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:
[1]

N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space," Pitman, Boston, 1981. Google Scholar

[2]

S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics," 2nd edition, Am. Math. Soc., Providence, Rhode Island, 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), 2198-2206.  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), 61-110; Ann. Phys., 110 (1978), 111-151.  Google Scholar

[5]

F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011-1014.  Google Scholar

[6]

J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.  Google Scholar

[7]

M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Spaces," Reidel, Dordrecht, Holland, 1987. Google Scholar

[8]

Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains, J. Math. Phys., 48 (2007), 082108.  Google Scholar

[9]

J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'' 2nd edition, Springer-Verlag, Berlin, 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), 322-331. 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), 700-718.  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), 314-347.  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), 249-261.  Google Scholar

[15]

J. W. Calkin, Abstract symmetric boundary conditions, Trans. Am. Math. Soc., 45 (1939), 369-342.  Google Scholar

[16]

A. Connes, "Noncommutative Geometry," Academic Press, New-York, 1994.  Google Scholar

[17]

C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics," Birkhäuser, Basel, 2009.  Google Scholar

[18]

N. C. Dias and J. N. Prata, Wigner functions with boundaries, J. Math. Phys., 43 (2002), 4602-4627.  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) 110-146.  Google Scholar

[21]

N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization", Ann. Phys., 321, (2006) 495-502.  Google Scholar

[22]

D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization," World Scientific, Singapore, 2000.  Google Scholar

[23]

W. Faris, "Self-Adjoint Operators," Lecture Notes in Mathematics 433, Springer-Verlag, Berlin, 1975.  Google Scholar

[24]

D. Fairlie, The formulation of quantum mechanics in terms of phase space functions, Proc. Camb. Phil. Soc., 60 (1964), 581-586.  Google Scholar

[25]

B. Fedosov, A simple geometric construction of deformation quantization, J. Diff. Geom., 40 (1994), 213-238.  Google Scholar

[26]

R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models, Phys. Rev., D 63 (2001), 105014.  Google Scholar

[27]

P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization, Am. J. Phys., 72 (2004), 924-933. 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.), Birkhäuser Verlag, 2009, 81-113.  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 "Perspectives in Partial Differential Equations, Harmonic Analysis and Applications'' (Proc. Sympos. Pure Math., 79), Amer. Math. Soc., Providence, Rhode Island, 2008, 105-173.  Google Scholar

[30]

V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations," Kluver, Dordrecht, 1991.  Google Scholar

[31]

M. de Gosson and F. Luef, A new approach to the $\star$-genvalue equation, Lett. Math. Phys., 85 (2008), 173-183.  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), 425-513.  Google Scholar

[33]

G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271-297.  Google Scholar

[34]

C. Isham, Topological and global aspects of quantum theory, in "Relativity, groups and topology II, Les Houches Session XL" (eds. B.S. DeWitt and R. Stora), North-Holland, Amsterdam, 1984, 1059-1290.  Google Scholar

[35]

M. Kontsevich, Deformation quantization of Poisson manifolds I, Lett. Math. Phys., 66 (2003), 157-216.  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), 431-495, (Russian).  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), 365-404, (Russian).  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), 032118-1-9.  Google Scholar

[39]

S. Kryukov and M. A. Walton, On infinite walls in deformation quantization, Ann. Phys., 317 (2005), 474-491.  Google Scholar

[40]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II, Ann. Institut Fourier, 11 (1961), 137-178.  Google Scholar

[41]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I," Springer-Verlag, Berlin, 1972. Google Scholar

[42]

J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications," 2nd edition, Cambridge University Press, Cambridge, 2000.  Google Scholar

[43]

M. A. Naimark, "Theory of Linear Differential Operators," Frederick Ungar Publishing Co., New York, 1967.  Google Scholar

[44]

J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren, Math. Ann., 102 (1929), 49-131. Google Scholar

[45]

J. von Neumann, "Mathematische Grundlagen der Quantenmechanik," Springer-Verlag, Berlin, 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), Paper 23, 14 pp.  Google Scholar

[47]

A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), 109-147.  Google Scholar

[48]

A. Posilicano, Self-adjoint extensions by additive perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1-20.  Google Scholar

[49]

A. Posilicano, Self-adjoint extensions of restrictions, Oper. Matrices, 2 (2008), 483-506.  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), 015204.  Google Scholar

[51]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness," Academic Press, London, 1975.  Google Scholar

[52]

V. Ryzhov, A general boundary value problem and its Weyl function, Opuscula Math., 27 (2007), 305-331.  Google Scholar

[53]

N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys., 9909 (1999), Paper 32, 93 pp.  Google Scholar

[54]

M. L. Višik, On general boundary problems for elliptic differential equations, Trudy Mosc. Mat. Obsv., 1 (1952), 186-246 (Russian); translated in Amer. Math. Soc. Trans., 24 (1963), 107-172.  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, Berlin, 1980.  Google Scholar

[57]

M. A. Walton, Wigner functions, contact interactions, and matching, Ann. Phys., 322 (2007), 2233-2248.  Google Scholar

[58]

M. W. Wong, "Weyl Transforms," Springer-Verlag, Berlin, 1998.  Google Scholar

show all references

References:
[1]

N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space," Pitman, Boston, 1981. Google Scholar

[2]

S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics," 2nd edition, Am. Math. Soc., Providence, Rhode Island, 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), 2198-2206.  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), 61-110; Ann. Phys., 110 (1978), 111-151.  Google Scholar

[5]

F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011-1014.  Google Scholar

[6]

J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.  Google Scholar

[7]

M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Spaces," Reidel, Dordrecht, Holland, 1987. Google Scholar

[8]

Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains, J. Math. Phys., 48 (2007), 082108.  Google Scholar

[9]

J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'' 2nd edition, Springer-Verlag, Berlin, 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), 322-331. 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), 700-718.  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), 314-347.  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), 249-261.  Google Scholar

[15]

J. W. Calkin, Abstract symmetric boundary conditions, Trans. Am. Math. Soc., 45 (1939), 369-342.  Google Scholar

[16]

A. Connes, "Noncommutative Geometry," Academic Press, New-York, 1994.  Google Scholar

[17]

C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics," Birkhäuser, Basel, 2009.  Google Scholar

[18]

N. C. Dias and J. N. Prata, Wigner functions with boundaries, J. Math. Phys., 43 (2002), 4602-4627.  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) 110-146.  Google Scholar

[21]

N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization", Ann. Phys., 321, (2006) 495-502.  Google Scholar

[22]

D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization," World Scientific, Singapore, 2000.  Google Scholar

[23]

W. Faris, "Self-Adjoint Operators," Lecture Notes in Mathematics 433, Springer-Verlag, Berlin, 1975.  Google Scholar

[24]

D. Fairlie, The formulation of quantum mechanics in terms of phase space functions, Proc. Camb. Phil. Soc., 60 (1964), 581-586.  Google Scholar

[25]

B. Fedosov, A simple geometric construction of deformation quantization, J. Diff. Geom., 40 (1994), 213-238.  Google Scholar

[26]

R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models, Phys. Rev., D 63 (2001), 105014.  Google Scholar

[27]

P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization, Am. J. Phys., 72 (2004), 924-933. 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.), Birkhäuser Verlag, 2009, 81-113.  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 "Perspectives in Partial Differential Equations, Harmonic Analysis and Applications'' (Proc. Sympos. Pure Math., 79), Amer. Math. Soc., Providence, Rhode Island, 2008, 105-173.  Google Scholar

[30]

V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations," Kluver, Dordrecht, 1991.  Google Scholar

[31]

M. de Gosson and F. Luef, A new approach to the $\star$-genvalue equation, Lett. Math. Phys., 85 (2008), 173-183.  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), 425-513.  Google Scholar

[33]

G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271-297.  Google Scholar

[34]

C. Isham, Topological and global aspects of quantum theory, in "Relativity, groups and topology II, Les Houches Session XL" (eds. B.S. DeWitt and R. Stora), North-Holland, Amsterdam, 1984, 1059-1290.  Google Scholar

[35]

M. Kontsevich, Deformation quantization of Poisson manifolds I, Lett. Math. Phys., 66 (2003), 157-216.  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), 431-495, (Russian).  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), 365-404, (Russian).  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), 032118-1-9.  Google Scholar

[39]

S. Kryukov and M. A. Walton, On infinite walls in deformation quantization, Ann. Phys., 317 (2005), 474-491.  Google Scholar

[40]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II, Ann. Institut Fourier, 11 (1961), 137-178.  Google Scholar

[41]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I," Springer-Verlag, Berlin, 1972. Google Scholar

[42]

J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications," 2nd edition, Cambridge University Press, Cambridge, 2000.  Google Scholar

[43]

M. A. Naimark, "Theory of Linear Differential Operators," Frederick Ungar Publishing Co., New York, 1967.  Google Scholar

[44]

J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren, Math. Ann., 102 (1929), 49-131. Google Scholar

[45]

J. von Neumann, "Mathematische Grundlagen der Quantenmechanik," Springer-Verlag, Berlin, 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), Paper 23, 14 pp.  Google Scholar

[47]

A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), 109-147.  Google Scholar

[48]

A. Posilicano, Self-adjoint extensions by additive perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1-20.  Google Scholar

[49]

A. Posilicano, Self-adjoint extensions of restrictions, Oper. Matrices, 2 (2008), 483-506.  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), 015204.  Google Scholar

[51]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness," Academic Press, London, 1975.  Google Scholar

[52]

V. Ryzhov, A general boundary value problem and its Weyl function, Opuscula Math., 27 (2007), 305-331.  Google Scholar

[53]

N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys., 9909 (1999), Paper 32, 93 pp.  Google Scholar

[54]

M. L. Višik, On general boundary problems for elliptic differential equations, Trudy Mosc. Mat. Obsv., 1 (1952), 186-246 (Russian); translated in Amer. Math. Soc. Trans., 24 (1963), 107-172.  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, Berlin, 1980.  Google Scholar

[57]

M. A. Walton, Wigner functions, contact interactions, and matching, Ann. Phys., 322 (2007), 2233-2248.  Google Scholar

[58]

M. W. Wong, "Weyl Transforms," Springer-Verlag, Berlin, 1998.  Google Scholar

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