Abstract
The purpose of this study is to investigate a new class of boundary value transmission problems (BVTPs) for a Sturm-Liouville equation on two separate intervals. We introduce a modified inner product in the direct sum space $L_{2}[a,c)\oplus L_{2}(c,b]\oplus C^{2}$ and define a symmetric linear operator in it in such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then, by suggesting own approaches, we construct the Green’s function for the BVTP under consideration and find the resolvent function for the corresponding inhomogeneous problem.
Highlights
Many interesting applications of Sturm-Liouville theory arise in quantum mechanics
Looking for separable solutions ψ(x, t) = φ(x)e–iEt/, we find that φ(x) satisfies the differential equation
The coefficient q is proportional to the potential V, and the eigenvalue parameter λ is proportional to the energy E
Summary
Many interesting applications of Sturm-Liouville theory arise in quantum mechanics. For instance, for a single quantum-mechanical particle of mass m moving in one space dimension in a potential V (x), the time-dependent Schrödinger equation is i ψt = – m ψxx + V (x)ψ.Looking for separable solutions ψ(x, t) = φ(x)e–iEt/ , we find that φ(x) satisfies the differential equation– φ + V (x)φ = Eφ. mThat is a Sturm-Liouville equation of the form –y + qy = λy.The coefficient q is proportional to the potential V , and the eigenvalue parameter λ is proportional to the energy E. Boundary value problems can be investigated through the methods of Green’s function and eigenfunction expansion. Green’s functions can often be found in an explicit way, and in these cases it is very efficient to solve the problem in this way.
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