We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic construction of the self-adjoint extensions of the minimal operator obtained by restricting the free Hamiltonian to wave-functions that vanish in the vicinity of the coincidence hyperplanes: all extensions thus model an interaction precisely supported at the spatial configurations where particles come on top of each other. Among them, we select the physically relevant ones, by implementing in the operator construction the presence of the specific short-scale structure suggested by formal physical arguments that are ubiquitous in the physical literature on zero-range methods. This is done by applying at different stages the self-adjoint extension schemes à la Kreĭn–Višik–Birman and à la von Neumann. We produce a class of canonical models for which we also analyze the structure of the negative bound states. Bosonicity and zero range combined together make such canonical models display the typical Thomas and Efimov spectra, i.e. sequence of energy eigenvalues accumulating to both minus infinity and zero. We also discuss a type of regularization that prevents such spectral instability while retaining an effective short-scale pattern. Besides the operator qualification, we also present the associated energy quadratic forms. We structured our analysis so as to clarify certain steps of the operator-theoretic construction that are notoriously subtle for the correct identification of a domain of self-adjointness.

中文翻译：

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玻色子三聚体的零范围相互作用模型

我们提出了三体系统的物理相关量子哈密顿量的数学构造，该系统由相同的玻色子组成，这些玻色子通过零范围的两体相互作用相互耦合。对于演示文稿的大部分内容，将考虑无限散射长度（统一方案）。该主题在数学文献中有几个先驱。我们通过将自由哈密顿量限制为在重合超平面附近消失的波函数而获得的最小算子的自伴随扩展的算子理论构造：因此，所有扩展都模拟了在空间配置中精确支持的相互作用粒子相互重叠的地方。其中，我们选择物理相关的，通过在算子构造中实现特定的短尺度结构的存在，这些结构由正式的物理参数所暗示，这些参数在零范围方法的物理文献中无处不在。这是通过在不同阶段应用自伴随扩展方案 à la Kreĭn–Višik–Birman 和à la von Neumann 来完成的。我们产生了一类规范模型，我们还分析了负束缚态的结构。玻色子性和零范围结合在一起使得这些规范模型显示了典型的 Thomas 和 Efimov 谱，即能量特征值序列累积到负无穷和零。我们还讨论了一种正则化，它可以防止这种光谱不稳定性，同时保持有效的短尺度模式。除了操作员资格，我们还提出了相关的能量二次形式。我们构建了我们的分析，以澄清算子理论构造的某些步骤，这些步骤对于正确识别自伴随域是出了名的微妙。