June  2015, 4(2): 233-240. doi: 10.3934/eect.2015.4.233

Relating systems properties of the wave and the Schrödinger equation

1. 

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands

2. 

FEMTO-ST AS2M, Université de Franche Comté/CNRS/ENSMM/UTBM, 24 rue Alain Savary, 25000 Besançon, France, France

Received  May 2014 Revised  September 2014 Published  May 2015

In this article we show that systems properties of the systems governed by the second order differential equation $\frac{d^{2}w}{dt^{2}}=-A_{0}w$ and the first order differential equation $\frac{dz}{dt}=iA_{0}z$ are related. This can be used to show that, for instance, exact observability of the $N$-dimensional wave equation implies the similar property for the $N$-dimensional Schrödinger equation.
Citation: Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233
References:
[1]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[2]

V. Duindam, A. Macchelli, S. Stramigioli and H. Bruyninckx, Modelling and Control of Complex Physical Systems - The Port-Hamiltonian Approach, Springer-Verlag, 2009. doi: 10.1007/978-3-642-03196-0.  Google Scholar

[3]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[4]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749, Springer-Verlag, Berlin, 1979.  Google Scholar

[5]

D. J. Griffiths, Introduction to Quantum Mechanics, Pearson, Prentice Hall, 2005. Google Scholar

[6]

B. Jacob and H. Zwart, Linear port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, 223, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.  Google Scholar

[7]

L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, Journal of Functional Analysis, 218 (2005), 425-444. doi: 10.1016/j.jfa.2004.02.001.  Google Scholar

[8]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[9]

A. J. van der Schaft and B. M. Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, J. of Geometry and Physics, 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3.  Google Scholar

[10]

H. Zwart, Sufficient conditions for admissibility, Systems & Control Lett., 54 (2005), 973-979. doi: 10.1016/j.sysconle.2005.02.009.  Google Scholar

show all references

References:
[1]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[2]

V. Duindam, A. Macchelli, S. Stramigioli and H. Bruyninckx, Modelling and Control of Complex Physical Systems - The Port-Hamiltonian Approach, Springer-Verlag, 2009. doi: 10.1007/978-3-642-03196-0.  Google Scholar

[3]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[4]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749, Springer-Verlag, Berlin, 1979.  Google Scholar

[5]

D. J. Griffiths, Introduction to Quantum Mechanics, Pearson, Prentice Hall, 2005. Google Scholar

[6]

B. Jacob and H. Zwart, Linear port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, 223, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.  Google Scholar

[7]

L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, Journal of Functional Analysis, 218 (2005), 425-444. doi: 10.1016/j.jfa.2004.02.001.  Google Scholar

[8]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[9]

A. J. van der Schaft and B. M. Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, J. of Geometry and Physics, 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3.  Google Scholar

[10]

H. Zwart, Sufficient conditions for admissibility, Systems & Control Lett., 54 (2005), 973-979. doi: 10.1016/j.sysconle.2005.02.009.  Google Scholar

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