Mathematician analyzes a quantum mechanics equation
Vladimir Peller, an invited professor from RUDN University, has worked on the Lifshits-Krein trace formula, which is used to solve important problems in quantum statistics and crystal theory. V. Peller and his foreign colleagues have presented new results of the analysis of expressions similar to the Lifshits-Krein formula in a paper published in Comptes Rendus Mathematique.
Several operations are possible with any mathematical objects (numbers, variables, functions): add, multiply, raise, integrate, differentiate, etc. These operations are set by appropriate formulas. If the transformation is applied to a number or variable, it is called a function. If it is applied to a function, it is usually called an operator. For example, the formula f (x) = 2x + 5 defines a function of the variable x, while the formula (Pf) (y) = ʃ f (x) k (x, y)) dx defines an operator integrating the product of the functions f (x) and k (x, y).
Operators are present in formulas and equations that describe physical processes. Problems with so-called "perturbed" operators often occur in quantum mechanics. There are two types of objects in such problems: an arbitrary initial operator and another one, not very different in its properties - a "perturbed" object. For example, this happens when we calculate free energy of a crystal lattice. If the substance is completely homogeneous, its energy is set by a special operator. When an impurity atom is introduced into the crystal lattice, the energy operator is modified and it becomes "perturbed".
In order to accurately solve physical equations, it is necessary to use and interpret operators correctly. One of the important indicators that characterize operator properties is the trace. The trace can be calculated in different ways, the specific formula depends on the problem under consideration. The expression for the trace of the difference of functions of the perturbed and unperturbed operators through the integral of the spectral shift function was first derived by a physicist I.M. Lifshits. Soviet mathematician M.G. Krein succeeded in generalizing and proving the equation.
Vladimir Peller, Doctor of Physical and Mathematical Sciences, invited professor of RUDN University, has solved the problem of trace evaluation for functions of special type, so called operator Lifshits functions. "We are now working with our colleagues in order to find functions and operators for which a trace formula similar to the Lifshits-Krein equation can be written. We are getting general theoretical results that will play an important role in specific examples and applications, in particular in differential equations and physics," Vladimir Peller concluded.
The study has been carried out jointly with Professor of the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Doctor of Physical and Mathematical Sciences Mark Malamud and Research Fellow at the Weierstrass Institute for Applied Analysis and Stochastics (Germany) Dr. Hagen Neidhardt.
More information:
doi.org/10.1016/j.crma.2017.06.003
Provided by RUDN University