Specific quantum mechanical solution of difference equation of hyperbolic type

• Difficulties in solving difference equation of hyperbolic type was analyzed and discussed.
• Results are compared to those of standard wave equation and same similarities.
• Method of equation solving is generalized: kernel expanded into polynomials.
• This analysis is caused by some new ideas concerning quantization of time.
• Examples are given: some specificity of excitons and phonons in thin crystalline films are obtained.

Difficulties connected to solving difference equations of hyperbolic type were analyzed in this work and discussed in detail. The results are compared to those of the standard wave equation and certain similarities were established. The method of solving the equation is generalized by means of kernel expanded into separable polynomials. The analysis was inspired by some new ideas concerning quantization of time. Two examples are given: excitons and phonons in thin crystalline films. The advanced methodology of Green’s function method and the application of this new methodology resulted in a set of interesting conclusions concerning thin film properties. The significance of the obtained spatial dependence of exciton concentration was discussed and it was concluded, on the basis of the found spatial dependence of exciton concentration, that such boundary conditions of a thin molecular film which lead to high exciton concentrations can be determined. It was also concluded that thin films possess high superconductive properties, that physical characteristics of thin films are spatially dependent and that the spatial dependence can be the basis for widening the field of application of nanostructures.