New PDF release: Singular Quadratic Forms in Perturbation Theory

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, == Q (y )). If both {

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.

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Singular Quadratic Forms in Perturbation Theory by Volodymyr Koshmanenko

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