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Schrödinger Operator (schrödinger + operator)
Selected AbstractsPRICING EQUITY DERIVATIVES SUBJECT TO BANKRUPTCYMATHEMATICAL FINANCE, Issue 2 2006Vadim Linetsky We solve in closed form a parsimonious extension of the Black,Scholes,Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets. [source] Regularization of the non-stationary Schrödinger operatorMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2009Paula Cerejeiras Abstract In this paper we prove a Lp -decomposition where one of the components is the kernel of a first-order differential operator that factorizes the non-stationary Schrödinger operator ,,,i,t. Copyright © 2008 John Wiley & Sons, Ltd. [source] Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in ,2MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2003Bénédicte Alziary Abstract The zero set {z,,2:,(z)=0} of an eigenfunction , of the Schrödinger operator ,V=(i,+A)2+V on L2(,2) with an Aharonov,Bohm-type magnetic potential is investigated. It is shown that, for the first eigenvalue ,1 (the ground state energy), the following two statements are equivalent: (I) the magnetic flux through each singular point of the magnetic potential A is a half-integer; and (II) a suitable eigenfunction , associated with ,1 (a ground state) may be chosen in such a way that the zero set of , is the union of a finite number of nodal lines (curves of class C2) which emanate from the singular points of the magnetic potential A and slit the two-dimensional plane ,2. As an auxiliary result, a Hardy-type inequality near the singular points of A is proved. The C2 differentiability of nodal lines is obtained from an asymptotic analysis combined with the implicit function theorem. Copyright © 2003 John Wiley & Sons, Ltd. [source] Unitary reduction for the two-dimensional Schrödinger operator with strong magnetic fieldMATHEMATISCHE NACHRICHTEN, Issue 4 2009Andrei EcksteinArticle first published online: 19 MAR 200 Abstract The spectrum of the two-dimensional Schrödinger operator with strong magnetic field and electric potential is contained in a union of intervals centered on the Landau levels. The study of the spectrum in any of these intervals is reduced by unitary equivalence to the study of a one-dimensional operator. We give a precise description of this operator in the case when the electric potential is periodic and analytic in a strip (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Weak asymptotics of the spectral shift functionMATHEMATISCHE NACHRICHTEN, Issue 11 2007Vincent Bruneau Abstract We consider the three-dimensional Schrödinger operator with constant magnetic field of strength b > 0, and with smooth electric potential. The weak asymptotics of the spectral shift function with respect to b , +, is studied. First, we fix the distance to the Landau levels, then the distance to Landau levels tends to infinity as b , +,. In particular we give explicitly the leading terms in the asymptotics and in some case we obtain full asymptotics expansions. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Three circles theorems for Schrödinger operators on cylindrical ends and geometric applicationsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2008Tobias H. Colding We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N × [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres ,,n for n , 1 and all Zoll surfaces. Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.© 2007 Wiley Periodicals, Inc. [source] Analytic smoothing effect for the Schrödinger equation with long-range perturbationCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2006Andre Martinez We study the microlocal analytic singularity of solutions to the Schrödinger equation with analytic coefficients. Using microlocal weight estimates developed for estimating phase space tunneling, we prove microlocal smoothing estimates that generalize results by Robbiano and Zuily. We show that the exponential decay of the initial state in a cone in the phase space implies microlocal analytic regularity of the solution at a positive time. We suppose the Schrödinger operator is a long-range-type perturbation of the Laplacian, and we employ positive commutator-type estimates to prove the smoothing property. © 2005 Wiley Periodicals, Inc. [source] | ||||||||||