2006/10/24 by C V Sukumar, C. V. Sukumar
Mathematics · Physics and Astronomy · #Domain (mathematical analysis) #Eigenfunction #Eigenvalues and eigenvectors #Harmonic oscillator #Mathematical functions and polynomials #Maxima and minima #Node (physics) #Quantum Mechanics and Non-Hermitian Physics #Simple (philosophy) #Spectral Theory in Mathematical Physics #Sum rule in quantum mechanics #quant-ph
paper · pdf · doi:10.1088/0305-4470/39/45/023
published as J.Phys. A: Math. Gen. 39 (2006) 14153-14163 · 17 pages
arxiv created 2006/10/24 · openalex publication_date 2006/10/24 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schrödinger equation has two linearly independent solutions one of which is normalizable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalizable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules gives conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on supersymmetric quantum mechanics are discussed.