The construction of Dirac delta type potentials has been achieved with the use of the theory of self adjoint extensions of non-self adjoint formally Hermitian (symmetric) operators. The application of this formalism to investigate the possible self adjoint extensions of the one dimensional kinematic operator K = −d 2 /dx 2 on the infinite square well potential is quite illustrative and has been given elsewhere. This requires the definition and use of four independent real parameters, which relate the boundary values of the wave functions at the walls. By means of a different approach, that fixes matching conditions at the origin for the wave functions, it is possible to define a perturbation of the type aδ(x) + bδ ′ (x), thus depending on two parameters, on the infinite square well. The objective of this paper is to investigate whether these two approaches are compatible in the sense that perturbations like aδ(x) + bδ ′ (x) can be fixed and determined using the first approach.

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The quantum Rabi model accepts analytical solutions in the so-called degenerate qubit and relativistic regimes with discrete and continuous spectrum, in that order. We show that solutions are the superposition of even and odd displaced number states, in the former, and infinitely squeezed coherent states, in the latter, of the boson field correlated to the internal states of the qubit. We propose a single parameter model that interpolates between these discrete and continuous spectrum regimes to study the spectral statistics for first and second neighbor differences before the so-called spectral collapse. We find two central first neighbor differences that interweave and fluctuate keeping a constant second neighbor separation. *