We consider isospectral perturbations of the Dirichlet problem for the Poisson equations in the unit square. Spectral properties of such problems are studied. We prove that eigenfunctions of the perturbed problem form a Riesz basis. Conditions of existence and uniqueness of the solution are established. Similar problems for ordinary differentional equations and operator-differentional equations was studied in [1,2,3].

[1] Baranetskyi Ya.O., Yarka U.B. On a class of boundary value problems for differential operator equations of even order // Mat. Methods and Phys.-Mechanical Fields. - 1999. - 42, N 4. - P.1-6.

[2] Baranetskyi Ya.O., Kalenyuk P.I., Yarka U.B. Perturbations of boundary value problems for ordinary differential equations of the second order / / Visn. State University "Lviv Polytechnic". - 1998. - N 337. - P.70-73.

[3] Yarka U.B. Spectral properties of a boundary value problem for an abstract differential equation // Visn. Lviv. University. Ser. Mech.-Mat. - 2000. - Issue 56. - P.185-192.

[4] Kalenyuk P.I., Baranetsky Ya.E., Nitrebich Z.N. Generalized method of separation of variables. - K.: Nauk. Dumka, 1993. - 229 p.