https://www.chenk.top/en/tags/unbounded-operators/Recent content in Unbounded-Operators on Chen Kai BlogHugoenSun, 17 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/09-unbounded-operators/Sun, 17 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/09-unbounded-operators/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/functional-analysis/figures/fig09_deficiency_indices.png" alt="Deficiency indices determine self-adjoint extensions" loading="lazy" decoding="async" class="content-image"> </figure> </p> <p>Two articles ago I was talking about how spectral theory is the linear-algebraic infrastructure of quantum mechanics. The trouble is that nearly every operator a physicist actually cares about &ndash; the position operator, the momentum operator, the Laplacian, the Schrodinger Hamiltonian &ndash; is <em>not bounded</em>. They are not defined on the whole Hilbert space. They are densely defined, with domains that depend on the regularity or decay of the input function. None of the previous spectral apparatus applies directly. We need to extend it.</p>