to check the additional perturbation Hamiltonian $H_{Int}$

$$\langle \Psi | H_{Int} | \Psi \rangle = \int\limits_{-\infty}^{\infty}\psi^{\dagger}\gamma^0 (-i\gamma^0\left(\frac{2M}{r}\right)^\frac{1}{2}(\partial_r+\frac{3}{4r}))\psi\ d^3r
\label{83}$$

$$\langle \Psi | H_{Int} | \Psi \rangle^{*} = \int\limits_{-\infty}^{\infty}\psi^{T}\gamma^0 (i\gamma^0\left(\frac{2M}{r}\right)^\frac{1}{2}(\partial_r+\frac{3}{4r}))\psi^{*}\ d^3r.
\label{84}$$

If we integrate the derivative term by parts and use the condition that the norm of the wave function vanishes at the boundary we find the expression

$$\begin{aligned}
\langle \Psi | H_{Int} | \Psi \rangle^{*} &=& 4\pi \int\limits_{0}^{\infty}\left(\left(\partial_r\psi^{T}\right)
\left(-i\left(\frac{2M}{r}\right)^\frac{1}{2}\right)\psi^{*}+\psi^{T}\left(i\left(\frac{2M}{r}\right)^\frac{1}{2}\frac{3}{4r}\right)\psi^{*}\right)r^2\ dr\nonumber\\
&&+4\pi\ \psi^{T} \left(i\left(2Mr^3\right)^\frac{1}{2}\right)\psi^{*} \mid_{r=0}+4\pi \int\limits_{0}^{\infty}\psi^{T}\left(-i\partial_r \left(2Mr^3\right)^\frac{1}{2}\right)\psi^{*}\ dr.\nonumber\\
\label{85}\end{aligned}$$

Here we have used that $(\gamma_0){}^2=1$. If we now use that $\phi^{T}\psi=\psi^{T}\phi$ we obtain

$$\begin{aligned}
\langle \Psi | H_{Int} | \Psi \rangle^{*} &=& \int\limits_{-\infty}^{\infty}\psi^{\dagger}\gamma^0
\left(-i\gamma^0\left(\frac{2M}{r}\right)^\frac{1}{2}\left(\partial_r+\frac{3}{4r}\right)\right)\psi\ d^3r\nonumber\\ 
&&+4\pi \int\limits_{0}^{\infty}\psi^{\dagger}\gamma^0 \left(-i\gamma^0\left(\partial_r \left(2Mr^3\right)^\frac{1}{2}-\left(2Mr^3\right)^\frac{1}{2}\frac{3}{2r}\right)\right)\psi\ dr \nonumber\\
&&+4\pi\ \psi^{\dagger} \left(i\left(2Mr^3\right)^\frac{1}{2}\right)\psi \mid_{r=0} \nonumber\\
&=&\langle \Psi | H_{Int} | \Psi \rangle+i\int\limits_{-\infty}^{\infty}\delta(r)\psi^{\dagger}\left(2Mr\right)^\frac{1}{2}\psi\ d^3 r.
\label{86}\end{aligned}$$

So we have $H^\dagger \neq H$ and the energy eigenvalues need not to be real and so we have to extend our search to complex numbers.

However if we have found the eigenstate we are mainly interested in the real part in the context of this article. The real part of the energy has to be smaller than the rest energy of the particle to provide a (if we abstain from instability) stationary state. 