the Quantum Theory undergoes modifications regarding the definition of the norm and the completeness relation [7]. In order to get a better understanding of energy-dependent potentials and the integrability of their associated Schrödinger equations, in this note we develop an exact quantization formula for potentials that are affine linear functions with respect to the energy.…”

Estimation of the Spatial Structure of \(\Lambda \)(1405) with Energy-Dependent Complex \(\bar{K}N\) Potential

“…[19]. The time-dependent and independent Schrödinger equation are expressed with the time-dependent wave function Ψ(r, t) and the eigenfunction of the Hamiltonian Ψ E (r, t) = e −iEt ψ E (r) as…”

“…For the application of such a potential, a special treatment is necessary because otherwise the orthogonality condition and the conservation of the norm are not guaranteed. For an energy-dependent real potential, a consistent treatment was developed from the continuity equation [19]. For an energy-independent complex potential, it is known that the use of the Gamow vector is needed to normalize the wave function [20][21][22].…”