ABSTRACT
In this chapter we discuss some simplest analytic methods of constructing one- and multi-dimensional quasi-exactly solvable models with rational or, more precisely, with quasi-polynomial potentials. We shall call a D-dimensional potential “quasi-polynomial” if it consists of two parts, one of which is an ordinary polynomial in D coordinates https://www.w3.org/1998/Math/MathML"> x i 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203741450/7518512b-63a4-45fb-bf1c-1871607b4d45/content/e0082_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , while the second is a linear combination of D singular terms https://www.w3.org/1998/Math/MathML"> x i - 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203741450/7518512b-63a4-45fb-bf1c-1871607b4d45/content/e0082_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Remember that these potentials naturally arise in many problems of quantum mechanics as a result of separation of variables in multi-dimensional toroidal coordinates. In particular, in the one-dimensional case they describe so-called radial Schrödinger equations appearing after the separation of variables in multidimensional anharmonic oscillators with spherical symmetry.
