Quantum mechanical bootstrap on the interval: Obtaining the exact spectrum

We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>H</a:mi><a:mo>=</a:mo><a:mi>S</a:mi><a:mi>Z</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:mo>−</a:mo><a:mi>Z</a:mi><a:mo stretchy="false">)</a:mo><a:mi>S</a:mi></a:math>, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Z</e:mi></e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>S</g:mi></g:math> satisfy canonical commutation relations. While this model may appear unusual, using an appropriate coordinate transformation, the Schrödinger equation can be cast into a standard form with a Pöschl-Teller-type potential. Since the system is defined on an interval, it is well-known that <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>S</i:mi></i:math> 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.