Geodesic
Regularization of the equations of the quantum theory for a scalar neutral field with selfinteraction in the case of two spatial degrees of freedom
Teoretičeskaâ i matematičeskaâ fizika, Tome 9 (1971) no. 3, pp. 355-364 
L. G. Zastavenko. Regularization of the equations of the quantum theory for a scalar neutral field with selfinteraction in the case of two spatial degrees of freedom. Teoretičeskaâ i matematičeskaâ fizika, Tome 9 (1971) no. 3, pp. 355-364. http://geodesic.mathdoc.fr/item/TMF_1971_9_3_a5/
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     title = {Regularization of the equations of the quantum theory for a~scalar neutral field with selfinteraction in the case of two spatial degrees of freedom},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The method developed earlier [1] is applied to the more complicated case of two spatial coordinates. The equations for the ground and excited states are of the same form as in the one-dimensional case, but the integrations are substituted by double ones. The divergencies more complicated than those in the one-dimensional case arise. However they can be avoided by means of extracting a certain special factor from the eigen-functional $\Omega$ of the energy operator $H$.

[1] L. G. Zastavenko, TMF, 7 (1971), 20

[2] N. N. Bogolyubov, D. V. Shirkov, Vvedenie v teoriyu kvantovannykh polei, Gostekhizdat, 1957 | MR