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Báo cáo toán học: "Gauge invariance of Schroedinger operators and related spectral properties "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Máy đo bất biến của các nhà khai thác Schroedinger và các tính chất quang phổ có liên quan. | J. OPERATOR THEORY 9 1983 163-179 Copyright by INCREST 1983 GAUGE INVARIANCE OF SCHRODINGER OPERATORS AND RELATED SPECTRAL PROPERTIES HERBERT LEINFELDER INTRODUCTION In this paper we consider Schrodinger operators _ 1 t -V 3 V V _ 31 7 i A 2 ia V iV fl a2 V where a V Rm - R xR is a measurable function and look for a most general condition on a V so that the essential spectrum of 3 a V coincides with the interval 0 oo . We set the spaces of potentials jf15jf2 2 and as follows Jf1 fl P SeZ oc PeZf0C 2 a P I ãe zfoc V flG zfoe Pg ZjU JS P I o V G the form JỂ fl V is closable on cs 5 2 a V I a P g Jf2 C a V is essentially self-adjoint on Co0 1 2 P I ã V sl i v_ is A-formbounded with relative bound less than 1 2 P I fl V G x2 V. is A-bounded with relative bound less than 1 . Here p_ denotes the nonnegative function max V 0 . We note that Xi respectively Jf2 is the largest possible set of potentials fl P such that the quadratic form fl P respectively the operator 3 a VJ can be defined on C that In this note a closable form is assumed to be bounded from below. 164 HERBERT LEINFELDER Jỉị c - ị c 3g J see 2 22 14 and if a F e there corresponds a unique self-adjoint operator H a V which we abbreviate simply by H a if V - 0. The purpose of this note is two-fold. On one hand we prove that any self--adjoint Schrodinger operator Hịấ V is invariant under gauge transformations 2 - fl V2. On the other hand we apply the freedom of gauge transformations to the investigation of the essential spectrum of H a V and show that ơ. s 77 2 Ụ 0 oo under very mild conditions on the potentials 2 V . We show about the gauge transformation that 1 is closed with respect to gauge transformations among xjs i.e. if 2 V G jSfj ồ V 6 and b a VẨ with some Ấ G Qi then b V e and the Schrổdinger operators H a V H b V are unitarily equivalent ẽlH ã K e-u - H b V . 2 On each 2 K e ĩ we may impose the so called Coulomb gauge condition V-2--0 i.e. for any 2 Vje.S i we can find a gauge transformation z G g .