Verified Numerical Computations for Eigenvalues of Non-Commutative Harmonic Oscillators

Abstract

We study the eigenvalue problem of the formally self-adjoint operator

The Heat Semigroup and Kernel Associated With Certain Non-Commutative Harmonic Oscillators

“…Its spectrum was studied in [12][13][14] by employing the representation-theoretic method, and extensively studied by many authors [2,3,[5][6][7][8][9][10][11]. In the present paper, we investigate the case when α = β from the point of view of stochastic analysis: we construct probabilistically the heat semigroup and kernel associated with…”

On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

“…As a result, the first eigenvalue λ 1 is positive, so that Q has a bounded inverse Q −1 . However, the value of any eigenvalue λ n is hardly computed explicitly (A numerical study of the spectrum was carried out in [9]). The two relations of non-commutativity, i.e.…”

“…Otherwise, it will possibly happen to have a multiplicity greater than 2. It was shown in [9] that the first eigenvalue λ 1 is simple for sufficiently large α, β. This fact was also observed by Parmeggiani [13] with perturbation theory.…”

On the essential spectrum of certain non-commutative oscillators

“…It is also worth mentioning that numerical study of the spectrum Q w (x, D) has been carried out by Nagatou, Nakao and Wakayama in [10], and that one can study the spectrum by Rellich's perturbation theory in the limit αβ → +∞ with α/β a fixed constant = 1 (see [15]). Furthermore, the study of Poisson-type relations for the spectral distribution, and clustering theorems of the spectrum were proved in Parmeggiani [16,17] (see also [18]).…”