Well, *they are all infinite*, since free scalar field theory is simply a compactly repackaged array of quantum harmonic oscillators. You are simply supposed to follow the systematic translation rules carefully, e.g. following [Jackiw 1988](https://books.google.com/books?hl=en&lr=&id=0m7mJEruqiUC&oi=fnd&pg=PA107&dq=R.+Jackiw,+%22Schrodinger+Picture+for+Boson+and+Fermion+Quantum+Field+Theories.%22+&ots=JzjfA9I3Bf&sig=Y1aZzH0fqKSGScfGyI94SFykFvI#v=onepage&q=2.16%20%20Schrodinger%20picture%20&f=false) or [Lüscher 1985](https://www.sciencedirect.com/science/article/pii/055032138590210X?via%3Dihub).

As for one oscillator, the energy eigen-functionals are Gaussian in 
form. I'll give you a jump-start:  the ground state eigen-functional $\langle\Phi|0\rangle$ is just
\begin{equation}
\Psi[ \Phi] =\exp\left(  -\frac{1}{2}\int d^3 x\,\Phi\left(
x\right)  \sqrt{m^{2}-\nabla_{x}^{2}}~ \Phi\left(  x\right)  \right).
\end{equation}
Boundary conditions are assumed such that the $\sqrt{m^{2}-\nabla_{x}^{2}}$
kernel in the exponent is naively self-adjoint. 
"Integrating by parts" one of the $\sqrt{m^{2}-\nabla_{z}^{2}}$ kernels,
functional derivation $\delta\Phi\left(  x\right)  /\delta\Phi\left(  z\right)=\delta^3\left(  z-x\right) $   leads to  
\begin{equation}  
 \frac{\delta}{\delta\Phi\left(  z\right)  }\Psi [\Phi] 
=-\left(  \sqrt{m^{2}-\nabla_{z}^{2}}\,\Phi\left(  z\right)  \right)
\Psi[ \Phi] ,
\end{equation} 
so that
 $$  
\frac{\delta^{2}\qquad }{\delta\Phi\left( w \right) \delta\Phi
\left( z\right)  }\Psi[ \Phi]  
=\left(  \sqrt{m^{2}-\nabla_{w}^{2}}\,
\Phi\left(  w\right)  \right)  \left(  \sqrt{m^{2}-\nabla_{z}^{2}}\,
\Phi\left(  z\right)  \right)  \,\Psi[ \Phi]\\ - \sqrt
{m^{2}-\nabla_{z}^{2}}\,\delta^3\left(  w-z\right)  \,\Psi[ \Phi]~. 
$$

Note that the divergent zero-point energy density, 
\begin{equation}  
E_0= { 1\over 2  }    \lim_{w\rightarrow z} 
\sqrt {m^{2}-\nabla_{z}^{2}}~ \delta^3\left(  w-z\right), 
\end{equation} 
may be handled rigorously using $\zeta$-function regularization. It is but the sum of all zero-point energies of the infinity of oscillators. The price of operator-valued distributions.

Leaving this zero-point energy present leads to the 
standard energy eigenvalue equation, again through integration by parts, 
\begin{equation}  
\frac{1}{2} \int d^3 z\,\left(  - \frac{\delta^{2}}
{\delta\Phi\left(  z\right)  ^{2} }+\Phi\left(  z\right)  
\left(  m^{2}-\nabla_{z}^{2}\right)  \Phi\left( z\right)-2E_0  \right)    
  ~\Psi[ \Phi]=0 ~, 
\end{equation} 
that is the lowest eigenvalue of *H* is $\int d^3 x ~E_0$. 

You might proceed to form functional ladder operators, etc... and pursue the finer aspects of [Schrödinger functional theory](https://en.wikipedia.org/wiki/Schrödinger_functional) to the bitter end...