3 problems

Let ${\bf A}$ be a semi-bounded self-adjoint operator in a Hilbert space $\mathcal{H}$, and let $\lambda$ be an eigenvalue of ${\bf A}$ with multiplicity $m$. Let ${\bf A}_r$ be th…

For $N\in\mathbb N^*$, let $\mathfrak A_N^{har}$ be the isotropic harmonic-oscillator array, let $\lambda_k^{har}$ denote its $k$-th eigenvalue, and let $m(k,\mathfrak A)$ denote t…

For $N\in\mathbb N^*$, let $\mathfrak M(N)$ be the family of admissible arrays $\mathfrak A$, and for $k\in\mathbb N^*$ let $\mathfrak m_k(N)$ be the supremum of the multiplicity o…