Completeness of stationary scattering states. II

✍ Scribed by N.G. Van Kampen

• Book ID: 104162272
• Publisher: Elsevier Science
• Year: 1955
• Weight: 496 KB
• Volume: 21
• Category: Article
• ISSN: 0031-8914
• DOI: 10.1016/s0031-8914(55)90413-4

✦ Synopsis

This result could also be obtained by using in the work of the previous section other solutions of {25) than ( ). The special solution ( ) is uniquely characterized by the property that S0{x)/0_(x) belongs to ~_. Indeed, a solution ho_(x) of the homogeneous equation cannot have this same property, for, on account of {32), he_(X) would then belong to both ~_ and ~+.

only the zero nearest to the real axis satisfies requirement (i), provided that no other zeros or poles are in the neighbourhood.)

From our present point of view these purely imaginary Av are not materially different form the other A,. Hence this definition is subject to the same limitations as the definition of resonance levels and excited states. Virtual levels can only be rigorously defined inside the neighbourhood of the real axis in which S has an analytic continuation. They can be approximately defined if S can be decomposed into an analytic S o and a sufficiently smoothly varying S 1.