Hamiltonian formulation of the ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> )-dimensional <mml:math xmlns:mml="http://www.w3.o

We apply the wavelet formalism of quantum field theory to investigate nonperturbative dynamics within the Hamiltonian framework. In particular, we employ Daubechies wavelets in momentum space, whose basis functions are labeled by resolution and translation indices, providing a natural nonperturbative truncation of both infrared and ultraviolet truncation of quantum field theories. As an application, we compute the energy spectra of a free scalar field theory and the interacting <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>1</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:math> -dimensional <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>ϕ</c:mi> <c:mn>4</c:mn> </c:msup> </c:math> theory. This approach successfully reproduces the well-known strong-coupling phase transition in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>m</e:mi> <e:mn>2</e:mn> </e:msup> <e:mo>&gt;</e:mo> <e:mn>0</e:mn> </e:math> regime. Our results indicate that with increasing resolution, the extracted critical coupling can approach toward its expected value, providing numerical evidence for the plausibility of the wavelet-based Hamiltonian formulation for nonperturbative field-theoretic calculations.