finitely many delta interactions with supports on concentric spheres using the theory of self adjoint extensions of symmetric operators the precise mathematical definition of the quantum hamiltonian describing a finite number of interactions with supports on concentric spheres is given its resolvent is also derived its spectral properties are described and it is shown how this hamiltonian can be obtained as a norm resolvent limit of a family of local scaled short range hamiltonians

a zero thickness limit of multilayer structures a resonant tunnelling δ potential a zero thickness limit for two terminal and three terminal devices from the quantum electronics domain is analysed the study is focused on heterostructures composed of a single barrier with adjacent one or two prewells the point interactions obtained in this limit are shown to be described by a family of resonant diagonal matrices that connect the two sided boundary conditions at the device origin which are a subclass of the whole four parameter family of point interactions transmission through such a device is absent almost everywhere except at a few points whose number and position can be controlled by a gate voltage applied externally to the barrier subsystem it is remarkable that the existence of resonances in the zero thickness limit occurs only if a squeezing sequence is constructed in a like way in this case the limit describes adequately the resonant behaviour of a barrier well heterostructure with realistic parameters simple analytical expressions obtained for resonance sets are supported by direct numerical calculations of the transmission being in agreement with the results of experiments on semiconductor devices the zero thickness like limiting procedure can be used in the design of nanodevices or contacting quantum wires

generalized point interactions in one dimensional quantum mechanics there is a four parameter family of point interactions in one dimensional quantum mechanics they represent all possible self adjoint extensions of the kinetic energy operator if time reversal invariance is imposed the number of parameters is reduced to three one of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known we present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so called interaction we emphasize that there is little resemblance between the interaction and what its name suggests

multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue abstract some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory

potential approximations to δ an inverse klauder phenomenon with norm resolvent convergence we show that there is a family schrodinger operators with scaled potentials which approximates the interaction hamiltonian in the norm resolvent sense this approximation based on a formal scheme proposed by cheon and shigehara has nontrivial convergence properties which are in several respects opposite to those of the klauder phenomenon

spectral asymptotics for schrodinger operators with periodic point interactions abstract spectrum of the second order differential operator with periodic point interactions in l 2 r is investigated classes of unitary equivalent operators of this type are described spectral asymptotics for the whole family of periodic operators are calculated it is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely it is proven that the spectrum of the operators with anomalous spectral asymptotics when the ratio between the lengths of the bands and gaps tends to zero at infinity can be approximated by standard periodic weighted operators with step wise density functions it is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal generalized resolvent of the periodic schrodinger operator with certain energy dependent boundary conditions the operator acting in an extended hilbert space such that its resolvent restricted to l 2 r coincides with the formal resolvent is constructed explicitly

a new class of solvable models in quantum mechanics describing point interactions on the line the authors provide a detailed analysis of properties of the schrodinger operator in l2 r which formally can be written as d2 dx2 sigma y in gamma nu y delta y where delta is the derivative of dirac s delta function and y contained in implied by r is discrete this model allows for an explicit calculation of spectral properties special emphasis is given to the periodic case y z nu nu j j in z where the spectrum and the density of states are explicitly computed also the spectrum for a half crystal is given the authors study in detail spectral consequences when various defects and impurities are added to the periodic case

energy dependent point interactions in one dimension we consider a new type of point interaction in one dimensional quantum mechanics it is characterized by a boundary condition at the origin that involves the second and or higher order derivatives of the wavefunction the interaction is effectively energy dependent it leads to a unitary s matrix for the transmission reflection problem the energy dependence of the interaction can be chosen such that any given unitary s matrix or the transmission and reflection coefficients can be reproduced at all energies generalization of the results to coupled channel cases is discussed

one dimensional schrodinger operators with interactions singular on a discrete set on etudie le caractere auto adjoint des operateurs de schrodinger sur y u j i x j y j ou on suppose y j x j 1

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

point interactions in one dimensional quantum mechanics with coupled channels we explore generalized point interactions in one dimensional quantum mechanics with two coupled channels they represent possible self adjoint extensions of the nonrelativistic kinetic energy operator assuming time reversal invariance we find a family of self adjoint extensions with seven parameters this can be compared with the one channel case in which the corresponding number of parameters is three

existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems abstract using the minimax methods in critical point theory and a generalized landesman lazer type condition we establish two existence results of solutions for asymptotically linear noncooperative elliptic systems at resonance besides this we obtain two solutions in the case of near resonance

prime order derangements in primitive permutation groups abstract let g be a transitive permutation group on a finite set of size at least 2 an element of g is a derangement if it has no fixed points on let r be a prime divisor of we say that g is r elusive if it does not contain a derangement of order r and strongly r elusive if it does not contain one of r power order in this note we determine the r elusive and strongly r elusive primitive actions of almost simple groups with socle an alternating or sporadic group

many body system with a four parameter family of point interactions in one dimension we consider a four parameter family of point interactions in one dimension this family is a generalization of the usual function potential we examine a system consisting of many particles of equal masses that are interacting pairwise through such a generalized point interaction we follow mcguire who obtained exact solutions for the system when the interaction is the function potential we find exact bound states with the four parameter family for the scattering problem however we have not been so successful this is because as we point out the condition of no diffraction that is crucial in mcguire s method is not satisfied except when the four parameter family is essentially reduced to the function potential

four parameter point interaction in 1d quantum systems the author constructs a four parameter point interaction for a nonrelativistic particle moving on a line as the limit of a short range interaction with a range tending toward zero for particular choices of the parameters one can obtain a delta interaction or the so called delta interaction the hamiltonian corresponding to the four parameter point interaction is shown to correspond to the four parameter self adjoint hamiltonian of the free particle moving on the line with the origin excluded

primitive permutation groups of odd degree and an application to finite projective planes one of the most beautiful and important results concerning finite projective planes is the ostrom wagner theorem 26 such a plane admitting a 2 transitive collineation group must be desarguesian it has long been conjectured that the same conclusion must hold if it is only assumed that there is a collineation group transitive on incident point line pairs ll pp 208 214 171 the starting point for this paper was a proof of this conjecture modulo a degenerate situation

characterization of one dimensional point interactions for the schrodinger operator by means of boundary conditions we find a new representation of all boundary conditions corresponding to the so called point interactions we show that there is one to one correspondence between one dimensional point interactions and boundary conditions of the form with satisfying 2 2 1

an exactly solvable periodic schrodinger operator the authors explicitly determine the band spectrum of a periodically strongly singular schrodinger operator in l2 r associated with the differential expression d2 dx2 s2 1 4 cos2 x s 2 1 4 sin2 x x epsilon r x epsilon pi z 2 0 s s 1 the corresponding density of states is calculated analytically

pt invariant point interactions in one dimension by using wu and yu s pseudo potential we construct point interactions in one dimension that are complex but conform to space time reflection invariance the resulting point interactions are equivalent to those obtained by albeverio fei and kurasov as self adjoint extensions of the kinetic energy operator

on partitioning the orbitals of a transitive permutation group let g be a permutation group on a set with a transitive normal subgroup m then g acts on the set orbl m of nontrivial m orbitals in the natural way and here we are interested in the case where orbl m has a partition p such that g acts transitively on p the problem of characterising such tuples m g p called tods arises naturally in permutation group theory and also occurs in number theory and combinatorics the case where p is a prime power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials the case where p 2 exactly corresponds to self complementary vertex transitive graphs while the general case corresponds to a type of isomorphic factorisation of complete graphs called a homogeneous factorisation characterising homogeneous factorisations is an important problem in graph theory with applications to ramsey theory this paper develops a framework for the study of tods establishes some numerical relations between the parameters involved in tods gives some reduction results with respect to the g actions on and on p and gives some construction methods for tods

comments on the schrodinger equation with delta interaction in one dimension the boundary conditions at the singular point for the schrodinger equation with delta interaction in one dimension are proposed it is also shown that the boundary conditions adopted by albeverio et al 1988 are irrelevant to the delta interaction

temporally stable coherent states for infinite well and poschl teller potentials this article is a direct illustration of a construction of coherent states which has been recently proposed by two of us jpg and jk we have chosen the example of a particle trapped in an infinite square well and also in poschl teller potentials of the trigonometric type in the construction of the corresponding coherent states we take advantage of the simplicity of the solutions which ultimately stems from the fact they share a common su 1 1 symmetry a la barut girardello many properties of these states are then studied both from mathematical and from physical points of view

on the residue fields of henselian valued stable fields abstract let k v be a henselian valued field satisfying the following conditions for a given prime number p i central division k algebras of finite p primary dimensions have schur indices equal to their exponents ii the value group v k properly includes its subgroup p v k the paper shows that if k is the residue field of k v and r is an intermediate field of the maximal p extension k p k then the natural homomorphism br k br r of brauer groups maps surjectively the p component br k p on br r p it proves that br k p is divisible if p 2 or k is a nonreal field and that br k 2 is of order 2 when k is formally real we also obtain that r embeds as a k subalgebra in a central division k algebra if and only if the degree r k divides the index of

on permutation groups with bounded movement abstract let g be a permutation group on a set with no fixed points in and let m be a positive integer if no element of g moves any subset of by more than m points that is if g m for every and g g then is finite and moreover 5m 2 if moreover g is transitive on then 3m and equality can be achieved if m 2 or m is a power of 3

point interactions of the dipole type defined through a three parametric power regularization a family of point interactions of the dipole type is studied in one dimension using a regularization by rectangles in the form of a barrier and a well separated by a finite distance the rectangles and the distance are parametrized by a squeezing parameter e 0 with three powers and describing the squeezing rates for the barrier the well and the distance respectively this parametrization allows us to construct a whole family of point potentials of the dipole type including some other point interactions such as e g potentials varying the power it is possible to obtain in the zero range limit the following two cases i the limiting potential is opaque the conventional result obtained earlier by some authors or ii this potential admits a resonant tunneling the opposite result obtained recently by other authors the structure of resonances if any also depends on a regularizing sequence the sets of the space where a non zero resonant or non resonant transmission occurs are found for all these cases in the zero range limit the transfer matrix is shown to be of the form with real parameters and g depending on a regularizing sequence those cases when 1 and g 0 mean that the corresponding potential is accompanied by an effective potential

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

hierarchy of qm susys on a bounded domain we systematically formulate a hierarchy of isospectral hamiltonians in one dimensional supersymmetric quantum mechanics on an interval and on a circle in which two successive hamiltonians form supersymmetry we find that boundary conditions compatible with supersymmetry are severely restricted in the case of an interval a hierarchy of at most three isospectral hamiltonians is possible with unique boundary conditions while in the case of a circle an infinite tower of isospectral hamiltonians can be constructed with a two parameter family of boundary conditions

a delta well with a mass jump in this paper it is shown that a one dimensional hamiltonian with an attractive delta potential at the origin plus a mass jump at the same point cannot have a bound state as is the case with the ordinary attractive delta potential with constant mass unless another term is added into the potential in the form of a derivative of the dirac delta at the origin the consistency of this singular potential with two terms is guaranteed by choosing suitable matching conditions at the singular point for the wavefunctions furthermore it is proved that the self adjointness of the hamiltonian with both singular interactions determines the coefficient of the derivative of the delta in a unique manner under these conditions the bound state and its energy are obtained and it is checked that the correct results in the limit of equal masses are obtained

on energy gaps in a new type of analytically solvable model in quantum mechanics abstract a new type of analytically solvable one dimensional model in quantum mechanics formally given by d2 dx2 j z j x yj j yj r is analyzed besides calculating the density of states of various periodic arrangements of such and models we also prove a generalization of the saxon hutner conjecture concerning the existence of gaps in the energy spectrum of certain substitutional alloys of or interactions in addition the spectrum of the corresponding random hamiltonians i e j j z ergodic stochastic processes is investigated and explicitly determined in certain cases

multiplicity results for a superlinear elliptic system with partial interference with the spectrum abstract in this work we consider an elliptic system of two equations in dimension greater than one with nonlinearities which are linear at and superlinear at we prove by variational techniques which involve a strongly indefinite functional the existence of two solutions for suitable forcing terms under a condition on the linear part which prevents resonance with the eigenvalues of the operator

spectral functions for the flat plasma sheet model the present work is based on bordag m et al 2005 j phys a math gen 38 11027 where the spectral analysis of the electromagnetic field on the background of an infinitely thin flat plasma layer is carried out the solutions to maxwell equations with the appropriate matching conditions at the plasma layer are derived and the spectrum of electromagnetic oscillations is determined the spectral zeta function and the integrated heat kernel are constructed for different branches of the spectrum in an explicit form the asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived the local heat kernels are considered also

multiple solutions for an elliptic system near resonance abstract we consider an elliptic system with linear part depending on two parameters and a sublinear perturbation we obtain the existence of at least two solutions when the linear part is near resonance the system is associated with a strongly indefinite functional and the solutions are obtained through saddle point theorem and galerkin approximation

an explicit realization of resonant tunnelling δ potentials a family of non trivial one point interactions is constructed by means of a two scale approximation with spatially symmetric piecewise constant functions the approximation is realized through two powers of a squeezing parameter the convergence is considered in terms of finite range transmission matrices which are given in an explicit form the limiting matrix that connects the two sided boundary conditions at the point of singularity appears to be the unity matrix defined at a countable set in the space of intensities of the potential everywhere beyond this resonance set the limiting point interaction separates the system into two subsystems with boundary conditions of the dirichlet type in other words any interaction of this family behaves as a fully transparent barrier on a resonance set and beyond this set as a fully reflecting wall among these interactions a special regularizing sequence that converges to the second derivative of dirac s delta function in the sense of distributions is found because of this example it is reasonable to term all the point interactions with the properties described above as resonant tunnelling potentials

the fermi pseudo potential in one dimension wu and yu recently examined point interactions in one dimension in the form of the fermi pseudo potential on the other hand there are point interactions in the form of self adjoint extensions saes of the kinetic energy operator we examine the relationship between the point interactions in these two forms in the one channel and two channel cases in the one channel case the pseudo potential leads to the standard three parameter family of saes in the two channel case the pseudo potential furnishes a ten parameter family of saes

multiple solutions for semilinear elliptic equations near resonance at higher eigenvalues abstract using the minimax methods in critical point theory and a generalized landesman lazer type condition we obtain two solutions for a class of semilinear elliptic equations near resonance at higher eigenvalues

on robin boundary conditions and the morse potential in quantum mechanics the physical origin of robin boundary conditions is investigated for wavefunctions at an infinite reflecting wall we consider both schrodinger and phase space quantum mechanics also known as deformation quantization for this simple example of a contact interaction a non relativistic particle moving freely on the half line is treated as moving on the full line in the presence of an infinite potential wall realized as a limit of a morse potential we show that the wavefunctions for the morse states can become those for a free particle on the half line with robin boundary conditions however dirichlet boundary conditions standard walls are obtained unless a mass dependent fine tuning to a reflection resonance is imposed this phenomenon was already observed for piece wise flat potentials so it is not removed by smoothing we argue that it explains why standard quantum walls are standard next we consider the wigner functions the symbols of both diagonal and off diagonal density operator elements of phase space quantum mechanics taking the fine tuned limit we show that our wigner functions do reduce to the expected ones on the half line this confirms that the wigner transform should indeed be unmodified for this contact interaction

on orbital partitions and exceptionality of primitive permutation groups let g and x be transitive permutation groups on a set such that g is a normal subgroup of x the overgroup x induces a natural action on the set orbl g of non trivial orbitals of g on in the study of galois groups of exceptional covers of curves one is led to characterizing the triples g x where x fixes no elements of orbl g such triples are called exceptional in the study of homogeneous factorizations of complete graphs one is led to characterizing quadruples g x p where p is a partition of orbl g such that x is transitive on p such a quadruple is called a tod transitive orbital decomposition it follows easily that the triple g x in a tod g x p is exceptional conversely if an exceptional triple g x is such that x g is cyclic of prime power order then there exists a partition p of orbl g such that g x p is a tod this paper characterizes tods g x p such that x is primitive and x g is cyclic of prime power order an application is given to the classification of self complementary vertex transitive graphs

distribution theory for schrodinger s integral equation much of the literature on point interactions in quantum mechanics has focused on the differential form of schrodinger s equation this paper in contrast investigates the integral form of schrodinger s equation while both forms are known to be equivalent for smooth potentials this is not true for distributional potentials here we assume that the potential is given by a distribution defined on the space of discontinuous test functions first by using schrodinger s integral equation we confirm a seminal result by kurasov which was originally obtained in the context of schrodinger s differential equation this hints at a possible deeper connection between both forms of the equation we also sketch a generalisation of kurasov s j math anal appl 201 1 297 323 1996 result to hypersurfaces second we derive a new closed form solution to schrodinger s integral equation with a delta prime potential this potential has attracted considerable attention including some controversy interestingly the derived propagator satisfies boundary conditions that were previously derived using schrodinger s differential equation third we derive boundary conditions for super singular potentials given by higher order derivatives of the delta potential these boundary conditions cannot be incorporated into the normal framework of self adjoint extensions we show that the boundary conditions depend on the energy of the solution and that probability is conserved this paper thereby confirms several seminal results and derives some new ones in sum it shows that schrodinger s integral equation is a viable tool for studying singular interactions in quantum mechanics

coexistence problem for the one dimensional schrodinger operators with the double or triple periodic δ 1 interactions abstract we discuss the coexistence problem for the one dimensional schrodinger operator with the double or triple periodic 1 interactions for each j n we determine whether or not the jth spectral gap is degenerate

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

limit theorems for random walks with dynamical random transitions let be an ergodic and conservative non singular transformation of m thedynamic environment let w be a random probability on a locally compact second countable groupg and define v w n mu theta n 1 w mu theta n 2 w mu theta w mu w

reducible boundary conditions in coupled channels we study hamiltonians with point interactions in spaces of vector valued functions using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several one dimensional problems it shown that such a reduction is closely connected with the invariance under channel permutations examples are provided by some model interactions in particular the so called and the kirchhoff couplings

semilinear elliptic problems near resonance with a nonprincipal eigenvalue abstract we consider the dirichlet problem for the equation u u f x u h x in a bounded domain where f has a sublinear growth and h l 2 we find suitable conditions on f and h in order to have at least two solutions for near to an eigenvalue of a typical example to which our results apply is when f x u behaves at infinity like a x u q 2 u with m a x 0 and 1 q 2

multiple solutions for a class of nonlinear boundary value problems near resonance a variational approach abstract a variational approach is given to the existence of multiple solutions for some nonlinear boundary value problems near resonance at the first eigenvalue

existence and multiplicity results for some elliptic systems at resonance abstract the existence and multiplicity of weak solutions for some resonant elliptic systems are obtained by using ekeland s variational principle the mountain pass theorem and the saddle point theorem in critical point theory

on the existence of resonances in the transmission probability for interactions arising from derivatives of dirac s delta function the scattering properties of regularizing finite range potentials constructed in the form of squeezed rectangles which approximate the first and second derivatives of the dirac delta function x are studied in the zero range limit particularly for a countable set of interaction strength values a non zero transmission through the point potential x defined as the weak limit in the standard sense of distributions of a special dipole like sequence of rectangles is shown to exist when the rectangles are squeezed to zero width a tripole sequence of rectangles which gives in the weak limit the distribution x is demonstrated to exhibit the total transmission for a countable sequence of the rectangle s width that tends to zero however this tripole sequence does not admit a well defined point interaction in the zero range limit making sense only for a finite range of the regularizing rectangular like potentials

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

one dimensional point interaction with three complex parameters for a pair of non hermitian hamiltonian h and its hermitian adjoint h there are situations in which their eigenfunctions form a biorthogonal system we illustrate such a situation by means of a one particle system with a one dimensional point interaction in the form of the fermi pseudo potential the interaction consists of three terms with three strength parameters gi i 1 2 and 3 which are all complex this complex point interaction is neither hermitian nor invariant in general the s matrix for the transmission reflection problem constructed with h or with h in the usual manner is not unitary but it conforms to the pseudo unitarity that we define the pseudo unitarity is closely related to the biorthogonality of the eigenfunctions the eigenvalue spectrum of h with the complex interaction is generally complex but there are cases where the spectrum is real in such a case h and h form a pseudo hermitian pair

comments on a recent note on the schrodinger equation with a delta interaction for original paper see bao heng zhao ibid vol 25 p l617 1992 it is argued that a recent letter by bao heng zhao 1992 on the one dimensional schrodinger equation with a delta interaction is seriously flawed and hence arrives at wrong conclusions

an existence result for a linear superlinear elliptic system with neumann boundary conditions abstract in this work we consider an elliptic system of two equations in dimension one with neumann boundary conditions where the nonlinearities are asymptotically linear at and superlinear at we obtain that under suitable hypotheses a solution exists for any couple of forcing terms in l 2 we also present a similar result in which the superlinearity is in only one of the two equations and we discuss the resonant problem too

transitive permutation groups with bounded movement having maximal degree abstract let g be a transitive permutation group on a set such that g is not a 2 group and let m be a positive integer it was shown by the fourth author that if g m for every subset of and all g g then 2 mp p 1 where p is the least odd prime dividing g if p 3 the upper bound for is 3 m and the groups g attaining this bound were classified in the work of gardiner mann and the fourth author here we show that the groups g attaining the bound for p 5 satisfy one of the following a g z p z 2 a p m p 1 2 where 2 a p 1 for some a 1 b g k p 2 s p m 2 s 1 p 1 where 1 s p k is a 2 group with p orbits of length 2 s each element of k moves at most 2 s p 1 points of and p z p is fixed point free on c g is a p group all groups in case a are examples in case b there exist examples for every p with s 1 in case c where g is a p group we also prove that the exponent of g is bounded in terms of p only each transitive group of exponent p is an example and it may be that these are the only examples in case c

dynamical symmetry approach to periodic hamiltonians we show that dynamical symmetry methods can be applied to hamiltonians with periodic potentials we construct dynamical symmetry hamiltonians for the scarf potential and its extensions using representations of su 1 1 and so 2 2 energy bands and gaps are readily understood in terms of representation theory we compute the transfer matrices and dispersion relations for these systems and find that the complementary series plays a central role as well as nonunitary representations

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

resonant tunnelling through short range singular potentials a three parameter family of point interactions constructed from sequences of symmetric barrier well barrier and well barrier well rectangles is studied in the limit when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite the zero range limit the limiting generalized potentials are referred to as the second derivative of dirac s delta function x with a renormalized coupling constant 0 or simply as like point interactions as a result a whole family of self adjoint extensions of the one dimensional schrodinger operator is shown to exist which results in full and partial resonant tunnelling through this class of singular potentials the resonant tunnelling occurs for countable sets of interaction strength values in the space which are the roots of several transcendental equations the comparison with the previous results for like point interactions is also discussed

generalized kronig penney hamiltonians abstract in this article we extend the study of generalized point interactions to the periodic case and we investigate analogues of the kronig penney model for the class of point interactions introduced in joint work with p chernoff in addition to periodic and potentials which follow as special cases several examples are studied in detail in general the models under consideration are shown to have purely absolutely continuous spectrum and in some cases all gaps in the spectrum close

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

the solvability of groups with nilpotent minimal coverings abstract a covering of a group is a finite set of proper subgroups whose union is the whole group a covering is minimal if there is no covering of smaller cardinality and it is nilpotent if all its members are nilpotent subgroups we complete a proof that every group that has a nilpotent minimal covering is solvable starting from the previously known result that a minimal counterexample is an almost simple finite group