Using extensions of the quadratic form in the potential term we construct Gaussian eigenstates of the free scalar quantum Hamiltonian operator acting in nontrivial functional space. Admissible positive closed extensions are generated at least by two external sources, the distance between them being limited by the extension parameter. The constructed functionals, in contrast to the ground state of the free theory, correspond to different boundary conditions and can be interpreted as eigenstates of some renormalized asymptotically free quantum Hamiltonian.
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T. A. Bolokhov (2024) studied this question.
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