New approach to (quasi)-exactly solvable Schrodinger equations with a position-dependent effective mass

By using the point canonical transformation approach in a manner distinct from previous ones, we generate some new exactly solvable or quasi-exactly solvable potentials for the one-dimensional Schr\"odinger equation with a position-dependent effective mass. In the latter case, SUSYQM techniques provide us with some additional new potentials.Comment: 11 pages, no figur

A general scheme for the effective-mass Schrodinger equation and the generation of the associated potentials

A systematic procedure to study one-dimensional Schr\"odinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals a new and interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schr\"odinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. This provides a unified approach to many exact results known in the literature, as well as to a lot of new ones.Comment: 16 pages + 1 figure; minor changes + new "free-particle" problem; version published in Mod. Phys. Lett.

Well-posedness for degenerate third order equations with delay and applications to inverse problems

[EN] In this paper, we study well-posedness for the following third-order in time equation with delay <disp-formula idoperators defined on a Banach space X with domains D(A) and D(B) such that t)is the state function taking values in X and u(t): (-, 0] X defined as u(t)() = u(t+) for < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue-Bochner spaces Lp(T,X), periodic Besov spaces Bp,qs(T,X) and periodic Triebel-Lizorkin spaces Fp,qs(T,X). A novel application to an inverse problem is given.The first, second and third authors have been supported by MEC, grant MTM2016-75963-P. The second author has been supported by AICO/2016/30. The fourth author has been supported by MEC, grant MTM2015-65825-P.Conejero, JA.; Lizama, C.; Murillo-Arcila, M.; Seoane Sepúlveda, JB. (2019). Well-posedness for degenerate third order equations with delay and applications to inverse problems. Israel Journal of Mathematics. 229(1):219-254. https://doi.org/10.1007/s11856-018-1796-8S2192542291K. Abbaoui and Y. Cherruault, New ideas for solving identification and optimal control problems related to biomedical systems, International Journal of Biomedical Computing 36 (1994), 181–186.M. Al Horani and A. Favini, Perturbation method for first- and complete second-order differential equations, Journal of Optimization Theory and Applications 166 (2015), 949–967.H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Mathematische Nachrichten 186 (1997), 5–56.U. A. Anufrieva, A degenerate Cauchy problem for a second-order equation. A wellposedness criterion, Differentsial’nye Uravneniya 34 (1998), 1131–1133; English translation: Differential Equations 34 (1999), 1135–1137.W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Mathematische Zeitschrift 240 (2002), 311–343.W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proceedings of the Edinburgh Mathematical Society 47 (2004), 15–33.W. Arendt, C. Batty and S. Bu, Fourier multipliers for Holder continuous functions and maximal regularity, Studia Mathematica 160 (2004), 23–51.V. Barbu and A. Favini, Periodic problems for degenerate differential equations, Rendiconti dell’Istituto di Matematica dell’Università di Trieste 28 (1996), 29–57.A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, Vol. 10, A K Peters, Wellesley, MA, 2005.S. Bu, Well-posedness of second order degenerate differential equations in vector-valued function spaces, Studia Mathematica 214 (2013), 1–16.S. Bu and G. Cai, Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces, Israel Journal of Mathematics 212 (2016), 163–188.S. Bu and G. Cai, Well-posedness of second-order degenerate differential equations with finite delay in vector-valued function spaces, Pacific Journal of Mathematics 288 (2017), 27–46.S. Bu and Y. Fang, Periodic solutions of delay equations in Besov spaces and Triebel–Lizorkin spaces, Taiwanese Journal of Mathematics 13 (2009), 1063–1076.S. Bu and J. Kim, Operator-valued Fourier multipliers on periodic Triebel spaces, Acta Mathematica Sinica 21 (2005), 1049–1056.G. Cai and S. Bu, Well-posedness of second order degenerate integro-differential equations with infinite delay in vector-valued function spaces, Mathematische Nachrichten 289 (2016), 436–451.R. Chill and S. Srivastava, Lp-maximal regularity for second order Cauchy problems, Mathematische Zeitschrift 251 (2005), 751–781.R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs of the American Mathematical Society 166 (2003).O. Diekmann, S. A. van Giles, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Applied Mathematical Sciences, Vol. 110, Springer, New York, 1995.K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194, Springer, New York, 2000.M. Fabrizio, A. Favini and G. Marinoschi, An optimal control problem for a singular system of solid liquid phase-transition, Numerical Functional Analysis and Optimization 31 (2010), 989–1022.A. Favini and G. Marinoschi, Periodic behavior for a degenerate fast diffusion equation, Journal of Mathematical Analysis and Applications 351 (2009), 509–521.A. Favini and G. Marinoschi, Identification of the time derivative coefficients in a fast diffusion degenerate equation, Journal of Optimization Theory and Applications 145 (2010), 249–269.A. Favini and A. Yagi, Degenerate differential equations in Banach spaces, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 215, Marcel Dekker, New York, 1999.X. L. Fu and M. Li, Maximal regularity of second-order evolution equations with infinite delay in Banach spaces, Studia Mathematica 224 (2014), 199–219.G. C. Gorain, Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in Rn, Journal of Mathematical Analysis and Applications 319 (2006), 635–650.P. Grisvard, Équations différentielles abstraites, Annales Scientifiques de l’école Normale Superieure 2 (1969), 311–395.J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations, Journal of Mathematical Analysis and Applications 178 (1993), 344–362.Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, 1991.B. Kaltenbacher, I. Lasiecka and M. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Moore-Gibson-Thomson equation arising in high intensity ultrasound, Mathematical Models & Methods in Applied Sciences 22 (2012), 1250035.V. Keyantuo and C. Lizama, Fourier multipliers and integro-differential equations in Banach spaces, Journal of the London Mathematical Society 69 (2004), 737–750.V. Keyantuo and C. Lizama, Maximal regularity for a class of integro-differential equations with infinite delay in Banach spaces, Studia Mathematica 168 (2005), 25–50.V. Keyantuo, C. Lizama and V. Poblete, Periodic solutions of integro-differential euations in vector-valued function spaces, Journal of Differential Equations 246 (2009), 1007–1037.C. Lizama, Fourier multipliers and periodic solutions of delay equations in Banach spaces, Journal of Mathematical Analysis and Applications 324 (2006), 921–933.C. Lizama and V. Poblete, Maximal regularity of delay equations in Banach spaces, Studia Mathematica 175 (2006), 91–102.C. Lizama and R. Ponce, Periodic solutions of degenerate differential equations in vector valued function spaces, Studia Mathematica 202 (2011), 49–63.C. Lizama and R. Ponce, Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces, Proceedings of the Edinburgh Mathematical Society 56 (2013), 853–871.R. Marchand, T. Mcdevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in highintensity ultrasound: structural decomposition, spectral analysis, exponential stability, Mathematical Methods in the Applied Sciences 35 (2012), 1896–1929.V. Poblete, Solutions of second-order integro-differential equations on periodic Besov spaces, Proceedings of the Edinburgh Mathematical Society 50 (2007), 477–492.V. Poblete and J. C. Pozo, Periodic solutions of an abstract third-order differential equation, Studia Mathematica 215 (2013), 195–219.J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser, Heidelberg, 1993.G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev type Equations and Degenerate Semigroups of Operators, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2003.L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Mathematische Annalen 319 (2001), 735–758

Exact controllability of a linear Euler-Bernoulli panel

The problem of control of flexible vibrations of a flexible space structure (such as solar cell array) modelled by a thin uniform rectangular panel is considered here. The flexural vibrations of such a panel satisfies the one dimensional fourth order Petrowsky equation or Euler-Bernoulli equation. The panel is held at one end by a rigid hub and the other end is free. By attaching the hub to one side of the panel the dynamics create a non-standard hybrid system of equations. It is shown that the vibrations of the overall system can be driven to rest by means of an active boundary control force applied on the rigid hub only. Also an estimate of the minimum time of control is obtained. A closed form approximate result is constructed by Galerkin's residual technique to support and implement the method

Stability of the boundary stabilised internally damped wave equation y" + λy"'= c<SUP>2</SUP>(Δy +μΔy') in a bounded domain in R<SUP>n</SUP>

The boundary stabilisation of the problem satisfying the differential equation y"+λy"' = c2(Δy +μΔy'), 0 &lt; λ &lt; μ, in a bounded domain Ω in Rn with smooth boundary Γ is studied. Such equations arise in the vibrations of flexible structures possessing internal material damping and modeled by the "Standard linear model" of viscoelasticity.Exponential energy decay rate is obtained for the solution of the above problem subject to mixed boundary conditions

Exact controllability and boundary stabilization of torsional vibrations of an internally damped flexible space structure

In this paper, we study the exact controllability and boundary stabilization of the torsional vibrations of a flexible space structure (such as a solar cell array) modeled by a rectangular panel, incorporating the material damping of the structure. The panel is hoisted at one end by a rigid hub and the other end is totally free. For the attachment of this hub on one side of the panel, the hub dynamics leads to a nonstandard boundary condition. To incorporate internal damping of the material, we assume Voigt-type viscoelasticity of the structure. Exact controllability theory is established using the Hilbert uniqueness method by means of a control torque applied only on the rigid hub of the panel. At the same time, uniform exponential energy decay rate is obtained directly for the solution of this problem

Correlation between Luders band formation and precipitation kinetics behaviour during the industrial processing of interstitial free high strength steels

Luders band formation in steels is critical to surface finish during automobile panel manufacturing. This research reports on the problem of Udders band formation in interstitial free high strength steel compositions (IFHS-steels). The study investigates the effect of chemical composition and processing parameters on the formation of Udders bands in IFHS-steels. It correlates the problem of Udders band formation with precipitation kinetics behaviour during the industrial processing of IFHS-steels. Four different compositions viz. Ti-stabilized, Ti-Nb stabilized, low Ti-low Nb, and high Ti-low Nb with high Al were investigated. Annealing parameters were similar to industrial practice followed for batch and continuous annealing lines in steel manufacturing plants. Stabilized IFHS-steel compositions possessing excess of stabilizing elements (Ti, Nb, etc.) for stabilization of interstitial elements (C, N) also showed the problem of Udders band formation. The new type of IFHS composition containing high Al, investigated in this research, showed no Luders band formation during batch annealing cycles along with adequate mechanical properties (YS: 190-202 MPa; Delta r-value 1.57; Delta r-value: 0.25). Thus, steel compositions with high Al content processed through batch annealing cycle offer a practical solution to the problem of Udders band formation in IFHS-steels. (C) 2018 Politechnika Wroclawska. Published by Elsevier B.V. All rights reserved