Boris N. Zakhariev

Exactly Solvable Models (Crum-Krein Transformations in the (λ2, E)-Plane)

A method is given of constructing new classes of potential V and corresponding wave functions in a closed analytical form using the known solutions to the Schrodinger equation with some potential V for energy E and angular momentum l along arbitrary lines in the (l + ½)2, E) plane. The Crum-Krein transformations which give solutions for fixed l and any E (or their modification for fixed E and different values l) are a particular case of the suggested formalism.

Instructive comparison of perturbations of potentials shifting energy levels and moving bound states in the configurational space

It is revealed that potential transformations shifting up or down energy levels of bound states have some common features with potential transformations changing space localization of the corresponding eigenstates. It is because every approach of two energy levels in the one-dimensional case requires some configurational shifts of the bound states (in the limit case of the level degeneration they must be removed to satisfy the contradictory requirement to remain orthogonal in spite of becoming more alike). Here we observe a new manifestation of the bricks of quantum design (of the elementary and universal constituents of arbitrary potential transformations).

The Principal Equations of Scattering Theory

The four chapters of Part I are devoted to problems related to the onedimensional motion of a single particle in an external field. The main concepts and equations are introduced in Chap. 1 within the framework of various formulations of the direct and inverse problems. Both the general aspects and special features of the respective formalisms are emphasized.

Multi-particle Systems

During the last twenty-five years significant progress in the three-body problem has been made which was stimulated by the appearance of the Faddeev integral equations (Belyaev 1990; Merkuryev, Faddeev 1985; Schmid, Ziegelman 1974). A similar formalism of integral equations was also being developed for more complicated systems (Komarov et al 1985; Faddeev, Merkuryev 1985). In parallel with the development of methods for solving integral equations, ex ploration of another approach to the problem for which the conventional differential form of the Schrodinger equation serves as a starting point continues. Here Faddeev’s idea concerning the separation of components of the wave function corresponding to different particle arrangements in fragments is utilized, or the Faddeev equations in the coordinate representation are solved directly (Merkuryev 1970–1980; Faddeev, Merkuryev 1985). The integral and differential approaches are to a large extent complementary, since the approximations applied in these approaches differ from each other. The Faddeev approach is not considered here. We refer the reader to Belyaev (1990) and Faddeev and Merkuryev (1985).

Multi-channel Equations

The term “channel” is used here in a narrow sense. If several one-dimensional Schrodinger equations are combined to form a single system, then each of these equations will be said to be related to one channel, while the components of the column of solutions will be called channel wave functions.1 The unified tech nique of strong coupling of channels is used for describing a large variety of physical objects. Their complicated equations of motion reduce to sets of coupled equations.

Multi-dimensional Problems

The special case of three-dimensional quantum systems with a spherically symmetric interaction V(r) = V(r) permitting separation of the variables in the Schrodinger equation when it reduces to separate equations for the partial waves was already considered in Part I. There all attention was concentrated on the aspects of radial motion in a single independent channel.

Exactly Solvable Models: Bargmann Potentials VB

The search for exact solutions to the Schrodinger equation was initiated immediately after its publication. Naturally, the first to be found were the solutions for the most simple potentials which reduce the equation of motion to differential equations for known elementary and special functions. Examples are provided by the potentials V(r) = cnrn where n = 2 represents the oscillatory, n = 1 the linear, n = 0 the constant, n = — 1 the Coulomb, and n = — 2 the centrifugal potentials (Flugge 1971).

The Levinson Theorem

In the development of any science the bulk of accumulated information gradually crystallizes into separate general laws or regularities which reflect especially deep relationships among the investigated objects. In quantum scattering theory the Levinson theorem, equally important for both the direct and inverse problems, is such a general law.1

The nonspherical-symmetric inverse scattering problem in spheroidal coordinates

Abstract It is shown that axially deformed potentials allowing the separation of variables in the Schrodinger equation in spheroidal coordinates can be reconstructed by the corresponding generalisation of the Newton-Sabatier and Hooshyar-Razavi methods. This formalism can also be extended to potentials with three-axial deformations.