Sturm’s theorem on zeros of linear combinations of eigenfunctions

on the effect of linear & non-linear texts on students comprehension and recalling

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Zeros of linear combinations of Laguerre polynomials from different sequences

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely Rn = L α n + aL α n and Sn = L α n + bL α n−1. Proofs and numerical counterexamples are given in situations where the zeros of Rn, and Sn, respectively, interlace (or do not in general) with the zeros of L α k , L ′ k , k = n or n− 1. The results we prove ho...

Zeros of eigenfunctions of some anharmonic oscillators

where P is a real even polynomial with positive leading coefficient, which is called a potential. The boundary condition is equivalent to y ∈ L(R) in this case. It is well-known that the spectrum is discrete, and all eigenvalues λ are real and simple, see, for example [3, 14]. The spectrum can be arranged in an increasing sequence λ0 < λ1 < . . .. Eigenfunctions y are real entire functions of o...

A Theorem on Spin-Eigenfunctions

A theorem is proved to the effect that a wave function for a set of N spins,, which is a product of single-spin wave functions and which is an eigenstate of the square of the total spin S*, must be a state with the maximum possible value of S* and of S 3 z being an arbitrary direction. This theorem has z been applied in a separate work by the authors to show strikingly that the imposition of sy...

Zeros of real analytic ergodic Laplace eigenfunctions