Quantum mechanical bootstrap on the interval: obtaining the exact spectrum

• URL: http://arxiv.org/abs/2402.03434v1
• Date: Mon, 5 Feb 2024 19:00:02 GMT
• Title: Quantum mechanical bootstrap on the interval: obtaining the exact spectrum
• Authors: Lewis Sword, David Vegh
• Abstract summary: We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian $H=SZ (1-Z)S$, where $Z$ and $S$ satisfy canonical commutation relations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian $H=SZ(1-Z)S$, where $Z$ and $S$ satisfy canonical commutation relations. While this model may appear unusual, using an appropriate coordinate transformation, the Schr\"odinger equation can be cast into a standard form with a P\"oschl-Teller-type potential. Since the system is defined on an interval, it is well-known that $S$ is not self-adjoint. Nevertheless, the bootstrap method can still be implemented, producing an infinite set of positivity constraints. Using a certain operator ordering, the energy eigenvalues are only constrained into bands. With an alternative ordering, however, we find that a finite number of constraints is sufficient to fix the low-lying energy levels exactly.

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