Minimal eigenvalue estimates for self-adjoint trace-class operators
- URL: http://arxiv.org/abs/2407.04478v1
- Date: Fri, 5 Jul 2024 12:56:20 GMT
- Title: Minimal eigenvalue estimates for self-adjoint trace-class operators
- Authors: Richárd Balka, Gábor Homa, András Csordás,
- Abstract summary: Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics.
We construct a monotone increasing sequence $q_n$ which tends to the minimal eigenvalue $lambda_min$ if $O$ is not positive semidefinite, and to $0$ otherwise.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics, and arguably the most important one is being positive semidefinite. For each self-adjoint, trace-class operator $O$ we construct a monotone increasing sequence $q_n$ which tends to the minimal eigenvalue $\lambda_{\min}$ if $O$ is not positive semidefinite, and to $0$ otherwise. This sequence only depends on the moments of $O$ and a concrete upper estimate of its $1$-norm; we also demonstrate that it can be effectively calculated for a large class of physically relevant operators. As a by-product, we obtain computable estimates for the $1$-norm of $O$, too. First assume that $O$ is positive semidefinite. Unfortunately, positivity tests fail to prove this in finitely many steps. However, $q_n$ gives a rigorous, monotone increasing lower estimate for all eigenvalues, providing a quantitative way of measuring positivity. In this case the speed of convergence is $q_n\approx -\frac cn$. Now suppose that $O$ is not positive semidefinite. Then $q_n$ monotonically converges to $\lambda_{\min}$ with super-exponential speed. Hence if $q_n$ stabilizes at a negative value, we obtain a strong indication that $O$ is in fact not positive semidefinite. We also construct an easier computable sequence $q_{n,0}$ which fails to be monotone, but converges to $\lambda_{\min}<0$ faster, providing an even better indicator of non-positivity.
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