SCIENTIFIC ABSTRACT LEVITAN, B.M. - LEVITAN, KH.N.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000929620010-2
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: July 12, 2001
• Sequence Number: 10
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

SUBJECT USSR/MATMUTICS/Yourier series CARD 1/2 PG - 449
AUTHOR LEVITAN B.M.
TITLE of the spectral function of the Laplace
operator.
PERIODICAL Mat. Sbornik, no Ser- 32j. 37-50 0956)
reviewed 12/1956
Let D be a finite simply connected domain of the n-dimensional Euclidean
space EX, lot B be the boundary of D. Let 2 P 29 ... 2,.... be the eigen-
P1 r2 rn
values and C01(3c),P 2(3c) j .... point of Z,) be the corresponding
eigenfunctions of the problem Au + 2u 0 (A - ~2 ~2
-aX2 + 'a x2
1 N
u 0. The author has obtained the asymptotic formula
n N
(1) Q(Xtyl 1 r) + 0(ik N-1
N
(2 1r)"Z r z

-Mat., Sbornik, n. Ser..U.L 37-50 (1956)
(Mat.Sbornik, n. Ser. 31L 267-316(1954)) where
CJLRD 2/2 PG - 449
Q(xl-vl 0-2: Con(x) Wn(y)
No), o(x,ylp) - - o0c,yl-,P) (r1 6, Col. 2 r.)
4 r
lo%ril4ing t.,.-; !he the Jlr,tr t. tht~orem.; the. c~)nv~!rov
F-, p r r) h 2 ,.m j ,i z i e v s-, d L t .,,3) be a% c~ompatible s-fotem,
a lirl!'"Jue 41,01uti in f,-,r cerl-Iin ,nitial conditions; under which
,.,,!J, t4ori;A! w.' 11 thit; so!,utir)n be an opperator of' uniform
tre, A number of qucstion3 in ~;onnlDr-
i, 1 4, 1 v p;) -A there are
I
Ink*rodu~ucj certain jj.-Lrators. correspovr.,,Jnc, to the
~.er.:;ru fjrdc~r in the il~;e of L:ie Croul, (c jp~, r
3 P n t r ~' il I I + r 0 1 5 Owo example,3 are considered.
-~i,;,,rv rir~, ..,c,vivt-bA,),c r.-nd 1 v~,,.)r.-5ovipt-bloc reference.
M"ar-A, 10.

C111YC444
AUTHORt Levitan, B. 9
TITLE: On a theorem of Titchmarsh and Sears
PERIODICALs Uspekhi matematicheakikh nauk, v. 16, no. 4, 1961,
.175-178
TFATt In the whole space R n the Schrddinger operator
Lu - - A u + q (xI1'-txn) u (1)
be considered, where q (xI9-#00xn) is real and continuous. Lot ()(Xfy9A
be the spectral function of (1) and R(xjyjz) the corresponding resolvent
dA-0, (x,y;A ) I----
R(x.ylz) j - z (2)
OD A
E. C. Titchmarsh (Ref. It On the uniqueness of the Green's function
associated with a second-order differential equation, Canad. J. Math.
1 (1949), 191-198) has showns if
Card 1/3
3/04 61/016/004/002/005

S/04 61/016/004/002/005
On a theorem of Titchmarsh and Sears C111YC444
q(x) >., Ar2 B (3)
where x E R n, r 1XI A,B are positive constants, then (1) possesses
a unique resolvent.
D, B. Sears (Ref. 21 Note on the uniqueness of Green's functions
associated with certain differential equations, Canad. Math. 2 (1950),
314-325) Improved this result by pointing out that the right hand of
(5) may be replaced by a function - Q(r) which has to satisfy certain
demunds.
The author gives a now proof of the mentioned results of [Ref.1,2] .
The proof is based on the estimation of the order of increase of the
solution of the Cauchy problem
2
,Au - q(x) u u (5)
j)t2 0
U U1 0 (6)
It-0 f(x), L
at t-O
Card 2/3

S/042/61/016/004/002/005
On a theorem of Titchmarsh and Sears C111/C444
where f (x) (--- L2(R ), for a fixed x and t--~o cD , and on the theorem of
uniqueness from B. M. Levitan, N. N. Meyman (Ref. 3t 0 teoreme yedinst-
vennostl fon the theorem of uniquenous3 DAN 81. no. 5 (1951),729-731).
There are 2 Soviet-bloc and 2 noti-Soviet-bloc references. The two
refeiences to English-language publication read as follows: E. C.
T!*rhmarsh. On the uniqueness of the Green's function associated
witb a second-ordor differential equation, Canad. J. Math. 1 (1949),
191-198~ D. B. Sears, Note on the uniqueness of Green's functions
associated with certain differential equations, Canad, Math. 2 (19JO),
314-525.
SUBMITTED; January 4, 1960
Card 5/5

ISVITAN# Boris_M9ineyeviob; MOLYANSKIY, M.L.,, red.; TMIAKOVAv Te.A.,
--tAld
in* red,
(Generalized displacive operators and sow of their applications]
Operatory obobsbohennogo adviga i nekotorye ikh primeDeniia. Mo-
skva, Goa, izd-vo fisiko-matem.lit-ry, 1962. 323 pe
(MIRA 15:5)
(Operators (Mathematics))

DDIIDOVICH, Boris Pavlovich; Inaak Abramovich; SOUVALOVA,
F.mma Zinov'yeva; LEVITk4, B.F.,, prof., retsenzent;
SVOLITSKIY, Kh.L. --pfof`., retsenzent; BMUK, G.I., red.;
AXIaAMOV, S.N.j i:khn. red.
[Numerical -methods of analysis; approximation of functions#
differential equations] Chielennye metody analiza; priblizhe-
nie fur&.tsiip differentsialfrqo uravneniia. Pod red. D.P.
Demidovicha. Moskva, Gos. izd-vo fiziko-matem. lit-ry,
1962. 367 (IMA .15:4)
(Functions (Differential equations)

IEVITAN, B.M.
Lie's theorems for generalized dieplacive operators*
Trudy Moak. mat. ob-va 11:128-19/7 162. (MIRA 15:10)
(Operators (Mathematics))

402(33
~;/020/62/146/oOl/col /rl 6
B1 121B108
j-: u'. 3 Levitan, B. L%
Continuation of solutions to partial differential nqudtilons
T D 1 1, j iikadusmiya nauk Z35R. Doklady, v. 146, no. 1, 1962, ZO
TLO elliptic equation a(X,Y)a 2ulay 2 + b(x9y)30Y * c(x,;r)u
++4(x)32u/,r)x2 + i~ (X au/3x + f (x,y) - 0 (8) is considered in a convex donaii.
D of th,-t upper sem1plane, -Mch contains an interval ~r of the x-axis. I,
i% de.:,onstr--ted that cach colution satisfying the boundary condition
(au/3y - hu): - 0 can be continued throughout the intervalCr if th,
1y"0
equation (0) has Linalytic coefficients. The continuation iv perfoi-,I.ed ',I
meun~i of transforryation operators. Thorv is 1 figure.
PH.-J:~-11-TA ---:, iApril~~2, 1962, by I. G. Petrovskiy, Academician
i i
13'4~ I T Tb iAarch,279 1962
Card 1/1

GUTER, le.S.; KUDDII-117UNY L.D.; LLVITAII", b.Y.; ULIYANOV, P.L.,
red.; LYUSTEPXIK, L.A., red.; YAIIFOLISKIY, A.R., red.;
GAFOSHKIV, V.F., rod.; KUYLOVA, A.11., red.; PIAKIM,
L.Yu., telJm. red.
[Elements of the theory of functions; functions of real
variables afproximatior. of functions; alrost periodic
functionsi Elerenty teorii funktsii; funktsii deistvitell-
nogo pererennoCo, priblizi-enic funktnii, pochti-periodi-
chookie funktsii. Moskva, Fizmatgiz,, 196~. 244 p,
(MI:(A 16:12)
(Functions)

WT(d)/nC(W)/J3rS AFFTC ZJP(C)
AcassION M AP3000733 s/oo20/63/150/00*474/076
AUTHORt Levitanj.-Kt ~Ij
TITLE: Determirwtion of a Stum_U.*uvUl~_4ff*rentU1 2glatlon over tv* spectra
SOURM AN SSSR. Voklady, v. 150, no. 3, 1963. 474-476
TOPIC TAGSi Sturm-Liouville differentlAl equation
ABSTRACT: The problem of constructing a Sturm-Liouville differential equation over
two spectra was engaged by M. 0. Krayn (DAN, 76. 345, 1951). In the present work
the author gives a different solution to this problem. This method allows him to
state necessary and sufficient conditions for two sequences of real numbers to be
two spectra over a Sturn-Liowii1le differential equation. Orig. art. has: 7 equa-
tions.
ASSOCIATION: none

-LI51
,~7243 EW-1(%FGG(v)/BD5 , AYFT1G11JP(C)
ACCfS_S_1_021VNRt AP3005, 25 S/6020/63/ 151/005/1014/1017
AUTHORSI QaG.V*m0V, M. G.; Levitan, B.
TITLE: Sum of the differances of the eigenvalueB of two singular
Sturm-Liouville operators 16
SOURCE: AN SSSR. Doklady*, v. W'., no. 5, 1963, iol4-lolT
TOPIC TAGSt eigenvalue, difference sum, perturbation, Sturm-
Liouville operator, boundary condItion
ABSTRACTz This is a continuation of a study carried out by M. G.
Gasy*mov (DAN, no. 5, 1963, P. 150) wherein a formula was proposed
for the case of two singular Sturm-Liouville operators with discrete
spectra differing from each other only by finite perturbation. Au-
t6rs studied the sum of the differences of the bigenvalues of two
singular Sturm-Liouville operators which differed from each other by
boundary conditions and finite perturbation. An analogue for Gasy*-
mov's formulas was obtained and some necessary conditions were proven
a nd
so that the two !scuuences of TTmbersil'~,3 U,,Iw(,r--- eigenvalues of one 3!n;,i1ar
Sturir,-Liouvilbt equation lj,.,.t with different LLundary conditions. Thme tl-.eo rr~mq ars
Card l/Vj prov-d. Orig. ait. ha-_z 2"' foniulas.

IZVIT,Afl~ D.H.
Lette r to the eGitOr- UOP- 'r-at- muk 18 no.4:239 Jl-Ag 163.
(MIRA 16:9)
'J,
FRI

GASYMOV, M.G.;_~~VITAN._Uj,_
Sum of differences between the eigenvalues of two singular Sturm-
Louiville operators. Dokl. AN SM 151 no.5tlO14-1017 Ag 163.
(MIRA 16s9)
1. Predstayleno akademikom I.G.Petrovskim.
(Operators (Mathematics))

IZVITAII, fi.mo
cRiculation of tiv! rogularizo-d trPcfw !,-r a
iisp.imit,naijk 19 no. 1:161-165 Je-F '64-
operator. (rP,4

ACCESSION NRs AP4031754 S/0042/64/019/002/0003/0063
:Aunim: I&vitang B. M.; Oasyftovp Ho Go
TITIZ:
Determination of a differential equation from two spectra
SOURCEs UsPekhi matematichaskikh nauk. Y, 19. no. 2. 196h. 3-63
:T01qC TAGSs differential equation., spectral functionj differential equation deter-i
minations differential operator, linear intogral equation, Parseval equalityp Sturm~
Liouville equation.. asymptotic formula.. Sturm Liouvine operator
IBSTRACTs Section titles ares
1. Determination of a differential equation from its spectral function
1. On the spectral function of a differential operator
2. Derivation of a linoar integral equation for the kernel K(x,,t~
3. Inverse problem. Solvability of the integral equation for the kernel K(xpt)
4. Derivation of the differential equation
So Parseval equality
6. Classic Sturm-Liouville problem
Card

'ACCESSION NR: AP4031754
:II. Determination of a regular Sturm-Liouvills equation from two spectra
j Is Expression of normalization numbs ire in terms of the spectrum
i 2. Asymptotic formulas for the numbers
3- Inverse Stum-Liouvine Problem
!in. Determination of the singular Sturm-Liouville problem from two spectra
1. Formulas for the differences of traces of two Sturm-Liouyilleo~erstors for
various bodndary conditions at zero
2. Expression of the numbers c4n~hj) in terms of the spectrum
3, Ono class of potentials
4. Solution of the inverse problem for the class.0
'Application 1. Proof of a theorem of V. A. Ambartsumyan
Application no Derivation of asymptotic formulas (1.6.6) andCL.6-7)
:Given two sequences of real nwobers1k @**A Ans #**Vo
i Of Alt ~**; /4 of 141 n
'the authors treat the problem of finding necessary mid sufficient conditions for
.these sequences to be two spectra of one Sturm-Liouvilla operator of the form
0- -.q W) Y
Card 2/5

ACCESSIMI NRt AP4031754
undor various boundary conditions. Here q(x) is a real function which is summsble
on each intorval (Oobt)p bt < b, In the first section the authors find a solution
or the inverse Sturm-Liouville problem from the spectral function# based on the
,following: Let, p(A) be the spectral function of the problem
Yft+OL-~-q(z))Y-O. ' 0'