The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

On the Multiplicity of Eigenvalues of the Laplacian on Surfaces

We show that the multiplicity of the eigenvalues of the Laplace Beltrami operator on compact Riemannian surfaces with genus zero is bounded by m(k) 2k – 3 for k 3. Here we label the eigenvalues in the following way: 0 = 1 < 2 3 . . ..     

Holomorphic Solutions of Some Functional Equations II

We study a certain type of functional equation, which is of significance from the view point of systems of difference equations. Let the characteristic values of the system be and The case that either || > 1 or 0 < || < 1 has been treated in a former paper. The case that = 1, || = 1 with 1 will be given in another paper. The present note deals with the case = = 1, the most difficult case.AMS Subject Classification (1991): 39A10 39B05     

Problème aux limites non-linéairey″ + λ2 f(y)=0:existence d'une solution et bornes pour λ2 (évaluations de la valeur critique)

Summary Le problème aux limites non-linéaire (1.1) possède une (ou plusieurs) solution seulement si < ^* = valeur critique. On donne dans cet article des bornes inférieures et supérieures de ^*. Pour certainsf, on détermine exactement ^*.Summary The nonlinear boundary value problem (1.1) possess one (or more) solution only if < ^* = critical value. We give in this paper lower and upper bounds of ^*. For somef, we determine ^* exactly.     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

A Recursive Construction for 2-Designs

This paper contains two main results: given a symmetric (v, k, ) design, D, and a resolvable design which has the parameters of a residual design of D, there exists a symmetric (dv + 1, v, k) design, where d = (v - k)/(k - ), and d is a prime power; given a symmetric (v, k, ) design, D, and a resolvable design with the parameters of a derived design of D, there exists a 2 - (ek + v, 2k, k) design, where e = k/,and e is a prime power.     

A minimax test for hypotheses on a spectral density

Assume that on the set of the values t = 1, , k there is given a realization of the Gaussian stationary process X_t with spectral density f(), (0, 1). There arises the problem of the minimax testing of the hypothesis H₀: f() = p().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 115–125, 1990.     

A Remark on Real Parameter Global Bifurcation

We study the nonlinear eigenvalue problem F(x,) = L()x +R(x,) = 0 where F : X × R X with X a Hilbert space. IfL() is a polynomial in , then it is shown that 0> 0 is a global bifurcation point of the eigenvalue problem provided astandard transversality condition is satisfied, the dimension of the nullspace of L(0) is an odd number and L() is composed of asequence of positive operators on the finite dimensional null space ofL(0).     

Spectral properties of λ-matrices

One investigates the spectral problem for polynomial -matrices of the general form (regular and singular). One establishes a relation between the elements of the complete spectral structure of the -matrix (i.e., its elementary divisors, its Jordan chains of vectors, its minimal indices and polynomial solutions) and of the matrices which occur in its transformation to the Smith canonical form. One establishes a correspondence between the complete spectral structures of a -matrix and of three linear accompanying matrix pencils. One notes the possibility of reducing the solution of the spectral problem for a -matrix to the solution of a similar problem for its accompanying pencil. One gives a factorization of a -matrix which allows to represent it by any of its considered accompanying pencils or by its Kronecker canonical form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 180–194, 1981.     

Minimality of derivative chains,corresponding to a boundary value problem on a finite segment

One investigates the minimality of derivative chains, constructed from the root vectors of polynomial pencils of operators, acting in a Hilbert space. One investigates in detail the quadratic pencil of operators. In particular, for L()=L₀+L₁+²L₂ with bounded operators L₀0, L₂0 and Re L₁0, one shows the minimality in the space_173-02 of the system {x_k, _kekx_k}, where x_k are eigenvectors of L(), corresponding to the characteristic numbers _kin the deleted neighborhoods of which one has the representation L^–1()=(–_k)^–1R_K+W_K() with one-dimensional operators R_k and operator-valued functions W_K(), k=1, 2, ..., analytic for =_k.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 195–205, February, 1990.     

Reverse Functional Inequalities and Their Applications to Nonlinear Elliptic Boundary Value Problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter which have two branches of solutions u (0) and U (>0) of which the first is continuous at the origin and the second increases indefinitely as 0.     

A Proof of the Jungnickel-Tonchev Conjecture on Quasi-Multiple Quasi-Symmetric Designs

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design.     

(v,k, λ)-Graphs and polarities of (v,k, λ)-designs

In 2.1 it is established that there is a one-to-one correspondence between (v, k, )-graphs and polarities, with no absolute points, of (v, k, )-designs. This is used to show that the parameters of a (v, k, )-graph are of the form ((s/a)((s + a)²–1), s(s+a), sa) where s and a are positive integers with a dividing s(s²–1) (Theorem 3.4) but strictly less than s(s²–1) (Proposition 4.3). Some consequences of this parametrization are discussed and in particular, it is shown that for fixed 2 there are only finitely many non-isomorphic (v, k, )-graphs. In 4. it is shown that (v, k, )-graphs can also be constructed using polarities, with all points absolute, of certain designs. In 5. isomorphisms and automorphisms of graphs and designs are discussed. Many examples of (v, k, )-graphs, including some apparently new ones, are given.Dedicated to Peter Dembowski, 28 January 1971