OPERATOR THEORY: ADVANCES AND APPLICATIONS
FINITENESS OF EIGENVALUES OF THE PERTURBED DIRAC OPERATOR A MATHEMATICAL STUDY OF QUANTUM REVIVALS AND QUANTUM FIDELITY ON RELATIONS BETWEEN STABLE AND ZENO DYNAMICS IN A LEAKY GRAPH DECAY MODEL ON THE SPECTRUM OF PARTIALLY PERIODIC OPERATORS FUNCTIONAL MODEL FOR SINGULAR PERTURBATIONS OF NON-SELF-ADJOINT OPERATORS TRACE FORMULAS FOR JACOBI OPERATORS IN CONNECTION WITH SCATTERING THEORY FOR QUASI-PERIODIC BACKGROUND DIRICHLET-TO NEUMANN TECHNIQUES FOR THE PLASMA-WAVES IN A SLOT-DIOD INVERSE SPECTRAL PROBLEM FOR QUANTUM GRAPHS WITH RATIONALLY DEPENDENT EDGES FUNCTIONAL MODEL OF A CLASS OF NON-SELFADJOINT EXTENSIONS OF SYMMETRIC OPERATORS LYAPUNOV EXPONENTS AT ANOMALIES OF SL(2, R)-ACTIONS UNIFORM AND SMOOTH BENZAID-LUTZ TYPE THEOREMS AND APPLICATIONS TO JACOBI MATRICES AN EXAMPLE OF SPECTRAL PHASE TRANSITION PHENOMENON IN A CLASS OF JACOBI MATRICES WITH PERIODICALLY MODULATED WEIGHTS ON CONNECTION BETWEEN FACTORIZATIONS OF WEIGHTED SCHUR FUNCTION AND INVARIANT SUBSPACES ?KDK + ?N+1 + Q, (1 1) WHERE DK = I ? ?XK (K = 1, , N), ?K (K = 1, , N+1) ARE M×M HERMITIAN MATRICES WHICH SATISFY THE ANTICOMMUTATION RELATIONS (OR, SO-CALLED CLI?ORD’S RELATIONS) ?J?K + ?K?J = 2?JK (J, K = 1, , N + 1), (1 2) M = N 2 FOR N EVEN AND M = (1 3) AND REPRESENTS THE OPERATOR OF MULTIPLICATION BY A GIVEN M×M HERMITIAN MATRIX- VALUED FUNCTION Q(X), X ?RN IN ACCORDANCE WITH OUR INTERESTS WE ASSUME THAT THE ELEMENTS QJK(X) (J, K = 1, , M) OF THE MATRIX Q(X) ARE MEASURABLE FUNCTIONS FROM THE SPACE L?(RN) THE OPERATORS H0 AND H ARE CONSIDERED IN THE SPACE L2(RN; CM) WITH THEIR MAXIMAL DOMAINS OF DE?NITION NAMELY, IT IS CONSIDERED THAT THE DOMAIN OF THE OPERATOR H0 IS THE SOBOLEV SPACE W 2 (RN; CM) AND, BECAUSE Q IS A BOUNDED OPERATOR, THE PERTURBED DIRAC OPERATOR H IS DE?NED ON THE SAME DOMAIN W 0 COVERS THE INTERVAL [1, ?] AND, SINCE THE SPECTRUM OF THE OPERATOR H0 IS A SYMMETRIC SET WITH RESPECT TO THE ORIGIN, IT RESULTS THAT ITS SPECTRUM COINCIDES WITH THE SET ?(H) = (??, ?1] ? [1, +?) WE NOTE THAT THE SYMMETRY OF THE SPECTRUM OF H0 CAN BE SHOWN EASILY BY INVOKING, FOR INSTANCE, ANOTHER MATRIX ? WHICH TOGETHER WITH ?K (K = 1, , N+1) THE ANTICOMMUTATION CONDITIONS (1 2) ARE SATIS?ED THEN (H0 + ?)? = ??(H0 ??) FOR EACH SCALAR ?, AND SO THE PROPERTY OF THE SYMMETRY OF ?(H0) BECOMES TO BE CLEAR THE UNPERTURBED OPERATOR H0 HAS NO EIGENVALUES (IN FACT THE SPECTRUM OF H0 IS ONLY ABSOLUTELY CONTINUOUS) IF THE ENTRIES OF THE MATRIX-VALUED FUNCTION Q(X) VANISH AT THE IN?NITE, THE CONTINUOUS SPECTRUM OF THE PERTURBED DIRAC OPERATOR H COINCIDES WITH ?(H0) AND THE PERTURBATION Q CAN PROVOKE A NON-TRIVIAL POINT SPECTRUM OUR PROBLEM IS TO STUDY THE POINT SPECTRUM OF THE PERTURBED DIRAC OPERATOR H THIS PROBLEM HAS BEEN STUDIED BY MANY RESEARCHERS IN CONNECTION WITH VARIOUS PROBLEMS (NOTE THAT THE MOST OF THE RESULTS WERE CONCERNED WITH THE CASE N = 3 AND M = 4) A GOOD DEAL OF BACKGROUND MATERIAL ON THE DEVELOPMENT AND PERSPECTIVES OF THE PROBLEM CAN BE FOUND IN [1], [2], [3], [5], [7], [10], [12], [13], [14] APART FROM THE ALREADY MENTIONED WORKS, WE REFER TO THE [15] AND THE REFERENCES GIVEN THEREIN FOR A PARTIAL LIST IN THIS PAPER, WE GIVE CONDITIONS ON Q(X) UNDER WHICH THE POINT SPECTRUM OF H (IF ANY) HAS ±1 AS THE ONLY POSSIBLE ACCUMULATION POINTS SPECI?CALLY, WE ASSUME THAT Q(X) SATIS?ES THE FOLLOWING ASSUMPTION (A) Q(X) = [QJK(X)], X ?RN, IS AN M × M HERMITIAN MATRIX-VALUED FUNCTION THE ENTRIES OF WHICH ARE ELEMENTS FROM THE SPACE L?(RN) AND LIM |X|??| X | QJK(X) = 0 (J, K = 1, , M) THE MAIN RESULTS ARE OBTAINED BY APPLYING THE ABSTRACT RESULTS FROM [6] (SEE ALSO ITS RE?NEMENT RESULTS MADE IN [9]) BELOW, WE CITE THE CORRESPONDING RESULT OPERATOR THEORY: ADVANCES AND APPLICATIONS VOL ?TK(?)U ?, (1 6) WHERE THE OPERATOR-VALUED FUNCTIONS TK(?) (K = 1, , N) SATISFY THE CONDITIONS (I) AND (II) AS WE ALREADY MENTIONED WE WILL APPLY THEOREM 1 1 TO THE STUDY OF THE PROBLEM OF THE DISCRETENESS OF THE SET OF EIGENVALUES OF THE PERTURBED DIRAC OPERATOR H THE MAIN RESULTS ARE PRESENTED IN THE NEXT SECTION 2 MAIN RESULTS LET H BE THE DIRAC OPERATOR DE?NED BY (1 1) IN WHICH THE MATRIX-VALUED FUNCTION SATIS?ES THE ASSUMPTION (A) THE UNPERTURBED DIRAC OPERATOR H0 REPRESENTS A MA- TRIX DI?ERENTIAL OPERATOR (OF THE DIMENSION M×M) OF ORDER 1 THE SYMBOL OF THE OP- ERATOR H0 IS A MATRIX-VALUED FUNCTION WHICH WE DENOTE BY H0(?), ? ?RN NOTE THAT BY APPLYING THE FOURIER TRANSFORMATION TO THE ELEMENTS OF THE SPACE L2(RN; CM) THE OPERATOR H0 IS TRANSFORMED (IN THE MOMENTUM SPACE) INTO A MULTIPLICATION OP- ERATOR BY THE MATRIX H0(?) THE FOURIER TRANSFORMATION IS DE?NED BY THE FORMULA ˆU(?) = (FU)(?) = (2?) N U(X)EI<X,?>DX (U ?L2(RN)) IN WHICH < X, ? > DESIGNATES THE SCALAR PRODUCT OF THE ELEMENTS X, ? ?RN (HERE AND IN WHAT FOLLOWS : RN) THE CORRESPONDING NORM IN RN (OR CM) WILL BE DENOTED AS USUALLY BY | | THE OPERATOR NORM OF M × M MATRICES CORRESPONDING TO THE NORM | | IN CM WILL BE DENOTED BY | |, AS WELL OPERATOR THEORY: ADVANCES AND APPLICATIONS VOL ˆF(?, ?) ? ?R(?) D?DS? ?ˆF ?? (T(? ?R(?)) + R(?), ?)DT D?DS? OPERATOR THEORY: ADVANCES AND APPLICATIONS VOL ?ˆF ?? (T(? ?R(?)) + R(?), ?)|2D?DS? ?ˆF ?? (?, ?) 2 ((N ?3)? + (N ?1)R(?))(? + R(?))?2ˆV(??) 2 (? + R(?))?1 ? ?? ˆV(??) QU(?) ˆV(?) U ?ˆV(?) |?| + R(?) U |? QU(?)|2D? QU(?)|2D? ?C(1 + H)?2 ?U ?2, AND SO (2 2) IS REALIZED OPERATOR THEORY: ADVANCES AND APPLICATIONS VOL THUS, TAKING INTO ACCOUNT (2 1), WE OBTAIN AN ESTIMATE LIKE THAT FROM (1 6) (SEE REMARK 1 2) AND, THEREFORE THEOREM 1 1 CAN BE APPLIED THIS COMPLETES THE PROOF OF THEOREM 2 1 ? AS AN APPLICATION OF THEOREM 2 1 WE GIVE A RESULT CONCERNING THE PARTICULAR CASE OF THE HAMILTONIAN OF A DIRAC PARTICLE IN AN ELECTROMAGNETIC ?ELD THE DIRAC OPERATOR IN THIS CASE IS TYPICALLY WRITTEN IN THE PHYSICS LITERATURE (SEE, FOR INSTANCE, [11], [15]) AS FOLLOWS HU = ?J(DJ ?AJ(X))U + ?4U + Q(X)U, U ?W 2 (R3; C4), (2 3) WHERE A(X) = (A1(X), A2(X), A3(X)) (THE VECTOR POTENTIAL) AND Q(X) (THE SCALAR POTENTIAL) ARE GIVEN FUNCTIONS ON R3 THEOREM 2 2 IF LIM |X|??|X|AJ(X) = 0 (J = 1, 2, 3), LIM |X|??|X|Q(X) = 0, THEN THE POINT SPECTRUM OF THE DIRAC OPERATOR DE?NED BY (2 1) IS DISCRETE HAVING ONLY ±1 AS ACCUMULATION POINTS EACH EIGENVALUE CAN BE ONLY OF A ?NITE MULTIPLICITY REFERENCES [1] A BERTHEIR, V GEORGESCU, ON THE POINT SPECTRUM FOR DIRAC OPERATORS, J FUNC ANAL 71 (1987), 309–338 [2] M S BIRMAN, ON THE SPECTRUM OF SCHRODINGER AND DIRAC OPERATORS, DOKL ACAD NAUK 129, NO 2 (1959), 239–241 [3] M S BIRMAN, ON THE SPECTRUM OF SINGULAR BOUNDARY-VALUE PROBLEMS, MAT SB (1961), NO 2, 125–174 [4] M S BIRMAN, G E KARADZHOV, M Z SOLOMYAK, BOUNDEDNESS CONDITIONS AND SPEC- TRUM ESTIMATES FOR THE OPERATORS B(X)A(D) AND THEIR ANALOGS, ADV SOVIET MATH , (1991), 85–106 [5] M S BIRMAN, A LAPTEV, DISCRETE SPECTRUM OF THE PERTURBED DIRAC OPERATOR, ARK MAT 32(1) (1994), 13–32 [6] P A COJUHARI, ON THE DISCRETE SPECTRUM OF A PERTURBED WIENER-HOPF OPERATOR, MAT ISSLED 54 (1980), 50–55 (RUSSIAN) [7] P A COJUHARI, ON THE POINT SPECTRUM OF THE DIRAC OPERATOR, DIF URAV , MINSK, DEP VINITI 24 06 87, N4611-B87, (1987), 1–17 (RUSSIAN) [8] P A COJUHARI, ON THE ?NITENESS OF THE DISCRETE SPECTRUM OF SOME MATRIX PSEUDODIF- FERENTIAL OPERATORS, VYSSH UCHEBN ZAVED MAT , 1 (1989), 42–50 (RUSSIAN) [9] P A COJUHARI, THE PROBLEM OF THE ?NITENESS OF THE POINT SPECTRUM FOR SELF-ADJOINT OP- ERATORS PERTURBATIONS OF WIENER-HOPF OPERATORS AND APPLICATIONS TO JACOBI MATRICES, OPERATOR THEORY, ADV AND APPL 154 (2004), 35–50 [10] W D EVANS, B J HARRIS, BOUNDS FOR THE POINT SPECTRA OF SEPARATED DIRAC OPERATORS, PROC ROY SOC EDINBURGH, SECT A 88 (1980), 1–15 OPERATOR THEORY: ADVANCES AND APPLICATIONS VOL