# American Institute of Mathematical Sciences

November  2011, 10(6): 1687-1706. doi: 10.3934/cpaa.2011.10.1687

## Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

 1 Universidade Lusófona de Humanidades e Tecnologias, Av. Campo Grande 376, 1749-024 Lisboa, Portugal, Portugal 2 Dipartimento di Scienze Fisiche e Matematiche, Università dell'Insubria, via valleggio 11, I-22100 Como, Italy

Received  March 2010 Revised  February 2011 Published  May 2011

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(R^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(R^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.
Citation: Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687
##### References:
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Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems,, Math. Nachr., 282 (2009), 314. Google Scholar [14] C. Cacciapuoti, R. Carlone and R. Figari, Spin dependent point potentials in one and three dimensions,, J. Phys. A: Math. Gen., 40 (2007), 249. Google Scholar [15] J. W. Calkin, Abstract symmetric boundary conditions,, Trans. Am. Math. Soc., 45 (1939), 369. Google Scholar [16] A. Connes, "Noncommutative Geometry,", Academic Press, (1994). Google Scholar [17] C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics,", Birkh\, (2009). Google Scholar [18] N. C. Dias and J. N. Prata, Wigner functions with boundaries,, J. Math. Phys., 43 (2002), 4602. Google Scholar [19] N. C. Dias, A. Posilicano and J. N. Prata, in, in preparation., (). Google Scholar [20] N. C. Dias and J. N. Prata, Admissible states in quantum phase space,, Ann. Phys., 313 (2004), 110. Google Scholar [21] N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization",, Ann. Phys., 321 (2006), 495. Google Scholar [22] D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization,", World Scientific, (2000). Google Scholar [23] W. Faris, "Self-Adjoint Operators,", Lecture Notes in Mathematics {\bf 433}, 433 (1975). Google Scholar [24] D. Fairlie, The formulation of quantum mechanics in terms of phase space functions,, Proc. Camb. Phil. Soc., 60 (1964), 581. Google Scholar [25] B. Fedosov, A simple geometric construction of deformation quantization,, J. Diff. Geom., 40 (1994), 213. Google Scholar [26] R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models,, Phys. Rev., D 63 (2001). Google Scholar [27] P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization,, Am. J. Phys., 72 (2004), 924. Google Scholar [28] F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 2: Differential Operators and Mechanics'', (eds. V. Adamyan et al.), (2009), 81. Google Scholar [29] F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 79 (2008), 105. Google Scholar [30] V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations,", Kluver, (1991). Google Scholar [31] M. de Gosson and F. Luef, A new approach to the $\star$-genvalue equation,, Lett. Math. Phys., 85 (2008), 173. Google Scholar [32] G. Grubb, A characterization of the non local boundary value problems associated with an elliptic operator,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 425. Google Scholar [33] G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains,, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271. Google Scholar [34] C. Isham, Topological and global aspects of quantum theory,, in, (1984), 1059. Google Scholar [35] M. Kontsevich, Deformation quantization of Poisson manifolds I,, Lett. Math. Phys., 66 (2003), 157. Google Scholar [36] M. G. Kre\u\i n, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications I,, Mat. Sbornik N.S., 20 (1947), 431. Google Scholar [37] M. G. Kreĭn, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications II,, Mat. Sbornik N.S., 21 (1947), 365. Google Scholar [38] K. Kowalski, K. Podlaski and J. Rembieliński, Quantum mechanics of a free particle on a plane with an extracted point,, Phys. Rev. A, 66 (2002), 032118. Google Scholar [39] S. Kryukov and M. A. Walton, On infinite walls in deformation quantization,, Ann. Phys., 317 (2005), 474. Google Scholar [40] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II,, Ann. Institut Fourier, 11 (1961), 137. Google Scholar [41] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I,", Springer-Verlag, (1972). Google Scholar [42] J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications,", 2$^{nd}$ edition, (2000). Google Scholar [43] M. A. Naimark, "Theory of Linear Differential Operators,", Frederick Ungar Publishing Co., (1967). Google Scholar [44] J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren,, Math. Ann., 102 (1929), 49. Google Scholar [45] J. von Neumann, "Mathematische Grundlagen der Quantenmechanik,", Springer-Verlag, (1932). Google Scholar [46] A. Pinzul and A. Stern, Absence of the holographic principle in noncommutative Chern-Simons theory,, J. High Energy Phys., 0111 (2001). Google Scholar [47] A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications,, J. Funct. Anal., 183 (2001), 109. Google Scholar [48] A. Posilicano, Self-adjoint extensions by additive perturbations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1. Google Scholar [49] A. Posilicano, Self-adjoint extensions of restrictions,, Oper. Matrices, 2 (2008), 483. Google Scholar [50] A. Posilicano and L. Raimondi, Krein's resolvent formula for self-adjoint extensions of symmetric second order elliptic differential operators,, J. Phys. A: Math. Theor., 42 (2009). Google Scholar [51] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness,", Academic Press, (1975). Google Scholar [52] V. Ryzhov, A general boundary value problem and its Weyl function,, Opuscula Math., 27 (2007), 305. Google Scholar [53] N. Seiberg and E. Witten, String theory and noncommutative geometry,, J. High Energy Phys., 9909 (1999). Google Scholar [54] M. L. Višik, On general boundary problems for elliptic differential equations,, Trudy Mosc. Mat. Obsv., 1 (1952), 186. Google Scholar [55] B. Voronov, D. Gitman and I. Tyutin, Self-adjoint differential operators associated with self-adjoint differential expressions,, preprint, (). Google Scholar [56] J. Weidmann, "Linear Operators in Hilbert Spaces,", Springer-Verlag, (1980). Google Scholar [57] M. A. Walton, Wigner functions, contact interactions, and matching,, Ann. Phys., 322 (2007), 2233. Google Scholar [58] M. W. Wong, "Weyl Transforms,", Springer-Verlag, (1998). Google Scholar

show all references

##### References:
 [1] N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space,", Pitman, (1981). Google Scholar [2] S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics,", 2$^{nd}$ edition, (2005). Google Scholar [3] G. A. Baker, Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space,, Phys. Rev., 109 (1958), 2198. Google Scholar [4] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. F. Sternheimer, Deformation theory and quantization, I and II,, Ann. Phys., 111 (1978), 61. Google Scholar [5] F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential,, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011. Google Scholar [6] J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains,, J. Funct. Anal., 243 (2007), 536. Google Scholar [7] M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Spaces,", Reidel, (1987). Google Scholar [8] Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains,, J. Math. Phys., 48 (2007). Google Scholar [9] J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'', 2$^{nd}$ edition, (2008). Google Scholar [10] G. Bonneau, J. Faraut and G. Valent, Self-adjoint extensions of operators and the teaching of quantum mechanics,, Am. J. Phys., 69 (2001), 322. Google Scholar [11] A. Bracken, G. Cassinelli and J. Wood, Quantum symmetries and the Weyl-Wigner product of group representations,, preprint, (). Google Scholar [12] B. M. Brown, M. Marletta, S. Naboko and I. G. Wood, Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices,, J. Lond. Math. Soc., 77 (2008), 700. Google Scholar [13] B. M. Brown, G. Grubb and I. G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems,, Math. Nachr., 282 (2009), 314. Google Scholar [14] C. Cacciapuoti, R. Carlone and R. Figari, Spin dependent point potentials in one and three dimensions,, J. Phys. A: Math. Gen., 40 (2007), 249. Google Scholar [15] J. W. Calkin, Abstract symmetric boundary conditions,, Trans. Am. Math. Soc., 45 (1939), 369. Google Scholar [16] A. Connes, "Noncommutative Geometry,", Academic Press, (1994). Google Scholar [17] C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics,", Birkh\, (2009). Google Scholar [18] N. C. Dias and J. N. Prata, Wigner functions with boundaries,, J. Math. Phys., 43 (2002), 4602. Google Scholar [19] N. C. Dias, A. Posilicano and J. N. Prata, in, in preparation., (). Google Scholar [20] N. C. Dias and J. N. Prata, Admissible states in quantum phase space,, Ann. Phys., 313 (2004), 110. Google Scholar [21] N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization",, Ann. Phys., 321 (2006), 495. Google Scholar [22] D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization,", World Scientific, (2000). Google Scholar [23] W. Faris, "Self-Adjoint Operators,", Lecture Notes in Mathematics {\bf 433}, 433 (1975). Google Scholar [24] D. Fairlie, The formulation of quantum mechanics in terms of phase space functions,, Proc. Camb. Phil. Soc., 60 (1964), 581. Google Scholar [25] B. Fedosov, A simple geometric construction of deformation quantization,, J. Diff. Geom., 40 (1994), 213. Google Scholar [26] R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models,, Phys. Rev., D 63 (2001). Google Scholar [27] P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization,, Am. J. Phys., 72 (2004), 924. Google Scholar [28] F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 2: Differential Operators and Mechanics'', (eds. V. Adamyan et al.), (2009), 81. Google Scholar [29] F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 79 (2008), 105. Google Scholar [30] V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations,", Kluver, (1991). Google Scholar [31] M. de Gosson and F. Luef, A new approach to the $\star$-genvalue equation,, Lett. Math. Phys., 85 (2008), 173. Google Scholar [32] G. Grubb, A characterization of the non local boundary value problems associated with an elliptic operator,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 425. Google Scholar [33] G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains,, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271. Google Scholar [34] C. Isham, Topological and global aspects of quantum theory,, in, (1984), 1059. Google Scholar [35] M. Kontsevich, Deformation quantization of Poisson manifolds I,, Lett. Math. Phys., 66 (2003), 157. Google Scholar [36] M. G. Kre\u\i n, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications I,, Mat. Sbornik N.S., 20 (1947), 431. Google Scholar [37] M. G. Kreĭn, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications II,, Mat. Sbornik N.S., 21 (1947), 365. Google Scholar [38] K. Kowalski, K. Podlaski and J. Rembieliński, Quantum mechanics of a free particle on a plane with an extracted point,, Phys. Rev. A, 66 (2002), 032118. Google Scholar [39] S. Kryukov and M. A. Walton, On infinite walls in deformation quantization,, Ann. Phys., 317 (2005), 474. Google Scholar [40] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II,, Ann. Institut Fourier, 11 (1961), 137. Google Scholar [41] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I,", Springer-Verlag, (1972). Google Scholar [42] J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications,", 2$^{nd}$ edition, (2000). Google Scholar [43] M. A. Naimark, "Theory of Linear Differential Operators,", Frederick Ungar Publishing Co., (1967). Google Scholar [44] J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren,, Math. Ann., 102 (1929), 49. Google Scholar [45] J. von Neumann, "Mathematische Grundlagen der Quantenmechanik,", Springer-Verlag, (1932). Google Scholar [46] A. Pinzul and A. Stern, Absence of the holographic principle in noncommutative Chern-Simons theory,, J. High Energy Phys., 0111 (2001). Google Scholar [47] A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications,, J. Funct. Anal., 183 (2001), 109. Google Scholar [48] A. Posilicano, Self-adjoint extensions by additive perturbations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1. Google Scholar [49] A. Posilicano, Self-adjoint extensions of restrictions,, Oper. Matrices, 2 (2008), 483. Google Scholar [50] A. Posilicano and L. Raimondi, Krein's resolvent formula for self-adjoint extensions of symmetric second order elliptic differential operators,, J. Phys. A: Math. Theor., 42 (2009). Google Scholar [51] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness,", Academic Press, (1975). Google Scholar [52] V. Ryzhov, A general boundary value problem and its Weyl function,, Opuscula Math., 27 (2007), 305. Google Scholar [53] N. Seiberg and E. Witten, String theory and noncommutative geometry,, J. High Energy Phys., 9909 (1999). Google Scholar [54] M. L. Višik, On general boundary problems for elliptic differential equations,, Trudy Mosc. Mat. Obsv., 1 (1952), 186. Google Scholar [55] B. Voronov, D. Gitman and I. Tyutin, Self-adjoint differential operators associated with self-adjoint differential expressions,, preprint, (). Google Scholar [56] J. Weidmann, "Linear Operators in Hilbert Spaces,", Springer-Verlag, (1980). Google Scholar [57] M. A. Walton, Wigner functions, contact interactions, and matching,, Ann. Phys., 322 (2007), 2233. Google Scholar [58] M. W. Wong, "Weyl Transforms,", Springer-Verlag, (1998). Google Scholar
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