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We consider the form of eigenfunction expansions associated with the time-independent Schrödinger operator on the line, under the assumption that the limit point case holds at both of the infinite endpoints. It is well known that in this situation the multiplicity of the operator may be one or two, depending on properties of the potential function. Moreover, for values of the spectral parameter in the upper half complex plane, there exist Weyl solutions associated with the restrictions of the operator to the negative and positive half-lines respectively, together with corresponding Titchmarsh-Weyl functions.
Gilbert, D. (2013) Eigenfunction Expansions Associated with the One-Dimensional Schrödinger Operator, Operator Methods in Mathematical Physics, Vol. 227, pp. 89-105. DOI 10.1007/978-3-0348-0531-5_4