A Necessary and Sufficient Discreteness Condition for the Spectrum of a two Term Differential Operator of Higher Order

Erich Müller-Pfeiffer, St. Werler
1985 Zeitschrift für Analysis und ihre Anwendungen  
Tm gewichteten Hilbertraum L2W(01 o) werden seibstadjungierte Diffcrentialoperatoren der Ordnung 2n betrachtet, die von dem Differentialausdruck Ay = . w'[(-l)" (py(fl))(fl) + qy] erzeugt werden. Unter Verwendung von Bedingungen für die Diskrctheit des Spektrums solcher Operatoren, wie sic von Kwong und Zetti angegeben worden sind, wird eine notwendige und hinreichende Bedingung für die Diskrethcit des Spektrums abgeleitet. B BecosoM rsLm6epT0u0M flOCTHCTBC L2W(01 oo) paccMaTpunaloTcn
more » ... punaloTcn caMoconpnxeHHb!e onepa'ropw nopua 2n, flOPOHWHHbIC All jaepeH[uaimIlhIM aupaeiiueM 4y (py(fl))(fl) + qy ]. flpuMeIInH ycioinui uii IrcHpeTIioc'ru dneKTpa, gallilbie Hnoiii'ori ii IeT-Te.rlbeM, BMBOIJ1TC1I Heo6xoLHMoe H JocTaro4IIoe yCJIOBHe JJII JiHCkpeTnOCTu cneicrpa. Self-adjoint differential operators of order 2n are considered that are associated with the expression Ay = w'[(-l)" (py(fl))(fl) + qy] and the weighted Hubert space L2 (O, oo). By use of discreteness conditions for the spectrum of such operators given by Kwong and Zetti a necessary and sufficient condition for the discreteness of the spectrum is established. Consider the differential expression ay = w1[(-1)' (pyü+ qy], 0 x < 00, where the weight functionw and the coefficients p and q are real-valued and w>0, wEC, pEW25 (0,X), qEL2(0,X) • for all X>,0. . The expression 4 determines the symmetric operator A 0 , -A 09, 4T, 99 ED(A 0 )=C0 (O,00), S in the weighted Hilbert space L2;,0(0, ) of all complex-valued measurable functions / satisfying j/,) =f /1 2w dx < + 00. It is known that all self-adjoint extensions A of A 0 have the same essential spectrum F(A). In the following we are interested in establishing a necessary and sufficient condition for the ease a(A) = 0, i.e., for the discreteness of the spectrum of A. In the special case where w 1, p = 1, and q 0, by a theorem of A. M. MOLRANOV [8] the spectrum is discrete if and only if x+h liin fq(t)dt=oo X-X for each h > 0. This theorem has been generalized in different directions by several authors. Relating to this we refer to [1, 5] (the case n = 1, w = 1, p = 1), [9]
doi:10.4171/zaa/150 fatcat:3rdwv7wrtjbazpjfury3jculfu