By Volodymyr Koshmanenko

ISBN-10: 9401059527

ISBN-13: 9789401059527

ISBN-10: 9401146195

ISBN-13: 9789401146197

The thought of singular quadratic shape appears to be like in mathematical physics as a device for the research of formal expressions resembling perturbations without operator experience. various actual versions are in response to using Hamiltonians containing perturba tion phrases with singular houses. general examples of such expressions are Schrodin ger operators with O-potentials (-~ + advert) and Hamiltonians in quantum box concept with perturbations given by way of operators of production and annihilation (P(

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28). • We emphasize that the condition Y1 ~ 1 used in the proof of the last theorem is not essential. Thus, if y I > m 1 > - 00, then we can replace this form by the form YI = YI + (I + ImIl)X, where X [

8), the equality \jf = t is possible only in the case where \jf = t = O. Thus, the form y is correctly defined on Q( y). Further, it is obvious that Section 4 Quadratic Forms in the Theory of Self-Adjoint Extensions Q( y) is dense in Ji and y ~ O. The form y is closed. Indeed, let {un} 53 E en Ly (here, we use the same notation as in Section 2,

A quadratic form y in a Hilbert space :J-{ is called y I-bounded (YI = Y; ~ m ~ 0), whenever Q(YI) ~ Q(y) and there exist nonnegative numbers a and b such that Section 2 Closed and Closable Quadratic Forms 25 *EQ(y). 26 (KLMN-theorem; Part I). 7). , in [Katl], [ReS2], and [BeK]. We now assume that the form YI is symmetric, closed, and bounded from below in = y~ = yfl ~ m l ). 27. A symmetric quadratic form 'Y in J{ is called 'YI-closable (or closable relative to 'Y I) if it is densely defined, bounded from below, and closable in J{I. *

### Singular Quadratic Forms in Perturbation Theory by Volodymyr Koshmanenko

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