NOTE ON VERMA BASES FOR REPRESENTATIONS OF SIMPLE LIE ALGEBRAS

We discuss the construction of the Verma basis of the enveloping algebra and of finite dimensional representations of the Lie algebra An. We give an alternate proof of so-called Verma inequalities to the one given in [1] by P. Littelmann

New method for computation of discrete spectrum of radical Schrödinger operator

summary:A new method for computation of eigenvalues of the radial Schrödinger operator $-d^2/dx^2+v(x), x\geq 0$ is presented. The potential $v(x)$ is assumed to behave as $x^{-2+\epsilon}$ if $x\rightarrow 0_+$ and as $x^{-2-\epsilon}$ if $x\rightarrow +\infty, \epsilon \geq 0$. The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function $z(x,\aleph)$. It is shown that the eigenvalues are the discontinuity points of the function $z(\infty, \aleph)$. Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison with other known methods used in numerical computations on computers

Hilbert Space Operators in Quantum Physics

The second edition of this course-tested book provides a detailed and in-depth discussion of the foundations of quantum theory as well as its applications to various systems. The exposition is self-contained; in the first part the reader finds the mathematical background in chapters about functional analysis, operators on Hilbert spaces and their spectral theory, as well as operator sets and algebras. This material is used in the second part to a systematic explanation of the foundations, in particular, states and observables, properties of canonical variables, time evolution, symmetries and various axiomatic approaches. In the third part, specific physical systems and situations are discussed. Two chapters analyze Schrödinger operators and scattering, two others added in the second edition are devoted to new important topics, quantum waveguides and quantum graphs. Some praise for the previous edition: "I really enjoyed reading this work. It is very well written, by three real experts in the field. It stands quite alone...." John R. Taylor, Professor of Physics and Presidential Teaching Scholar, University of Colorado at Boulde