Anton Kutsenko

"Operators in the 'Quantum Cubic World' or in the 'Minecraft World'"

 Date: Thu, November 7, 2019 Time: 15:00 Place: Research I Seminar Room

Abstract: "Minecraft World" (MW) consists of a finite number of disjoint cubes of the same size. The original (Lego?) computer interpretation of this world can be found in https://www.minecraft.net/. Sometimes, this is a good approximation of the real world, especially when the size of the cubes tends to $$0$$. What operators are possible in the "Minecraft World"? These are averaging (integral) operators along the edges of the cubes, and operators that move values from one cube to another. Usually, various combinations of such elementary operators cover most of the needs of practical applications.

The typical problems for these operators are as follows:

• How to find the spectrum of the operators?
• How to find unknown $$u$$ in the operator problem $$Au=f$$, or how to find the inverse $$A^{-1}$$?
• How to find the square root of $$A$$, or, generally, how to construct the functional calculus on the algebra of operators in MW? (Yes, obviously, the algebra contains all the possible combinations of elementary operators in MW.)
In order to answer these questions, we need to have a good representation of the algebra $$\mathscr{A}$$ generated by the operators acting on MW. One of the results we would like to present is that the $$C^*$$-algebra $$\mathscr{A}$$ is isomorphic to the direct product of simple matrix algebras ${\mathscr A}\cong(\mathbb{C}^{p\times p})^{2^N},$ where $$p$$ is the number of cubes in MW and $$N$$ is the dimension (number of non-parallel edges of the cube) of MW. The isomorphism has a non-trivial but explicit form. Thus, the operator problems in the "Minecraft World" are reduced to the matrix problems. Note, that most of the operators, including some of integral operators, are non-compact, but the algebra $${\mathscr A}$$ generated by them is finite-dimensional. As an application, we find explicitly the wave function for 3D Schrödinger operator acting on the infinite medium with planar, guided, local potential defects, and point sources. We also apply the results to construct the functional calculus on the algebra of extended Fredholm integral operators. Finally, we discuss the problems of approximation of continuous integral kernels by piecewise constant ones. Maybe, we provide a categorial classification (don't be scared:)))) of the direct limit of $$C^*$$-algebras of operators in MW when the cubic size tends to $$0$$. Do you remember what are Glimm-Bratteli symbols for AF-algebras?

Some of the results are recently published in Kutsenko, A. A. (2020). Matrix representations of multidimensional integral and ergodic operators, Appl. Math. Comput. 369, 124818.