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Schrödinger equation for Sturm-Liouville operator with singular propagation and potential

M. Ruzhansky, A. Yeskermessuly

Abstract

In this paper we consider an initial/boundary value problem for the Schrödinger equation with a right-hand side involving the fractional Sturm-Liouville operator with singular propagation and potential. To construct a solution, first considering the coefficients in a regular sense, the method of separation of variables is used, which leads the solution of the equation to the eigenvalue and eigenfunction problem of the Sturm-Liouville operator. Next, using the Fourier series expansion in eigenfunctions, a solution to the Schrödinger equation is constructed. Important estimates related to the Sobolev space are also obtained. In addition, the equation is studied in the case where the initial data, propagation and potential are strongly singular. For this case, the concept of\, ``very weak solutions'' is used. The existence, uniqueness, negligibility and consistency of very weak solution of the Schrödinger equation are established.

Schrödinger equation for Sturm-Liouville operator with singular propagation and potential

Abstract

In this paper we consider an initial/boundary value problem for the Schrödinger equation with a right-hand side involving the fractional Sturm-Liouville operator with singular propagation and potential. To construct a solution, first considering the coefficients in a regular sense, the method of separation of variables is used, which leads the solution of the equation to the eigenvalue and eigenfunction problem of the Sturm-Liouville operator. Next, using the Fourier series expansion in eigenfunctions, a solution to the Schrödinger equation is constructed. Important estimates related to the Sobolev space are also obtained. In addition, the equation is studied in the case where the initial data, propagation and potential are strongly singular. For this case, the concept of\, ``very weak solutions'' is used. The existence, uniqueness, negligibility and consistency of very weak solution of the Schrödinger equation are established.
Paper Structure (3 sections, 7 theorems, 146 equations)

This paper contains 3 sections, 7 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Non-homogeneous Schrödinger equation
  3. Very weak solutions

Key Result

Theorem 2.1

Assume that $q \in L^\infty(0,1)$, $q\geq0$, $a(t)\geq a_0>0$ for all $t\in [0,T],$ and $a\in L^\infty[0,T]$. For any $k\in \mathbb{R}$, if the initial condition satisfies $u_0 \in W^k_\mathcal{L}$ then the Schrödinger equation (C.p1) with the initial/boundary conditions (C.p2)-(C.p3) has a unique s When $s=1$, we also have where the constants in these inequalities are independent of $u_0$, $\nu$

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 10 more