On the Spectrum of the Two-Particle Schrödinger Operator With Point Potential: One Dimentional Case

Abstract

In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

“…Differential equations of mathematical physics have direct applications in the theory of nanosystems (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13], and [14]). Partial differential and integro-differential equations of parabolic type with initial and boundary conditions are investigated widely by large number of scientists and have different applications in sciences and technology (see, for example, [15][16][17][18][19][20][21][22][23][24][25][26][27][28]).…”