conditionally solvable path integral problems ii natanzon potentials new classes of exactly solvable potentials are discussed within the path integral formalism they are constructed from the hypergeometric and confluent natanzon potentials respectively it is found that they allow incorporation of four free parameters which give rise to fractional power behaviour long range and strongly anharmonic terms we find six different classes of such potentials

acoustic propagation in a refracting ocean waveguide with an irregular interface abstract a method is derived for computing the acoustic field in a cylindrically symmetric ocean with an irregular water sediment interface and a general depth dependent sound speed profile the approach is based on a stepwise coupled mode solution which was originally developed for a homogeneous water column the extension to a general sound speed profile is carried out with the aid of galerkin s method numerical examples are given for a shallow water waveguide with an upward refracting sound speed profile the first example shows that propagation loss is significantly greater in a rough bottomed waveguide than in a smooth bottomed waveguide in the second example it is shown that upward refraction reduces the effect of bottom roughness on sound propagation by reducing the insonification of the rough water sediment interface

on the gibbsian thermostatics of mixtures in 1970 noll n developed a modern version of the elements of gibbsian thermostatics for describing the equilibrium of mixtures an important aspect of noll s work is that he considered the old fundamental basis of thermostatics within a modern variational setting and with mild hypotheses gave proofs to some central theorems in the subject noll considered the stable states of a mixture to be those that maximize an entropy functional over a certain convex set of measures and for the analysis of this variational problem he employed an elegant albeit not standard mathematical theory while his work could be considered a precursor to much of the present day studies on coexistent phases in continuum thermomechanics nevertheless it has remained relatively and undeservedly obscure perhaps because of its unusual mathematical setting

tunneling through a singular potential barrier quantum tunneling of a nonrelativistic particle through a singular potential barrier v is studied on the line the hamiltonian is a self adjoint extension of the operator h1 d2 dx2 v x if h1 is essentially self adjoint on its natural domain the tunneling is forbidden for a class of potentials that includes all semiclassically impenetrable barriers if h1 is not essentially self adjoint the friedrichs extension of h1 yields no tunneling for another class of potentials which again includes the semiclassically impenetrable ones in general the occurrence of tunneling is not excluded and depends on the self adjoint extension we choose as the hamiltonian of our problem as an example we evaluate the transmission coefficient for all self adjoint extensions of the operator h1 referring to v x gx 2 with 0 g 3 4

on solvable potentials related to so 2 2 ii natanzon potentials general natanzon potentials related to the so 2 2 group are studied the s matrices for systems under consideration are related to intertwining operators between weyl equivalent most degenerate principal series representations of so 2 2

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

a note on differential equations of coupled mode propagation in fluids there is a significant discrepancy between bathymetric contributions to ocean acoustic mode coupling matrix elements derived by various authors in this paper the discrepancy is shown to be due to incorrectness of the traditional approach to mode coupling theory originated by a d pierce j acoust soc am 37 19 27 1965 and d m milder j acoust soc am 46 1259 1263 1969 in the presence of a sloping interface or a range dependent rigid boundary physically their approach is consistent with neither energy conservation nor reciprocity mathematically the problem with the traditional approach consists of formal unjustified term by term differentiation of a series representing the acoustic pressure in terms of local normal modes the series is poorly convergent when a nonhorizontal interface of fluids with distinct densities is present a rigorous derivation of the acoustic mode coupling equations in media with piecewise continuous density and sound speed is given here on the basis of the reciprocity principle necessary corrections to the results presented recently by b e mcdonald j acoust soc am 100 219 224 1996 are discussed

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

algebraic solution for the natanzon hypergeometric potentials an algebraic method based on a strategy that makes use of a realization of the algebra so 2 1 in terms of differential operators is used to solve the bound state problem for the most general natanzon potentials for which the schrodinger equation can be reduced to hypergeometric form hence hypergeometric potentials

positon solutions of the sine gordon equation the concept of positons is introduced for the sine gordon equation exploiting its covariance under darboux transformations positon solutions represent potentials in the associated linear system corresponding to a vanishing reflection coefficient and a transmission coefficient identically one the mutual interaction of positons and the interaction of positons with solitons is considered

on scattering systems related to the so 2 1 group ii we study a class of solvable hamiltonians h which have strongly broken potential group structure however the scattering matrices of the systems under consideration are also related to the intertwining operator of so 2 1

mapping of shape invariant potentials under point canonical transformations the authors give explicit point canonical transformations which map twelve types of shape invariant potentials which are known to be exactly solvable into two potential classes the eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively

quantum motion on a half line connected to a plane in this paper the free motion of a particle on a manifold that consists of a one dimensional and a two dimensional part connected in one point is discussed the class of admissible hamiltonians is found using the theory of self adjoint extensions particular attention is paid to those hamiltonians that allow the particle to pass through the point singularity the reflection coefficient and other quantities characterizing scattering on the connection point are calculated a possible application is also discussed

complex potential description of the damped harmonic oscillator the multidimensional damped harmonic oscillator is treated by means of a non self adjoint hamiltonian with complex potential the propagator referring to the evolution semigroup is evaluated from the lie trotter formula the one dimensional case is discussed in detail with the following results a the nondamped limit gives the correct propagator including the maslov phase factor b for some initial conditions the classical limit of the solution can differ from the behavior of the classical damped oscillator the difference being negligible in the case of weak damping and c the point spectrum of the considered pseudo hamiltonian is found

supertransparent potentials for the dirac equation an extension of the classical darboux transformations is applied to the one dimensional dirac equation in order to construct von neumann wigner potentials allowing embedded eigenvalues these potentials lead to a novel type of scattering problem with a trivial s matrix composed of vanishing reflection coefficients and a trivial transmission coefficient related topics like the underlying symmetry of the dirac equation and the connection with positon solutions of nonlinear evolution equations are discussed

the new integrable deformations of a short pulse equation and sine gordon equation and their solutions we first derive an integrable deformed hierarchy of a short pulse equation and its lax representation then we concentrate on the solution of the integrable deformed short pulse equation idspe by proposing a generalized reciprocal transformation we find a new integrable deformed sine gordon equation idsge and its lax representation the multisoliton solutions negaton solutions and positon solutions for the idsge and the n loop soliton solutions n negaton and n positon solutions for the idspe are presented in the reduced case the new n positon solutions and n negaton solutions for the short pulse equation are obtained

supersymmetry operator transformations and exactly solvable potentials a large class of potentials can be solved algebraically by using supersymmetry and shape invariance the authors apply operator transformations f transformations to these algebraically solvable problems to obtain a larger class of solvable potentials the natanzon class of potentials which are not shape invariant the important condition which is independent of supersymmetry for finding new potentials from old ones using operator transformations is that the resulting schrodinger equation has a potential which does not depend on the state as a special case of the f transformation they rederive the previously known connection between the 3d harmonic oscillator the hydrogen atom and the morse potential they also discuss the lack of commutivity of susy and the f transformations

path integral treatment for the one dimensional natanzon potentials the green s function relative to the one dimensional natanzon potentials is exactly calculated by the path integral approach the energy spectrum the wave functions and the scattering function s are deduced

on the application of the reciprocity principle to coupled mode propagation l the reciprocity principle is applied to the derivation of coefficients describing the coupling of acoustic normal modes in waveguides characterized by range dependent density gradients and discontinuities the requirement of reciprocity alone is found insufficient to determine the mode coupling coefficients the stronger requirement of continuity of pressure and particle velocity at horizontal discontinuities of the waveguide is shown to provide coupling coefficients in agreement with those of fawcett j acoust soc am 92 290 295 1992

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

on solvable potentials related to so 2 2 scattering problems in one dimension associated with the so 2 2 group are studied the s matrices for the systems under consideration are calculated by using the theory of intertwining operators for semisimple lie groups the wave functions are expressed in terms of matrix elements of principal series representation of the so 2 1 group

special measure function separability and reflectionless potentials a relationship is found between the separability of the spectral measure function and the poschl teller pt reflectionless potential rlp the limits on this relationship are briefly discussed

ambiguities and almost reflectionless potentials this paper considers possible ambiguities associated with perturbations on reflectionless solutions of the gel fand levitan equation by considering perturbations on the spectral measure function and the potential it is shown that no ambiguity exists with respect to spectral measure perturbations however it is shown that first order effects in the potential can lead to second order effects in the spectral measure function thereby demonstrating an ambiguity with respect to first order approximations

the periodic extension of stepwise coupled modes the method of stepwise coupled modes for studying acoustic propagation is extended to apply to periodic repetitions of a sequence of depth variations the extension is based on an eigenvalue analysis of the propagator matrix for one period of the depth sequence numerical examples are given which compare propagation loss in a waveguide with periodic bottom roughness to that in a waveguide with a smooth bottom it is shown that propagation loss is significantly greater for the rough bottom waveguide the increased loss is attributed to scattering of energy out of the waveguide

potentials of the heun class we review different methods of generating potentials such that the one dimensional schrodinger equation odse can be transformed into the hypergeometric equation we compare our results with previous studies and complement the subject with new findings our main result is to derive new classes of potentials such that the odse can be transformed into the heun equation and its confluent cases the generalized heun equation is also considered

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

a method to construct reflectionless potentials this paper gives a method with an example of constructing a potential which is near reflectionless for some range of energies the assumed structure of the desired reflection coefficient r k is known function that is filtered by an inverted gaussian this gives an r k that is zero at particular energy and is exponentially near zero in a surrounding range

the solutions of the nls equations with self consistent sources we construct the generalized darboux transformation with arbitrary functions at time t for the akns equation with self consistent sources aknsescs which in contrast with the darboux transformation for the akns equation provides a non auto backlund transformation between two aknsescss with different degrees of sources the formula for n times repeated generalized darboux transformation is proposed by reduction the generalized darboux transformation with arbitrary functions at time t for the nonlinear schrodinger equation with self consistent sources nlsescs is obtained and enables us to find the dark soliton bright soliton and positon solutions for nls escs and nls escs the properties of these solution are analysed

coupled mode perturbation theory of range dependence the conventional coupled mode solution is combined with perturbation theory to give a fast accurate range dependent normal mode solution for deep water acoustic propagation perturbation theory is used to calculate the new normal modes at each range step the new modes are obtained as a linear combination of the modes for the previous step without requiring a numerical solution of the depth separated wave equation the process may be repeated for many steps and yields normal modes and eigenvalues which are sufficiently accurate for solution of practical problems in deep water the method is applied to long range propagation through oceanic fronts

group theoretical approach to reflectionless potentials we examine the general form of potentials with zero reflection coefficient in one dimensional hamiltonians connected with casimir invariants of non compact groups

negaton and complexiton solutions of sine gordon equation abstract the negaton and complexiton solutions are introduced for the sine gordon equation by means of the darboux transformation dt the mutual interactions of multinegaton and interactions of negaton soliton and negaton positon are discussed in detail the superposition formulae for negations and complexitons are constructed noticing the relation of the sine gordon equation and the sinh gordon equation we also obtain the real valued singular negaton and complexiton solution for the sinh gordon equation

positon and negaton solutions of the mkdv equation with self consistent sources by using the generalized binary darboux transformation with arbitrary functions at time t for the negative modified kdv equation with self consistent sources mkdv escss which offers a non auto backlund transformation between two mkdv escss with different degrees of sources some new solutions for the mkdv escss such as singular multisoliton multipositon multinegaton multisoliton positon multisoliton negaton and multipositon negaton solutions are found by taking the special initial solution for auxiliary linear problems and the special functions of t time at the same time the properties of these solutions are analyzed in detail

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

infinite dimensional estabrook wahlquist prolongations for the sine gordon equation we are looking for the universal covering algebra for all symmetries of a given partial differential equation pde using the sine gordon equation as a typical example for a nonevolution equation for nonevolution equations estabrook wahlquist prolongation structures for nonlocal symmetries depend on the choice of a specific subideal of the contact module to define the pde for each inequivalent such choice we determine the most general solution of the prolongation equations as subalgebras of the infinite dimensional algebra of all vector fields over the space of nonlocal variables associated with the pde in the style of vinogradov covering spaces we show explicitly how previously known prolongation structures known to lie within the kac moody algebra a 1 1 are special cases of these general solutions although we are unable to identify the most general solutions with previously studied algebras we show the existence of gauge transformations between prolongation structures viewed as determinin

the potential group approach and hypergeometric differential equations this paper proposes a generalized realization of the potential groups so 2 1 and so 2 2 to describe the confluent hypergeometric and the hypergeometric equations respectively it implies that the classes of schrodinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure

negatons positons rational like solutions and conservation laws of the korteweg de vries equation with loss and non uniformity terms solitons negatons positons rational like solutions and mixed solutions of a non isospectral equation the korteweg de vries equation with loss and non uniformity terms are obtained through the wronskian technique the non isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using mathematica we also derive an infinite number of conservation laws of the equation

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

generic aspects of convexification with applications to thermodynamic equilibrium this work presents a mathematical analysis of the process of convexification of a smooth function based on singularity theory the theory developed is applied to the problem of thermodynamic phase equilibrium the central notion introduced here is that of phase simplex which we use to discuss phase equilibrium and phase transition in an abstract framework one of the by products of the results of this paper is a rigorous proof of gibbs phase rule for multicomponent systems in which a well accepted mathematical notion of genericity is used to account for ostensible exceptions to the rule also many other features known from theoretical or experimental thermodynamics can be rediscovered through purely mathematical arguments from the notions introduced here such features include among other things the existence of saturation pressures the existence of multiple or critical points and the existence of spontaneous or continuous changes in the composition of the phases at phase transitions

exact discrete nonlocal boundary conditions for high order pade parabolic equations the nonlocal boundary conditions nlbcs for high order finite difference parabolic equations pes are obtained by z transformation of the discrete pe in a homogeneous medium the considered nlbcs include the free space radiation condition possibly with a density jump at the nlbc interface the nlbc at an arbitrary impedance interface and the nlbcs for sources and the starting field beyond the nlbc interface the derivation is presented for the multiterm pade pe model owwe one way wave equation but the developed technique is applicable to a broad class of finite difference pes the obtained nlbcs are exact for the given finite difference scheme they are not limited by the order of pade approximation or by the pe steps in range and depth the nlbc convolution coefficients are calculated by the numerical inverse z transformation the accuracy and performance of the algorithm are analyzed for several benchmark problems the solution is robust for the range steps over 25 wavelengths and or the approximations up to the tenth order the nlbcs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution precomputation of the nlbc coefficients may be required in time demanding applications the results are compared with earlier proposed nlbcs for high order pes

connection conditions and the spectral family under singular potentials to describe a quantum system whose potential is divergent at one point one must provide proper connection conditions for the wavefunctions at the singularity generalizing the scheme used for point interactions in one dimension we present a set of connection conditions which are well defined even if the wavefunctions and or their derivatives are divergent at the singularity our generalized scheme covers the entire u 2 family of quantizations self adjoint hamiltonians admitted for the singular system we use this scheme to examine the spectra of the coulomb potential v x e2 x and the harmonic oscillator with square inverse potential v x m 2 2 x2 g x2 and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra we further show that for any parity invariant singular potential v x v x the spectrum is determined solely by the eigenvalues of the characteristic matrix u u 2

consistent coupled mode theory of sound propagation for a class of nonseparable problems this article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of a d pierce j acoust soc am 37 19 27 1965 and d m milder j acoust soc am 46 1259 1263 1969 are applied to propagation problems involving range dependent boundaries this boundary condition approximation requires that the depth derivative rather than the normal derivative of the field be continuous across a sloping interface the approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory this article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy it is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope moreover the corrections are not prohibitive in terms of added computational complexity numerical examples are presented which illustrate the nonconservation of energy effect and the corrections to the theory

environmentally adaptive wedge modes the notion of environmentally adaptive curvilinear uncoupled normal modes is developed by writing the helmholtz equation in curvilinear coordinates the depth dependent vertical scaling function provides some coordinate freedom missing from the cartesian formulation this freedom permits adaptation of the normal mode approach to variable depth and variable sound speed problems for environments that change slowly in range a w k b approximation to the eigenfunctions shows modal decoupling can be realized when the vertical scaling function is chosen equal to the reciprocal of the depth dependent vertical wave number of the lowest mode numerical simulations demonstrate that a requirement for constant inter modal ratios of the vertical wave number is valid throughout several wedges with different bottom boundary conditions the horizontal refraction of such modes agrees closely with analytic solutions for a wide angle wedge the acoustic field agrees qualitatively well with the benchmark solution of dean and buckingham j acoust soc am 93 1319 1328 1993

algebraic approach to non central potentials an algebraic approach to class of separable non central hamiltonians is presented we show that the bound states of quantum systems under consideration are described by unitary representations of the algebra

positons for the toda lattice and related spectral problems the concept of positons is introduced for the toda lattice equation it is shown that these multiparametric oscillating and slowly decaying solutions when inserted as potentials in the finite difference schrodinger equation of the corresponding lax pair lead to a trivial s matrix the resulting eigenvalues are embedded in the continuous spectrum of this infinite jacobi matrix the singularities connected with the one positon solution are discussed and compared with those of the positons of continuous integrable models the special features of the soliton positon interaction are analysed

pt symmetric potentials and the so 2 2 algebra starting from a differential realization of the generators of the so 2 2 algebra we connect the eigenvalue equation of the casimir invariant either with the hypergeometric equation or the schr odinger equation in the latter case we consider problems for whichso 2 2 appears as a potentia la lgebra connecting states with the same energy in different potentials we analyse the role of the two so 2 1 subalgebras and point out their importance for pt symmetric problems where the doubling of bound states is known to occur we present two mechanisms for this and illustrate them with the example of the scarf and the p oschl teller ii potentials we also analyse scattering states transmission

asymptotics of the multipositon soliton τ function of the korteweg de vries equation and the supertransparency the asymptotics of the function generating the n positon m soliton solution of the korteweg de vries equation is calculated this allows to prove that solitons do not experience any phase shift in a collision with positons the positons themselves survive mutual collisions unchanged this phenomenon is called the supertransparency or super reflectionless property of the multipositon solutions the linear aspects of this phenomenon are also discussed it is demonstrated that positons acquire two additional but always finite phase shifts in collision with solitons this result admits a natural extension to any number of solitons and positons in interaction

on a class of solutions of the sine gordon equation in this paper we consider solutions of the sine gordon equation that are nonlinear analogs of solutions of the klein gordon equation just like the latter may be combined together by means of the fourier integral to form more complicated solutions of the klein gordon equation the former may also interact to produce more complicated solutions of the sine gordon equation by means of a nonlinear analog of the fourier integral also derived in the paper

complexitons of the modified kdv equation by darboux transformation abstract darboux transformation dt a comprehensive approach to construct the explicit solutions of the nonlinear evolutionary equation is applied to construct the new complexiton solution of the negative mkdv equation we find that the complexiton solution is related to the single complex spectral parameter and that it is completely different from the breather solution for the positive mkdv equation consequently we generalize the concept of complexiton of the kdv equation to the mkdv equation in addition the relationship of the spectral parameter and the known solutions of the mkdv equation is clarified and the multi complexiton solution multi complexiton positon multi complexiton negaton and multi complexiton soliton solutions are obtained in the uniform manner by dt the interaction of complexiton and soliton is also discussed in detail it is shown that the new complexiton and soliton remain unchanged except for phase shifts after their interaction at the same time the superreflectionless property of one positon potential for the mkdv equation is shown in detail

models with sl 2 c symmetry and their s matrices we study quantum systems whose scattering modes are governed by unitary representation of the algebra the s matrices of the systems under consideration are defined from intertwining relations for the weyl equivalent representations of the group sl 2 c or its lie algebra